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+CLASS-INCREMENTAL LEARNING WITH REPETITION
+Hamed Hemati1, Andrea Cossu2, Antonio Carta3, Julio Hurtado3, Lorenzo Pellegrini4
+Davide Bacciu3, Vincenzo Lomonaco3, Damian Borth1
+hamed.hemati@unisg.ch, andrea.cossu@sns.it, antonio.carta@unipi.it,
+julio.hurtado@di.unipi.it, l.pellegrini@unibo.it, davide.bacciu@unipi.it,
+vincenzo.lomonaco@unipi.it, damian.borth@unisg.ch
+1University of St. Gallen, Switzerland
+2 Scuola Normale Superiore, Italy
+3University of Pisa, Italy
+4University of Bologna, Italy
+ABSTRACT
+Real-world data streams naturally include the repetition of previous concepts.
+From a Continual Learning (CL) perspective, repetition is a property of the en-
+vironment and, unlike replay, cannot be controlled by the user. Nowadays, Class-
+Incremental scenarios represent the leading test-bed for assessing and comparing
+CL strategies. This family of scenarios is very easy to use, but it never allows
+revisiting previously seen classes, thus completely disregarding the role of repe-
+tition. We focus on the family of Class-Incremental with Repetition (CIR) sce-
+narios, where repetition is embedded in the definition of the stream. We propose
+two stochastic scenario generators that produce a wide range of CIR scenarios
+starting from a single dataset and a few control parameters. We conduct the first
+comprehensive evaluation of repetition in CL by studying the behavior of existing
+CL strategies under different CIR scenarios. We then present a novel replay strat-
+egy that exploits repetition and counteracts the natural imbalance present in the
+stream. On both CIFAR100 and TinyImageNet, our strategy outperforms other
+replay approaches, which are not designed for environments with repetition.
+1
+INTRODUCTION
+Continual Learning (CL) requires a model to learn new information from a stream of experiences
+presented over time, without forgetting previous knowledge (Parisi et al., 2019; Lesort et al., 2020).
+The nature and characteristics of the data stream can vary a lot depending on the real-world en-
+vironment and target application. Class-Incremental (CI) scenarios (Rebuffi et al., 2017) are the
+most popular ones in CL. CI requires the model to solve a classification problem where new classes
+appear over time. Importantly, when a set of new classes appears, the previous ones are never
+seen again. However, the model still needs to correctly predict them at test time. Conversely, in a
+Domain-Incremental (DI) scenario (van de Ven & Tolias, 2019) the model sees all the classes at the
+beginning and continue to observe new instances of the classes over time.
+The CI and DI scenarios have been very helpful to promote and drive CL research in the last few
+years. However, they strongly constrain the properties of the data stream in a way that it sometimes
+considered unrealistic or very limiting (Cossu et al., 2021). Recently, the idea of Class-Incremental
+with Repetition (CIR) scenarios has started to gather some attention in CL (Cossu et al., 2021).
+CIR scenarios are arguably more flexible in the definition of the stream, since they allow both the
+introduction of new classes and the repetition of previously seen classes. Crucially, repetition is a
+property of the environment and cannot be controlled by the CL agent. This is very different from
+Replay strategies (Hayes et al., 2021), where the repetition of previous concepts is heavily structured
+and can be tuned at will.
+CIR defines a family of CL scenarios which ranges from CI (new classes only, without repetition) to
+DI (full repetition of all seen classes). Although appealing, currently there exists neither a quantita-
+1
+arXiv:2301.11396v1  [cs.LG]  26 Jan 2023
+
+tive analysis nor an empirical evaluation of CL strategies learning in CIR scenarios. Mainly, because
+it is not obvious how to build a stream with repetition, given the large amount of variables involved.
+How to manage repetition over time? How to decide what to repeat? What data should we use?
+In this paper, we provide two generators for CIR that, starting from a single dataset, allow to build
+customized streams by only setting few parameters. The generators are as easy to use as CI or DI
+ones.
+We leveraged our generators to run an extensive empirical evaluation of the behavior of CL strategies
+in CIR scenarios. We found out that knowledge accumulation happens naturally in streams with
+repetition. Even a naive fine-tuning, subjected to complete forgetting in CI scenarios, is able to
+accumulate knowledge for classes that are not always present in an experience. We observed that
+Replay strategies still provide an advantage in terms of final accuracy, even though they are not
+crucial to avoid catastrophic forgetting. On one side, distillation-based strategies like LwF (Li &
+Hoiem, 2018) are competitive in streams with a moderate amount of repetition. On the other side,
+existing Replay strategies are not specifically designed for CIR streams. We propose a novel Replay
+approach, called Frequency-Aware Replay (ER-FA) designed for streams with unbalanced repetition
+(few classes appear rarely, the other very frequently). ER-FA surpasses by a large margin other
+Replay variants when looking at infrequent classes and it does not lose performance in terms of
+frequent classes. This leads to a moderate gain in the final accuracy, with a much better robustness
+and a reduced variance across all classes. Our main contributions are:
+1. The design of two CIR generators, able to create streams with repetition by only setting few
+control parameters. We built both generators with Avalanche (Lomonaco et al., 2021) and
+we will make them publicly available to foster future research. The generators are general
+enough to fit any classification dataset and are fully integrated with Avalanche pipeline to
+run CL experiments.
+2. We perform an extensive evaluation of the properties of CIR streams and the performance
+of CL strategies. We study knowledge accumulation and we showed that Replay, although
+still effective, is not crucial for the mitigation of catastrophic forgetting. Some approaches
+(e.g., LwF) look more promising than others in CIR scenarios. We consolidate our results
+with an analysis of the CL models over time through Centered Kernel Alignment (CKA)
+(Kornblith et al., 2019) and weights analysis.
+3. We propose a novel Replay variant, ER-FA, which is designed based on the properties of
+CIR scenarios. ER-FA surpasses other Replay strategies in unbalanced streams and provide
+a more robust performance on infrequent classes without losing accuracy on the frequent
+ones.
+2
+CLASS-INCREMENTAL LEARNING WITH REPETITION GENERATORS
+𝑒!
+CI
+DI
+Concepts:
+𝑒"
+𝑒#
+𝑒!
+𝑒"
+𝑒#
+𝑒$
+…
+𝑒!
+𝑒"
+𝑒#
+𝑒$
+…
+CIR
+New Inst.
+New/Old 
+Inst.
+Figure 1: Illustration of scenario types
+that can be generated with episodic par-
+tial access to a finite set of concepts.
+The shape colors indicate whether in-
+stances are new in each episode or can
+be a mixture of old and new instances.
+CL requires a model to learn from a stream of N expe-
+riences S = {e1, e2, ..., eN}, where each experience ei
+brings a dataset of examples Dei = {Xi, Yi} . Many
+CL scenarios, like CI or DI, are generated from a fixed
+dataset D = {(x, y); x ∈ X, y ∈ Y }, where x is the in-
+put example, y is the target and Y = {1, · · · , C} is the la-
+bel space (closed-world assumption). Depending on how
+classes from the entire dataset D are shown or revisited
+in the stream, this configuration can lead to CI, CIR or DI
+scenarios (Figure 1). In Table 1, we formally present and
+compare the properties of the three scenario types.
+In CIR, streams with repetition are characterized by mul-
+tiple occurrences of the same class over time. To study
+this scenario, we propose two stream generators designed
+to create a stream from a finite dataset: the Slot-Based
+Generator (Gslot) and the Sampling-Based Generator
+(Gsamp). Gslot generate streams by enforcing constraints
+on the number of occurrences of classes in the stream us-
+ing only two parameters. Gslot does not repeat already observed samples, therefore the stream length
+is limited by the number of classes. However, it guarantees that all samples in the dataset will be
+2
+
+Property
+CI
+DI
+CIR
+Instance Repetition *
+Xei ∩ Xj = Ø
+Xi ∩ Xj = Ø
+|Xi ∩ Xj| ≥ 0
+Domain Coverage
+�i=N
+i=1 Xi = X
+�i=N
+i=1 Xi = X
+�i=N
+i=1 Xi ∈ P(X) \ Ø
+Concept Repetition *
+Yi ∩ Yj = Ø
+Y1 = . . . = YN = Y
+|Yi ∩ Yj| ≥ 0
+Codomain Coverage
+�i=N
+i=1 Yi = Y
+�i=N
+i=1 Yi = Y
+�i=N
+i=1 Yi ∈ P(Y ) \ Ø
+Table 1: Comparison of scenario properties in CI, DI and CIR. P(A) and |A| represent the power
+set and the cardinality of set A. *: ∀1 ≤ i, j ≤ N , i ̸= j.
+observed exactly once during the lifetime of the model. Instead, Gsamp generates streams according
+to several parametric distributions that control the stream properties. It can generate arbitrarily long
+streams in which old instances can also re-appear with some probability.
+2.1
+SLOT-BASED GENERATOR
+The Slot-Based Generator Gslot allows to carefully control the class repetitions in the generated
+stream with a single parameter K. Gslot takes as input a dataset D, the total number of experiences
+N and the number of slots per experience K. It returns a CIR stream composed by N experiences,
+where each of the K slots in each experience is filled with samples coming from a single class.
+1
+10
+20
+...
+C
+Number of Experiences (N)
+1
+10
+20
+C
+Number of Slots (K)
+CIR
+Class-Incremental
+Domain Incremental
+Figure 2: Illustration of how various
+scenarios can be generated by Gslot, by
+changing K and N. The red area under
+the blue curve represents invalid scenar-
+ios.
+Gslot constrains the slot-class association such that all the
+samples present in the dataset are seen exactly once in the
+stream. Therefore, Gslot considers repetition at the level
+of concepts. To implement this logic, Gslot first partitions
+all the samples in the dataset into the target number of
+slots. Then, it randomly assigns without replacement K
+slots per experience. At the end, the N mod K blocks
+remaining are assigned to the first experience, such that
+the rest of the stream is not affected by a variable number
+of slots.
+The Slot-Based Generator is useful to study the transition
+from CI scenarios to DI scenarios, obtained by simply
+changing the parameter K (Figure 2). For example, let
+us consider a dataset with 10 classes such as MNIST. By
+choosing N = 5 and K = 2 we obtain the popular Split-
+MNIST, a CI scenario with no repetition and 2 classes for
+each experience. Conversely, by setting N = 5 and K =
+10 we obtain a DI stream where all the 10 classes appear
+in each experience with new unseen samples. More in
+general, given a dataset with C classes, we obtain a CI scenario by setting K = C
+N (N must divide
+C). We obtain a DI scenario by setting K = C. In Appendix B we illustrate the overall steps of
+stream generation (Figure 12), and provide a step-by-step formal definition of Gslot (Algorithm 2).
+2.2
+SAMPLING-BASED GENERATOR
+The Sampling-Based Generator (Gsamp) generates arbitrarily long streams and controls the repe-
+titions via probability distributions. The stream generator allows to control the first occurrence of
+new classes and the amount of repetitions of old classes. Unlike Gslot, it allows to generate infinite
+and even unbalanced streams.
+Gsamp parameters:
+• N: Stream length, i.e. number of experiences in the stream.
+• S: Experience size which defines the number of samples in each experience.
+• Pf(S): Probability distribution over the stream S used for sampling the experience ID of
+the first occurrence in each class.
+• Pr: List of repetition probabilities for dataset classes.
+3
+
+Scenario 
+Matrix 
+Generator
+𝑁
+𝐾
+CL Scenario
+ℙ!(𝒮)
+𝑃"
+Scenario Matrix
+𝐺!"#$
+Parameters
+Sampler
+…
+𝑒#
+𝑒! 𝑒" 𝑒# 𝑒$ 𝑒%
+𝑐!
+𝑐"
+𝑐#
+𝑐$
+𝑐%
+𝑐&
+𝑐'
+𝑐(
+𝑐)
+…
+…
+…
+…
+…
+…
+…
+…
+…
+…
+𝒟
+𝒮$"%&'
+𝒮$()$
+𝑒*
+…
+𝑒#
+𝑒*
+Instances
+Concepts
+Figure 3: Schematic view of Gsamp generator. Each concept is shown with a different color.
+Note that Pf is a probability mass function over the stream S which means it sums up to 1.0 and
+determines in which parts of the stream it is more likely to observe new classes for the first time.
+However, the list of probabilities {p1, p2, ..., pC} in Pr are independent and each probability value
+0.0 ≤ pi ≤ 1.0 indicates how likely it is for each class to repeat after its first occurrence.
+For each experience, Gsamp samples instances from the original dataset D according to a two step
+process. First, Gsamp defines a C × N binary matrix T called Occurrence Matrix that determines
+which classes can appear in each experience. Then, for each experience ei, 1 ≤ i ≤ N we use the
+i-th column of T to sample data points for that experience. The generator uses the inputs N, Pf(S)
+and Pr to generate T. Therefore, it first initializes T as a zero C × N matrix. Then for each class
+c in the dataset, it uses Pf(S) to sample the index of the experience in which class c appears for
+the first time. Different probability distributions can be used to either spread the first occurrence
+along the stream or concentrate them at the beginning, which allows a good flexibility in the stream
+design. After sampling the first occurrences, the classes are then repeated based on Pr probability
+values to finalize matrix T. In the simplest case, Pr can be fixed to the same value for all classes to
+generate a balanced stream.
+Once the matrix T is constructed, a random sampler is used to sample patterns for each experience.
+Since each experience may contain an arbitrary number of classes, another control parameter that
+could be added here is the fraction of samples per class in experience size S. For simplicity we keep
+the fractions equally fixed and thus the number of datapoints sampled from each class in experience
+ei is ⌊ S
+|Ci|⌋ where |Ci| indicates the number of classes present in ei. Since the sampler is stochastic,
+each time we sample from a class, both new and old patterns can appear for that class. Given a large
+enough stream length N, the final stream will cover the whole dataset with a measurable amount of
+average repetition. In Figure 3 we demonstrate the schematic of the generator Gsamp. We provide
+the pseudo-code for Gsamp in Appendix C (Algorithm 2).
+Although we assume a fixed number of instances per class in D, Gsamp can be easily extended
+to settings where the number of instances can grow over time. Moreover, the sampler can also be
+designed arbitrarily depending on the stochasticity type, e.g., irregular or cyclic.
+3
+FREQUENCY-AWARE REPLAY
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+Figure 4: Ratio of buffer slots for infre-
+quent classes for three random seeds.
+Experience Replay (ER) is the most popular CL strategy
+due to its simplicity of use and high performance in class-
+incremental scenarios. The storage policy, which deter-
+mines which samples to keep in a limited buffer, is the
+major component of ER methods. Class-Balanced (CB)
+and Reservoir Sampling (RS) Vitter (1985) are the most
+popular storage policies in ER methods. CB keeps a fixed
+quota for each class, while RS samples randomly from the
+stream, which leads to the class frequency in the buffer
+being equal to the frequency in the stream. CB and RS
+are great choices for balanced streams such as class incremental scenarios, where the number of
+samples per class is the same over the whole stream. However, as in most real-world scenarios, CIR
+scenarios are naturally unbalanced, and different classes may have completely different repetition
+frequencies. Accordingly, CB and RS storage policies may suffer a big accuracy drop in the infre-
+quent classes of an unbalanced stream. For example, in highly unbalanced streams, RS will store
+an unbalanced buffer replicating the the original distribution of the stream, which is sub-optimal be-
+4
+
+cause the less frequent classes will require more repetition to prevent forgetting, while the frequent
+classes will be repeated naturally via the stream occurrences.
+We propose Frequency-Aware (FA) storage policy that addresses the imbalance issue in CIR streams
+by online adjustment of the buffer slots in accordance with the amount of repetition for each class.
+Given a buffer B with a maximum size of M, a list of previously observed classes P initialized as
+P = {} with a corresponding list O indicating the number of observations per class c in C, and a
+dataset Di from experience ei, the algorithm first checks the present classes Pi in Di and adds them
+to P (P ← P ∪ Pi). Then, for each class c in Pi it increments the number of observations O[c] by
+1 if the class was previously seen, otherwise it initializes O[c] = 1. After updating the number of
+observations, FA computes the buffer quota Q for all observed classes by inverting the number of
+observations (Q = [
+1
+O[c]∀c ∈ C]) and normalizes it . This way, the algorithm offers the less frequent
+classes a larger quota. Finally, a procedure ensures the buffer is used to its maximum capacity by
+filling unused slots with samples from more frequent classes sorted by their observation times. This
+is a crucial step since it is possible that an infrequent class which is not present ei will be assigned
+with a larger quota than its current number of samples in B, and therefore the buffer will remain
+partially empty. In Figure 4 we show how our method assigns higher ratio of samples for infrequent
+classes to overcome the imbalance issue in the stream. For further analysis and pseudo-code of FA
+policy refer to Appendix E. We present examples of unbalanced scenarios in Appendix D.
+4
+EMPIRICAL EVALUATION
+We study CIR scenarios by leveraging our generators Gsamp and Gslot. First, by using Gslot we
+provide quantitative results about forgetting in CL strategies when transitioning from CI to DI sce-
+narios (Sec. 4.1). Then, by using Gsamp we focus on long streams with 500 experiences and
+study the performance of Replay and Naive (Sec. 4.2). The long streams give us the opportunity
+to study knowledge accumulation over time in the presence of repetition. We also provide an intu-
+itive interpretation of the model dynamics over long streams (Sec. 4.3). Finally, we show that our
+Frequency-Aware Replay is able to exploit the repetitions present in the stream and to surpass the
+performance of other replay approaches not specifically designed for CIR scenarios (Sec. 4.4).
+The experiments were conducted using the CIFAR-100 Krizhevsky et al. (2009) and Tiny-ImageNet
+LeCun et al. (1998) datasets with the ResNet-18 model. For Gslot, we run experiments for Naive (in-
+cremental fine tuning), LwF Li & Hoiem (2018), EWC Kirkpatrick et al. (2017), Experience Replay
+with reservoir sampling Kim et al. (2020) (ER-RS) and AGEM Chaudhry et al. (2018) strategies.
+For Gsamp we run experiments for Naive and ER (CB/RS/FA) strategies. We set the default buffer
+size for CIFAR-100 to 2000, and for Tiny-ImageNet to 4000 in the replay strategies. We evaluate
+all strategies on the Average Test Accuracy (TA).
+4.1
+TRANSITION FROM CLASS-INCREMENTAL TO DOMAIN INCREMENTAL
+DI and CI scenarios are heavily studied in the CL literature. However, little is known about what
+happens to the performance of popular CL strategies when gradually transitioning from one scenario
+to the other. By changing the value of K in Gslot, we provide a quantitative analysis of such
+behavior in CIR scenarios. Figure 5 shows the Average Test Accuracy over all classes for different
+CL strategies when transitioning from CI (left-most point of each plot) to DI (right-most point of
+each plot).
+Replay is one of the most effective strategies in CI scenarios. As expected, in CIR scenarios the
+advantage provided by ER-RS with respect to other CL strategies diminishes as the amount of rep-
+etition increases. However, in order for the other strategies to match the performance of ER-RS, the
+environment needs to provide a large amount of repetition.
+LwF guarantees a consistent boost in the performance, both in CIFAR-100 and Tiny-ImageNet.
+In particular, and quite surprisingly, on Tiny-ImageNet LwF is able to quickly close the gap with
+ER-RS and even surpass it as the amount of repetition increases. The positive interplay between
+distillation and repetition provides an effective way to mitigate forgetting in CIR scenarios, without
+the need to explicitly store previous samples in an external memory. EWC showed different sen-
+sitivity to the regularization hyper-parameter λ. We experimented with λ = 0.1, 1, 10, 100. While
+on MNIST we did not see any difference in performance, on CIFAR-100 and Tiny-ImageNet large
+values of λ lead to a dramatic decrease, dropping as low as Naive. We found 0.1 to be the best value
+5
+
+2
+3
+5
+8
+10
+K
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Average Test Accuracy
+MNIST
+ER-RS
+LwF
+EWC
+A-GEM
+Naive
+10
+30
+50
+80
+100
+K
+0.0
+0.2
+0.4
+0.6
+CIFAR-100
+20
+60
+100
+160
+200
+K
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Tiny-ImageNet
+Figure 5: Average Test Accuracy for different values of K in CIR scenarios generated with Gslot.
+Class-Incremental scenarios are represented by the left-most point of each plot, Domain-Incremental
+scenarios by the right-most point. Results averaged over 3 seeds.
+50
+100
+150
+200
+250
+300
+350
+400
+450
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Class Accuracy
+Class Status
+Present
+Missing
+Figure 6: Accuracy of a particular class over the stream. The target class is either present or absent
+in the experiences indicated by the blue and orange points, respectively.
+on both CIFAR-100 and Tiny-ImageNet. This configuration only provides a low amount of regular-
+ization. Overall, the role played by the natural repetition already guarantees a sufficient stability of
+the model, which is additionally boosted only in the case of LwF.
+4.2
+IMPACT OF REPETITION IN LONG STREAMS
+We investigate the impact of repetition in long streams (N = 500) generated with Gsamp. For
+the long-stream experiments we also report the missing-classes accuracy (MCA) and seen-classes
+accuracy (SCA). MCA measure the accuracy over the classes that were seen before but are missing
+in the current experience, and SCA measure the accuracy over all seen classes up to the current
+experience.
+Missing Class Accuracy Increases Over Time
+In CI scenarios, a Naive strategy catastrophically
+forgets each class as soon as it starts learning on new classes. Surprisingly, we found that in CIR
+scenarios there is knowledge accumulation over time for all the classes. Figure 6 shows the accuracy
+of a single class over time, highlighting whether the class is present or not in the current experience.
+At the beginning of the stream missing classes are completely forgotten, which can be noticed by
+the instant drop of the accuracy to zero. However, over time the model accumulate knowledge
+and the training process stabilizes. As a result, the accuracy of missing classes tends to increase
+over time, suggesting that the network becomes more resistant to forgetting. Notice that this is an
+example of continual learning property that is completely ignored when testing on CI scenarios.
+This finding prompts the question, ”What is happening to allow knowledge accumulation even for
+Naive finetuning?”. We investigate this question by analysing the model’s accuracy over time and
+the properties of the learned model in the next experiments.
+Accuracy Gap Between Naive and Replay Decreases Over Time
+To study the impact of long
+streams with repetitions we monitor the accuracy gap between ER and Naive fine-tuning by com-
+paring their accuracy after each experience. For the scenario configuration, we set Pf(S) as a
+Geometric distribution with a coefficient of 0.01 and fix the probability of repetition Pr as 0.2 for
+all classes. For more details and illustrations on distribution types refer to Appendix C. In such
+scenarios, the majority of classes occur for the first time in the first quarter of the stream, and then
+repeat with a constant probability of 0.2 which makes them appropriate for our experiments since
+all classes are observed before the middle of the stream and the repetition probability is low enough.
+As can be seen in Figure 7, while the accuracy of ER saturates over time, the accuracy of Naive
+increases, closing the gap between the two strategies from around 25% in experience 100 to 7% in
+6
+
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+Average Test Accuracy
+Strategy
+ER-CB
+Naive
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+Average Seen Class Accuracy
+Strategy
+ER-CB
+Naive
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+Average Missing Class Accuracy
+Strategy
+ER-CB
+Naive
+Figure 7: Average test accuracy and average missing class accuracy plots for long streams streams
+with 500 experiences.
+experience 500. This supports our hypothesis that neural network’s ability to consolidate knowledge
+is significantly influenced by ”natural” repetition in the environment.
+The Role of Repetition
+The amount of repetition is one of the key aspects of a CIR scenario.
+To find out how strategies perform under different repetition probabilities, we consider a setting
+where all components of a scenario are fixed except for Pr. For this experiment, we set Pf(S)
+as geometric distribution with p = 0.2 and let Pr change. In Figure 8 we demonstrate the seen
+class accuracy (SCA) for the Naive and ER-CB strategies in CIFAR-100. It is clear from the plots,
+that the model’s rate of convergence can be significantly affected by the amount of repetition in the
+scenario. Although, it may seem obvious that higher repetition leads to less forgetting, it is not very
+intuitive to what extent different strategies may gain from the environment’s repetition. While the
+Naive strategy gains a lot from increased repetition, the replay strategy saturates after some point
+for higher repetitions and the gaps close between different repetition values.
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+ER-CB
+p
+0.1
+0.3
+0.7
+1.0
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+Naive
+p
+0.1
+0.3
+0.7
+1.0
+Figure 8: Retained accuracy for different values of p in Pr.
+4.3
+MODEL SIMILARITY AND WEIGHT SPACE ANALYSIS
+Weight Interpolation
+Based on the ”gradual loss drop” observation in missing classes, we study
+how the loss surface changes over time if we perturb the weights. We interpolate between the model
+weights from two consecutive checkpoints with an interval of 10 experiences in various segments
+of the stream. Assuming that w∗
+t and w∗
+t+10 are the obtained solutions for experiences t and t + 10
+respectively, we generate eight in-between models wk = α ∗ w∗
+t + (1 − α) ∗ w∗
+t+50 by by increasing
+α from zero to one, and then compute the accuracy of wk for the data of experience t. We show
+the interpolation accuracy for various pairs of experiments in different segments of the stream for
+the Naive strategy in Figure 9 (left). In the beginning of the stream, the accuracy of experience t
+in each pair drops significantly, while we observe a milder loss drop towards the end of the stream.
+The findings suggest that, towards the end of the stream, even a relatively big perturbation does not
+have a large negative effect on the model’s accuracy and the optimal solutions of the consecutive
+experiments are connected with a linear low-loss path.
+Weight changes
+Another approach to analyzing the gradual drop of the accuracy is by dissecting
+how much, when, and where the weight changes occurs. As shown in Figure 9 (right), we can
+observe that within the first experiences, there is a significant difference for blocks 0, 1, and 2. This
+difference then stalls, showing that as we continue training experiences, the weights of these blocks
+stabilize. On the other hand, blocks 3 and 4 show a linear increase in the difference with the number
+of experiences. An explanation for this phenomenon, is that the first layers of the model capture
+knowledge that can be useful for several classes (more general), so it is unnecessary to change them
+after several experiences. On the other hand, the last blocks are the ones that memorize or learn
+more specific patterns, so they adapt to each experience.
+7
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Interpolation Weight
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Accuracy
+Accuracy over Linear Path
+Start
+End
+50
+100
+100
+150
+200
+250
+300
+350
+400
+450
+0
+100
+200
+300
+400
+500
+Experience
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+Difference
+Difference over the Experience
+Blocks
+Block 0
+Block 1
+Block 2
+Block 3
+Block 4
+Figure 9: (left) Interpolation accuracy. (right) Weight changes in each block. The difference used in
+(right) is calculated as Dj =
+1
+|θ0|
+�θb
+i
+��� (θ0,i−θj,i)
+∥θ0,i∥2
+���, where the weights of experience j are compare
+with the initialization θ0 for each block i
+CKA Analysis
+Finally, we show the CKA Kornblith et al. (2019) of the model in the beginning,
+middle and the end of the stream with an interval difference of 50 experiences. As shown in the
+visualizations in Figure 10, the longer the model is trained on more experiences, the less significant
+the changes in the representations become especially for the final layers. We can see that the diag-
+onal of the CKA becomes sharper propagating forward with more experiments. This indicates that
+although the model is trained on different subsets of classes in each experiment, the representations
+change less after some point in the stream.
+Experience-15
+Experience-5
+5
+15
+Experience-35
+Experience-25
+25
+35
+Experience-65
+Experience-55
+55
+65
+Experience-200
+Experience-190
+190
+200
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Figure 10: CKA of the model in different parts of the stream.
+4.4
+FREQUENCY-AWARE REPLAY IN UNBALANCED SCENARIOS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Accuracy
+Strategy
+ER-FA
+ER-CB
+ER-RS
+Naive
+Figure 11:
+Accuracy of Infrequent
+Classes.
+We conduct experiments for bi-modal unbalanced scenar-
+ios where classes can have a high frequency of 1.0 or a
+low frequency of 0.1. We use a fraction factor that de-
+termines the amount of infrequent classes in the scenario,
+e.g., Fraction=0.3 means that 30% of the classes are infre-
+quent. In Table 2 we compare ER-FA with the Naive, ER-
+RS and ER-CB strategies. The numbers show the MCA
+and average Test Accuracy (TA) metrics for each strategy
+in the end of the stream averaged over three runs. Our
+strategy outperforms all other scenarios in almost all set-
+tings in both CIFAR-100 and TinyImageNet datasets in
+terms of TA, and significantly outperforms other methods
+in terms of MCA (in the last experience). Moreover, we
+illustrate the accuracy of infrequent classes in CIFAR-100 experiments for Fraction=0.3 in Figure
+11 where ER-FA achieves considerably higher accuracy in the whole stream by assigning larger
+quota to infrequent classes without losing its performance on frequent classes (refer to Appendix G
+for further illustrations).
+5
+RELATED WORK
+Current CL methods are mainly focused on two types of benchmarks namely, Multi Task (MT) and
+Single Incremental Task (SIT) Maltoni & Lomonaco (2019). MT divides training data into distinct
+tasks and labels them during training and inference. SIT splits a single task into a sequence of
+unlabeled experiences. SIT can be further divided into Domain-Incremental (DI) where all classes
+are seen in each experience, and Class-Incremental (CL) where each experience contains only new
+8
+
+DS
+Strategy
+Fraction= 0.1
+Fraction= 0.3
+Fraction= 0.5
+MCA
+TA
+MCA
+TA
+MCA
+TA
+C-100
+Naive
+5.0 ± 0.7
+58.0 ± 0.1
+7.3 ± 2.2
+49.0 ± 0.8
+8.0 ± 2.0
+40.8 ± 1.4
+ER-RS
+11.4 ± 0.9
+57.7 ± 0.7
+16.7 ± 3.6
+51.1 ± 0.4
+20.5 ± 1.9
+45.6 ± 0.8
+ER-CB
+30.9 ± 2.7
+59.5 ± 0.1
+34.5 ± 1.7
+55.3 ± 0.1
+35.7 ± 0.5
+52.0 ± 1.5
+ER-FA
+52.2 ± 1.1
+60.8 ± 0.3
+44.7 ± 1.5
+57.8 ± 0.4
+40.9 ± 1.2
+54.2 ± 1.2
+TIN
+Naive
+2.0 ± 0.8
+33.5 ± 0.4
+2.0 ± 0.1
+29.1 ± 0.1
+2.0 ± 0.1
+24.0 ± 0.4
+ER-RS
+3.7 ± 0.6
+31.8 ± 1.2
+4.4 ± 0.7
+28.1 ± 0.2
+6.0 ± 0.1
+24.0 ± 0.1
+ER-CB
+10.4 ± 0.2
+32.2 ± 0.7
+10.0 ± 1.0
+28.8 ± 0.2
+11.0 ± 0.2
+26.0 ± 0.3
+ER-FA
+22.0 ± 1.0
+33.0 ± 0.9
+15.3 ± 1.0
+30.4 ± 0.1
+13.6 ± 0.1
+27.0 ± 0.1
+Table 2: Unbalanced scenario results for the CIFAR-100 (C-100) and TinyImageNet (TIN) dataset.
+“Fraction” refers to the fraction of infrequent classes having repetition probability of only 10%.
+(unseen) classes van de Ven & Tolias (2019). Both DI and CI are extreme cases and are unlikely to
+hold in real-world environments Cossu et al. (2021). In a more realistic setting, the role of natural
+repetition in CL scenarios was studied in the context of New Instances and Classes (NIC) scenario
+Lomonaco et al. (2020) and the CRIB benchmark Stojanov et al. (2019). NIC mainly focuses on
+small experiences composed of images of the same object, and repetitions in CRIB are adapted to
+a certain dataset and protocol. The Class-Incremental with Repetition (CIR) scenario was initially
+formalized in Cossu et al. (2021), however the work lacks a systematic study of CIR scenarios as
+the wide range of CIR scenarios makes them difficult to study.
+To counter the lack of repetition in CI, replay has been extensively used as a CL strategy (Rebuffi
+et al., 2017; Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2018; Wu et al., 2019; Castro et al.,
+2018; Belouadah & Popescu, 2019; Kim et al., 2020; Douillard et al., 2020). In such methods,
+natural repetition is artificially simulated by storing past data in an external memory, and replaying
+them alongside the scenario stream data. Repetition reduces catastrophic forgetting through implicit
+regularization of model’s weights Hayes et al. (2021). In CI benchmarks, replay seems to be the
+only working strategy van de Ven et al. (2020). In other words, replay seems to be a necessity when
+no natural repetition happens.
+Although replay can be seen as a method to simulate natural repetitions artificially, the two concepts
+are fundamentally different. Repetition in replay strategies occurs with the same data seen in pre-
+vious experiences, which is neither realistic nor biologically plausible Gupta et al. (2010). On the
+other hand, natural repetitions of already seen objects occur in different real-world environments,
+and better fit the CIR scenario studied in this paper. Recently, Lesort et al. (2022) scaled the number
+of tasks in a finite world setting (Boult et al., 2019; Mundt et al., 2022) where the model has access to
+a random subset of classes in each experience. The authors proposed naive fine-tuning with masking
+techniques to improve retained accuracy. Our work is different in the sense that we compare among
+different strategies and study various types of repetitions with two flexible generators.
+6
+DISCUSSION AND CONCLUSION
+We defined CIR scenarios which represent CL environments where repetition is naturally present in
+the stream. Although the concept of repetition is quite intuitive, it is not obvious how to realize it in
+practice for research purposes. Therefore, we proposed two CIR generators that can be exploited to
+address this issue. Through empirical evaluations, we showed that, unlike CI scenarios, knowledge
+accumulation happens naturally in CIR streams, even without applying any CL strategy. This raised
+the question of whether the systematic repetition provided by Replay is critical in all CIR scenarios.
+With several experiments in long streams, we demonstrated that although Replay provides an ad-
+vantage in general, even random repetition in the environment can be sufficient to induce knowledge
+accumulation given a long enough lifetime.
+Moreover, we found that existing Replay strategies are exclusively designed for classical CI scenar-
+ios. Thus, we proposed a novel strategy, ER-FA, to exploit the properties of CIR scenarios. ER-FA
+accumulates knowledge even in highly unbalanced stream in terms of class frequency. ER-FA out-
+performs by a large margin other Replay approaches when monitoring the accuracy for infrequent
+classes while preserving accuracy for the frequent ones. Overall, ER-FA guarantees a more ro-
+bust performance on a wide range of real-world scenarios where classes are not homogeneously
+distributed over time.
+9
+
+The framework defined in this work opens new research directions which depart from the existing
+ones, mainly focused on the mitigation of forgetting in CI scenarios. We hope that our experiments
+and results will promote the study of CIR scenarios and the development of new CL strategies, able
+to exploit the inner semantics of repetition, a natural trait of real-world data streams, in many other
+clever and imaginative ways.
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+12
+
+A
+RELATED WORK (CONTINUED)
+Considering benchmark formalization frameworks, De Lange & Tuytelaars (2021) recently pro-
+posed a subdivision aimed at framing continual learning setups by categorizing them based on the
+batch and observable horizon that the learning agent is able to access at each time. With this frame-
+work, the authors aim to better formalize the online learning setup. While the concept of observ-
+able horizon may be useful in evaluating the significance and local (in time) usefulness of natural
+repetition in a training stream, this work does not consider the concept of natural repetition in its
+framework.
+Recently, Koh et al. (2022) proposed to introduce blurry task boundaries in class incremental bench-
+marks. Their proposal is based on previous works Bang et al. (2021); Aljundi et al. (2019) that tried
+to produce more realistic benchmarks by blurring the class-incremental scenario, which however
+resulted in a setup in which no classes are added to new tasks. They argue that this idea moves the
+focus too far away from the class-incremental setup and it is still not quite realistic. The resulting
+setup, named i-Blurry, aims at resolving the aforementioned issues and moving toward a more re-
+alistic scenario by partitioning the classes available in the source dataset into two groups: Disjoint
+and Blurry. Classes of the disjoint group are gradually added in successive experiences while sam-
+ples of classes from the blurry group always appear in all experiences with their numerosity being
+controlled through a blur ratio M. The authors show that, based on the degree of disjunction N
+and blurry M, this framework can produce class-incremental (no blurring), domain-incremental (no
+disjunction), and blurred setups. This setup is the one that most moved towards the direction of
+introducing repetition in Continual learning benchmarks in a controlled way so far. However, the
+proposed blurring mechanism is too coarse-grained to simulate a natural repetition of concepts as
+the significance of the repetition introduced by blurring relies too much on i) a random (uniform)
+sampling of the concepts to be repeated, ii) the static subdivision of classes in the Disjoint and Blurry
+groups.
+B
+SLOT-BASED GENERATOR
+Following the properties of CIR scenarios in Section 2, Gslot generates a subset of CIR streams that
+hold the assumptions below in the defined properties:
+• |Xi ∩ Xj| = 0 : new instances appear in each experiences
+• �i=N
+i=1 Xi = X: all samples are used
+• |Yi ∩ Yj| ≥ 0, ∀1 ≤ i, j ≤ N where i ̸= j
+• �i=N
+i=1 Yi = Y : all classes are used
+• X is constant.
+These assumptions allow transitioning through different CIR scenario types between the two ex-
+tremes of CI and DI.
+B.1
+ALGORITHM
+The overall steps of Gslot are illustrated in Figure 12. In Algorithm 2 we present all steps of Gslot
+used to generate arbitrary CIR scenarios given a dataset D, number of experiences N and number
+of slots K. The output of the algorithm is a CL stream.
+B.2
+TRANSITIONING
+Transitioning in Gslot for a scenario with a fixed number of experiences can be done by increasing
+K. When K = 1 the generated scenario will be class-incremental and as K get closer to the total
+number of classes in D, the scenarios moves towards a domain incremental setting. In Figure 13 we
+show an example of how generated scenarios change by increasing K.
+13
+
+…
+Concepts:
+Replicate each concept !×#
+$
+times
+and shuffle the sequence. 
+𝒆𝟏
+𝒆𝟐
+𝒆𝟑
+…
+Example: 𝐾 = 4
+Slot 1
+Slot 2
+Slot 3
+Slot 4
+New Instances
+𝒆𝑵
+𝓢𝒕𝒓𝒂𝒊𝒏
+Figure 12: Illustrations of the overall steps of Gslot. Each shape represents a concept, and the green
+color means that new instances of that concept are used in each experience.
+Class
+Experience
+Figure 13: From left to right: transitioning from CI to DI in Gslot. Each class is represented with a
+unique color.
+C
+SAMPLING-BASED GENERATOR
+Following the properties of CIR scenarios in Section 2, Gsamp generates a subset of CIR scenarios
+that hold all defined properties. Additionally, for any stream S = {e1, e2, ..., eN}, Gsamp defines
+a probability distribution for the first occurrence of concepts over S and per-class probabilities for
+each concept c ∈ Y . Gsamp can generate arbitrarily long stream (N ≥ 1) and even from a growing
+set of samples X where Y remains constant.
+C.1
+ALGORITHM
+The overall steps of the Gslot are shows in Algorithm 2.
+C.2
+DISTRIBUTION TYPES
+In this section we show some examples of different discrete distributions that can be used for Pf(S)
+and Pr. For Pr we use the unnormalized version of the final distribution. Distributions used for
+Gsamp can be any arbitrary discrete distribution and are not limited to the ones we describe here.
+C.2.1
+ZIPFIAN
+Given the number of elements N and scalar e ≥ 0 , the probability mass function of a Zipfian
+distribution over a list of N elements is defined in Equation 1. When used for the probability of first
+occurrence, the distribution can be defined over the experiences of a stream. For example, N can
+be considered as the number of experiences and i can indicate the ith experience in the stream. By
+increasing e, the distribution over the stream will be skewed towards the beginning. In figure 14 we
+demonstrate some examples of first occurrence probabilities over a stream of length 10 generated
+with Zipf distribution with increasing values of e. Many natural distribution follow Zipf distribution
+14
+
+Algorithm 1 Slot-Based Generator (Gslot) Pseudo-Code.
+Require: Dataset D = {(xi, yi)}i=1,...,P with C classes, number of experiences N, experience
+size S present in each experience.
+Ensure: K ≤ N
+Ensure: C mod N = 0
+Ensure: NK mod C = 0
+cls-idxs = {}
+▷ Empty dictionary
+for y ∈ set({yi}i=1,...,P ) do
+cls-idxs[y] = []
+▷ Empty list init
+end for
+for i = 1, . . . , P do
+cls-idxs[yi].append(i)
+end for
+slots={}
+▷ Empty dictionary
+for y ∈ cls-idxs do
+slots[y] = []
+ksample = int(len(cls-idxs[y]) /K)
+for k = 1, . . . , N×K
+C
+do
+subset-idxs = pop(cls-idxs[y], ksample)
+subset-samples = [xidx for idx ∈ subset-idxs]
+slots[y].append(subset-samples)
+end for
+end for
+stream = []
+for n = 1, . . . , N do
+experience = dataset()
+seen-classes = []
+for k=1,...,K do
+repeat
+y = sample(slots)
+until y /∈ seen-classes
+seen-classes.append(y)
+experience.add(pop(slots[y], 1))
+end for
+stream.append(experience)
+end for
+return stream
+0
+5
+10
+0.0
+0.5
+1.0
+Probability
+e = 0.0
+0
+5
+10
+0.0
+0.5
+1.0
+e = 0.5
+0
+5
+10
+Experience
+0.0
+0.5
+1.0
+e = 1.0
+0
+5
+10
+0.0
+0.5
+1.0
+e = 1.5
+0
+5
+10
+0.0
+0.5
+1.0
+e = 2.0
+Figure 14: Zipf distribution with varying values of e.
+and it can be used to generate highly skewed distributions both for first occurrence and repetition
+probabilities.
+f(i; e, N) =
+1
+ie
+�N
+n=1
+1
+ne
+(1)
+15
+
+Algorithm 2 Sampling-Based Generator (Gsamp) Pseudo-Code.
+Require: Dataset D = {(xi, yi)}i=1,...,P with C classes, number of experiences N, number of
+slots K, probability distribution for first occurrence Pf(S), and list of repetition probabilities Pr.
+T = {0}C×N
+▷ Initialize occurrence matrix with zeros
+for c ∈ {0, 1, . . . C} do
+i ∼ Pf(S)
+▷ Samples the first occurrence of class c
+T[c, i] = 1
+for j ∈ {i, i + 1, . . . N} do
+r ∼ U(0, 1)
+▷ U: uniform distribution over [0, 1.0]
+if r < Pr[c] then T[c, j] = 1
+end if
+end for
+end for
+E = {}
+for ei ∈ {1, 2, . . . N} do
+Ci = RetrieveClasses(ei)
+Dei = Sample(D, Ci, S)
+▷ Sample S instances from dataset D for classes Ci
+E ← E ∪ Dei
+end for
+Strain, Stest = GenerateStream(E)
+▷ Generate streams using E
+return Strain, Stest
+C.2.2
+POISSON
+The PMF for Poisson distribution is given in Equation 2 where µ ≥ 0. Poisson with larger values
+of µ can be used for distributions where the probability of occurrence/repetition first rises and then
+gradually decreases over time.
+f(i; µ) = µie−µ
+i!
+(2)
+0
+5
+10
+0.0
+0.5
+1.0
+Probability
+= 0.0
+0
+5
+10
+0.0
+0.5
+1.0
+= 0.5
+0
+5
+10
+Experience
+0.0
+0.5
+1.0
+= 1.0
+0
+5
+10
+0.0
+0.5
+1.0
+= 1.5
+0
+5
+10
+0.0
+0.5
+1.0
+= 2.0
+Figure 15: Poisson distribution with varying values of µ.
+C.2.3
+GEOMETRIC
+Another useful distribution that can be used for the first occurrence probabilities over a stream is
+Geometric distribution with its PMF given in Equation 3. This distribution is in particular inter-
+esting for transitioning from domain incremental to class incremental. By setting p = 1, only the
+probability of experience i = 0 will be equal to 1.0 and the rest will be zero, and by decreasing p,
+the probability will spread over the stream. In figure 17 we show examples for generated scenarios
+with Gsamp with Geometric first occurrence and fixed probability of repetition.
+f(i, p) = (1 − p)i−1p
+(3)
+16
+
+0
+5
+10
+0.0
+0.5
+1.0
+Probability
+p = 0.01
+0
+5
+10
+0.0
+0.5
+1.0
+p = 0.2
+0
+5
+10
+Experience
+0.0
+0.5
+1.0
+p = 0.5
+0
+5
+10
+0.0
+0.5
+1.0
+p = 0.7
+0
+5
+10
+0.0
+0.5
+1.0
+p = 1.0
+Figure 16: Geometric distribution with varying values of p.
+Experience
+Class
+Experience
+Class
+Experience
+Class
+Figure 17: Scenarios generated with Geometric first occurrence and probability of repetition equal
+to 1.0 for all classes. The p values for the Geometric distributions from left to right are 0.01, 0.2 and
+1.0 respectively.
+D
+UNBALANCED SCENARIOS
+In this section we present a particular type of unbalanced scenarios where a subset of classes in
+the stream have a low probability of repetition and the rest repeat very often. We refer to such
+scenarios bi-modal scenarios, where each mode refers to a subset of classes with a distinct repetition
+probability. More specifically, we have a stream of experiences S = {e1, e2, ..., eN} where Y S =
+�N
+1 Yei indicates the set of all available concepts in S. In bi-modal scenarios Y = Y if ∪Y fr where
+Y if and Y fr are the set of frequent and infrequent concepts respectively, and Y if ∪ Y fr = Ø. In
+Figure 18 we show examples of unbalanced bi-modal scenarios.
+Experience
+Class
+Experience
+Class
+Experience
+Class
+Figure 18: Unbalanced scenarios with two modes of repetition. The fractions of infrequent classes
+from left to right are 0.2, 0.4 and 0.6 respectively. Repetition probabilities for frequent and infre-
+quent classes are set to 0.2 and 0.9 accordingly.
+E
+FREQUENCY-AWARE REPLAY
+E.1
+ALGORITHM
+In Algorithm 3 we present the steps for updating the buffer in FA storage policy.
+E.2
+ANALYSIS: VARYING THE FRACTION OF INFREQUENT CLASSES
+In this section, we study the behavior of FA, CB, and RS storage policies by changing the fraction
+of infrequent classes. In our analysis, we consider an unbalanced stream generated with Gsamp
+where N = 100 and the probability of repetition for frequent and infrequent classes are 0.9 and
+0.1, respectively. In such streams, the large probability gap between frequent and infrequent classes
+17
+
+Algorithm 3 Frequency-Aware Buffer.
+Require: Current Buffer Set B, Maximum buffer size M, List of Seen Classes C, Number of
+Observation per Seen Class O
+D = GetExperienceDataset(ei)
+P = DetectPresentClasses(D)
+C ← C ∪ P
+for c ∈ P do
+▷ For each present class, increment the number of observations
+if c ∈ O then O[c]+ = 1
+else O[c] = 1
+end if
+end for
+Q = [
+1
+O[c]∀c ∈ C]
+▷ Calculate quota per class
+ˆQ =
+Q
+|Q|
+▷ Normalize quota values
+S = {⌈Q[c] ∗ M⌉∀c ∈ C}
+▷ Calculate buffer slot size for each class
+UpdateSlots(S)
+▷ Update assigned slots according to the current state of B
+UpdateBuffer(B, D, S)
+return B, M, C, O
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+0
+20
+40
+60
+80
+100
+Experience
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ratio
+Storage Policy
+FA (Ours)
+CB
+RS
+Figure 19:
+Ratio of samples for infrequent classes in unbalanced scenarios for the FA,
+CB and RS policies.
+Fraction of infrequent classes from top-left to bottom-right are
+20%, 40%, 60%, 80%, 100%.
+helps us observe the difference more clearly. We report the ratio of samples assigned to infrequent
+classes in the buffer in the lifetime of the model in the stream for scenarios where the fraction of
+infrequent classes is equal to {20%, 40%, 60%, 80%, 100%}. For this experiment, we set the buffer
+size to 500 for all methods.
+As demonstrated in Figure 19, when the fraction of infrequent classes is equal to 20%, i.e. only 20%
+of classes are infrequent, the ratio is very low for RS policy as it tries to replicate the true distribution
+of the stream while CB assigns exactly 20% of the buffer space to the infrequent samples. However,
+we can observe that FA starts to assign more samples over time to the infrequent classes over time as
+it adapts the buffer slots based on the frequency of repetition. Moreover, it is evident in the plots that,
+by increasing the fraction of infrequent classes, the ratio gap between FA and CB gets smaller as
+the quota for CB stays the same while the number of infrequent classes increases. Eventually, when
+the fraction of infrequent classes is equal to 100%, i.e. all classes have the same (low) probably of
+repetition, all buffers have exactly the same ratio since all classes are infrequent.
+In conclusion, FA buffer slots can be very helpful in highly unbalanced streams where a smaller
+fraction of classes have a low probability of repetition. When the stream moves towards becoming
+18
+
+balanced, the FA and CB get closer, and all methods become similar in the extreme case of a fully
+balanced stream with similar probability of repetition.
+F
+CHANGING Pf(S)
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+ER-CB
+p
+0.1
+0.3
+0.7
+1.0
+0
+100
+200
+300
+400
+500
+Experience
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+Naive
+p
+0.1
+0.3
+0.7
+1.0
+Figure 20: Average test accuracy for different values of p in first occurrence.
+We conduct experiments to find the differences between situations in which all classes occur early
+versus those in which new classes also appear late in the stream in order to analyze the role of first
+occurrence type. How early or late in the stream we observe all classes of a dataset, depends on
+the the parameters that control Pf(S). In this experiment, we fix the probability of repetition Pr
+and change Pf(S)’s parameters. In particular, we opt the geometric distribution for Pf(S) and
+choose the values {0.1, 0.3, 0.7, 1.0} for its only parameter 0 < p ≤ 1.0. Increasing p is inversely
+proportional to the spread factor in the first occurrence distribution, i.e. when p is close to 0 all
+classes happen in the first experience and as we move p toward 1.0 the classes start to spread along
+the stream. Figure 20 shows the CIFAR-100 results for the Naive and ER-CB strategies. The results
+suggest that when the spread factor is low, the model initially has difficulty to learn since there
+are more classes in the initial experiments and thus the model has to learn from fewer instances.
+However, with more experiences, all first occurrence types, reach almost the same SCA.
+G
+ER-FA RESULTS
+Results in Figure 21 illustrate the total test accuracy and accuracy of frequent classes over time.
+Although the discrepancy between the accuracies of frequent classes is very small, the total test ac-
+curacy can significantly vary due to the difference in the accuracy of infrequent classes as presented
+in Section 4.4.
+0
+20
+40
+60
+80
+100
+Experience
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+Strategy
+ER-FA
+ER-CB
+ER-RS
+Naive
+0
+20
+40
+60
+80
+100
+Experience
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+Strategy
+ER-FA
+ER-CB
+ER-RS
+Naive
+Figure 21: TA over all classes (left) and frequent classes (right) in a bi-modal unbalanced scenario
+with Fraction=0.3.
+19
+
diff --git a/4tFIT4oBgHgl3EQf7Ctv/content/tmp_files/load_file.txt b/4tFIT4oBgHgl3EQf7Ctv/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..76bbd4cfd497b8dd1610e7e3b210bb2556fc22a2
--- /dev/null
+++ b/4tFIT4oBgHgl3EQf7Ctv/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf,len=1049
+page_content='CLASS-INCREMENTAL LEARNING WITH REPETITION  Hamed Hemati1, Andrea Cossu2, Antonio Carta3, Julio Hurtado3, Lorenzo Pellegrini4  Davide Bacciu3, Vincenzo Lomonaco3, Damian Borth1  hamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='hemati@unisg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='ch, andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='cossu@sns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='it, antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='carta@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='it,  julio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='hurtado@di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='it, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='pellegrini@unibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='it, davide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='bacciu@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='it,  vincenzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='lomonaco@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='it, damian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='borth@unisg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='ch  1University of St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Gallen, Switzerland  2 Scuola Normale Superiore, Italy  3University of Pisa, Italy  4University of Bologna, Italy  ABSTRACT  Real-world data streams naturally include the repetition of previous concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  From a Continual Learning (CL) perspective, repetition is a property of the en-  vironment and, unlike replay, cannot be controlled by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Nowadays, Class-  Incremental scenarios represent the leading test-bed for assessing and comparing  CL strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This family of scenarios is very easy to use, but it never allows  revisiting previously seen classes, thus completely disregarding the role of repe-  tition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We focus on the family of Class-Incremental with Repetition (CIR) sce-  narios, where repetition is embedded in the definition of the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We propose  two stochastic scenario generators that produce a wide range of CIR scenarios  starting from a single dataset and a few control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We conduct the first  comprehensive evaluation of repetition in CL by studying the behavior of existing  CL strategies under different CIR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We then present a novel replay strat-  egy that exploits repetition and counteracts the natural imbalance present in the  stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' On both CIFAR100 and TinyImageNet, our strategy outperforms other  replay approaches, which are not designed for environments with repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  1  INTRODUCTION  Continual Learning (CL) requires a model to learn new information from a stream of experiences  presented over time, without forgetting previous knowledge (Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Lesort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  The nature and characteristics of the data stream can vary a lot depending on the real-world en-  vironment and target application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Class-Incremental (CI) scenarios (Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2017) are the  most popular ones in CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' CI requires the model to solve a classification problem where new classes  appear over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Importantly, when a set of new classes appears, the previous ones are never  seen again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However, the model still needs to correctly predict them at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Conversely, in a  Domain-Incremental (DI) scenario (van de Ven & Tolias, 2019) the model sees all the classes at the  beginning and continue to observe new instances of the classes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  The CI and DI scenarios have been very helpful to promote and drive CL research in the last few  years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However, they strongly constrain the properties of the data stream in a way that it sometimes  considered unrealistic or very limiting (Cossu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Recently, the idea of Class-Incremental  with Repetition (CIR) scenarios has started to gather some attention in CL (Cossu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  CIR scenarios are arguably more flexible in the definition of the stream, since they allow both the  introduction of new classes and the repetition of previously seen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Crucially, repetition is a  property of the environment and cannot be controlled by the CL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This is very different from  Replay strategies (Hayes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2021), where the repetition of previous concepts is heavily structured  and can be tuned at will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  CIR defines a family of CL scenarios which ranges from CI (new classes only, without repetition) to  DI (full repetition of all seen classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Although appealing, currently there exists neither a quantita-  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='11396v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='LG]  26 Jan 2023  tive analysis nor an empirical evaluation of CL strategies learning in CIR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Mainly, because  it is not obvious how to build a stream with repetition, given the large amount of variables involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  How to manage repetition over time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' How to decide what to repeat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' What data should we use?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  In this paper, we provide two generators for CIR that, starting from a single dataset, allow to build  customized streams by only setting few parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The generators are as easy to use as CI or DI  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  We leveraged our generators to run an extensive empirical evaluation of the behavior of CL strategies  in CIR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We found out that knowledge accumulation happens naturally in streams with  repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Even a naive fine-tuning, subjected to complete forgetting in CI scenarios, is able to  accumulate knowledge for classes that are not always present in an experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We observed that  Replay strategies still provide an advantage in terms of final accuracy, even though they are not  crucial to avoid catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' On one side, distillation-based strategies like LwF (Li &  Hoiem, 2018) are competitive in streams with a moderate amount of repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' On the other side,  existing Replay strategies are not specifically designed for CIR streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We propose a novel Replay  approach, called Frequency-Aware Replay (ER-FA) designed for streams with unbalanced repetition  (few classes appear rarely, the other very frequently).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' ER-FA surpasses by a large margin other  Replay variants when looking at infrequent classes and it does not lose performance in terms of  frequent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This leads to a moderate gain in the final accuracy, with a much better robustness  and a reduced variance across all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Our main contributions are:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The design of two CIR generators, able to create streams with repetition by only setting few  control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We built both generators with Avalanche (Lomonaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2021) and  we will make them publicly available to foster future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The generators are general  enough to fit any classification dataset and are fully integrated with Avalanche pipeline to  run CL experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We perform an extensive evaluation of the properties of CIR streams and the performance  of CL strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We study knowledge accumulation and we showed that Replay, although  still effective, is not crucial for the mitigation of catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Some approaches  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', LwF) look more promising than others in CIR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We consolidate our results  with an analysis of the CL models over time through Centered Kernel Alignment (CKA)  (Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2019) and weights analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We propose a novel Replay variant, ER-FA, which is designed based on the properties of  CIR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' ER-FA surpasses other Replay strategies in unbalanced streams and provide  a more robust performance on infrequent classes without losing accuracy on the frequent  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  2  CLASS-INCREMENTAL LEARNING WITH REPETITION GENERATORS  𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  CI  DI  Concepts:  𝑒"  𝑒#  𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  𝑒"  𝑒#  𝑒$  …  𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  𝑒"  𝑒#  𝑒$  …  CIR  New Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  New/Old  Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Figure 1: Illustration of scenario types  that can be generated with episodic par-  tial access to a finite set of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  The shape colors indicate whether in-  stances are new in each episode or can  be a mixture of old and new instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  CL requires a model to learn from a stream of N expe-  riences S = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', eN}, where each experience ei  brings a dataset of examples Dei = {Xi, Yi} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Many  CL scenarios, like CI or DI, are generated from a fixed  dataset D = {(x, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' x ∈ X, y ∈ Y }, where x is the in-  put example, y is the target and Y = {1, · · · , C} is the la-  bel space (closed-world assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Depending on how  classes from the entire dataset D are shown or revisited  in the stream, this configuration can lead to CI, CIR or DI  scenarios (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Table 1, we formally present and  compare the properties of the three scenario types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  In CIR, streams with repetition are characterized by mul-  tiple occurrences of the same class over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' To study  this scenario, we propose two stream generators designed  to create a stream from a finite dataset: the Slot-Based  Generator (Gslot) and the Sampling-Based Generator  (Gsamp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Gslot generate streams by enforcing constraints  on the number of occurrences of classes in the stream us-  ing only two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Gslot does not repeat already observed samples, therefore the stream length  is limited by the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However, it guarantees that all samples in the dataset will be  2  Property  CI  DI  CIR  Instance Repetition *  Xei ∩ Xj = Ø  Xi ∩ Xj = Ø  |Xi ∩ Xj| ≥ 0  Domain Coverage  �i=N  i=1 Xi = X  �i=N  i=1 Xi = X  �i=N  i=1 Xi ∈ P(X) \\ Ø  Concept Repetition *  Yi ∩ Yj = Ø  Y1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' = YN = Y  |Yi ∩ Yj| ≥ 0  Codomain Coverage  �i=N  i=1 Yi = Y  �i=N  i=1 Yi = Y  �i=N  i=1 Yi ∈ P(Y ) \\ Ø  Table 1: Comparison of scenario properties in CI, DI and CIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' P(A) and |A| represent the power  set and the cardinality of set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' *: ∀1 ≤ i, j ≤ N , i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  observed exactly once during the lifetime of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Instead, Gsamp generates streams according  to several parametric distributions that control the stream properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' It can generate arbitrarily long  streams in which old instances can also re-appear with some probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  SLOT-BASED GENERATOR  The Slot-Based Generator Gslot allows to carefully control the class repetitions in the generated  stream with a single parameter K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Gslot takes as input a dataset D, the total number of experiences  N and the number of slots per experience K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' It returns a CIR stream composed by N experiences,  where each of the K slots in each experience is filled with samples coming from a single class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  1  10  20  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  C  Number of Experiences (N)  1  10  20  C  Number of Slots (K)  CIR  Class-Incremental  Domain Incremental  Figure 2: Illustration of how various  scenarios can be generated by Gslot, by  changing K and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The red area under  the blue curve represents invalid scenar-  ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Gslot constrains the slot-class association such that all the  samples present in the dataset are seen exactly once in the  stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Therefore, Gslot considers repetition at the level  of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' To implement this logic, Gslot first partitions  all the samples in the dataset into the target number of  slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Then, it randomly assigns without replacement K  slots per experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' At the end, the N mod K blocks  remaining are assigned to the first experience, such that  the rest of the stream is not affected by a variable number  of slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  The Slot-Based Generator is useful to study the transition  from CI scenarios to DI scenarios, obtained by simply  changing the parameter K (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For example, let  us consider a dataset with 10 classes such as MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' By  choosing N = 5 and K = 2 we obtain the popular Split-  MNIST, a CI scenario with no repetition and 2 classes for  each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Conversely, by setting N = 5 and K =  10 we obtain a DI stream where all the 10 classes appear  in each experience with new unseen samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' More in  general, given a dataset with C classes, we obtain a CI scenario by setting K = C  N (N must divide  C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We obtain a DI scenario by setting K = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Appendix B we illustrate the overall steps of  stream generation (Figure 12), and provide a step-by-step formal definition of Gslot (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  SAMPLING-BASED GENERATOR  The Sampling-Based Generator (Gsamp) generates arbitrarily long streams and controls the repe-  titions via probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The stream generator allows to control the first occurrence of  new classes and the amount of repetitions of old classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Unlike Gslot, it allows to generate infinite  and even unbalanced streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Gsamp parameters:  N: Stream length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' number of experiences in the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  S: Experience size which defines the number of samples in each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Pf(S): Probability distribution over the stream S used for sampling the experience ID of  the first occurrence in each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Pr: List of repetition probabilities for dataset classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  3  Scenario  Matrix  Generator  𝑁  𝐾  CL Scenario  ℙ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (𝒮)  𝑃"  Scenario Matrix  𝐺!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' "#$  Parameters  Sampler  …  𝑒#  𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' 𝑒" 𝑒# 𝑒$ 𝑒%  𝑐!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  𝑐"  𝑐#  𝑐$  𝑐%  𝑐&  𝑐\'  𝑐(  𝑐)  …  …  …  …  …  …  …  …  …  …  𝒟  𝒮$"%&\'  𝒮$()$  𝑒*  …  𝑒#  𝑒*  Instances  Concepts  Figure 3: Schematic view of Gsamp generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Each concept is shown with a different color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Note that Pf is a probability mass function over the stream S which means it sums up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 and  determines in which parts of the stream it is more likely to observe new classes for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  However, the list of probabilities {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', pC} in Pr are independent and each probability value  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ≤ pi ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 indicates how likely it is for each class to repeat after its first occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  For each experience, Gsamp samples instances from the original dataset D according to a two step  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' First, Gsamp defines a C × N binary matrix T called Occurrence Matrix that determines  which classes can appear in each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Then, for each experience ei, 1 ≤ i ≤ N we use the  i-th column of T to sample data points for that experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The generator uses the inputs N, Pf(S)  and Pr to generate T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Therefore, it first initializes T as a zero C × N matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Then for each class  c in the dataset, it uses Pf(S) to sample the index of the experience in which class c appears for  the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Different probability distributions can be used to either spread the first occurrence  along the stream or concentrate them at the beginning, which allows a good flexibility in the stream  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' After sampling the first occurrences, the classes are then repeated based on Pr probability  values to finalize matrix T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In the simplest case, Pr can be fixed to the same value for all classes to  generate a balanced stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Once the matrix T is constructed, a random sampler is used to sample patterns for each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Since each experience may contain an arbitrary number of classes, another control parameter that  could be added here is the fraction of samples per class in experience size S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For simplicity we keep  the fractions equally fixed and thus the number of datapoints sampled from each class in experience  ei is ⌊ S  |Ci|⌋ where |Ci| indicates the number of classes present in ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Since the sampler is stochastic,  each time we sample from a class, both new and old patterns can appear for that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Given a large  enough stream length N, the final stream will cover the whole dataset with a measurable amount of  average repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Figure 3 we demonstrate the schematic of the generator Gsamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We provide  the pseudo-code for Gsamp in Appendix C (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Although we assume a fixed number of instances per class in D, Gsamp can be easily extended  to settings where the number of instances can grow over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Moreover, the sampler can also be  designed arbitrarily depending on the stochasticity type, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', irregular or cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  3  FREQUENCY-AWARE REPLAY  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  Ratio  Storage Policy  FA (Ours)  CB  RS  Figure 4: Ratio of buffer slots for infre-  quent classes for three random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Experience Replay (ER) is the most popular CL strategy  due to its simplicity of use and high performance in class-  incremental scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The storage policy, which deter-  mines which samples to keep in a limited buffer, is the  major component of ER methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Class-Balanced (CB)  and Reservoir Sampling (RS) Vitter (1985) are the most  popular storage policies in ER methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' CB keeps a fixed  quota for each class, while RS samples randomly from the  stream, which leads to the class frequency in the buffer  being equal to the frequency in the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' CB and RS  are great choices for balanced streams such as class incremental scenarios, where the number of  samples per class is the same over the whole stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However, as in most real-world scenarios, CIR  scenarios are naturally unbalanced, and different classes may have completely different repetition  frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Accordingly, CB and RS storage policies may suffer a big accuracy drop in the infre-  quent classes of an unbalanced stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For example, in highly unbalanced streams, RS will store  an unbalanced buffer replicating the the original distribution of the stream, which is sub-optimal be-  4  cause the less frequent classes will require more repetition to prevent forgetting, while the frequent  classes will be repeated naturally via the stream occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  We propose Frequency-Aware (FA) storage policy that addresses the imbalance issue in CIR streams  by online adjustment of the buffer slots in accordance with the amount of repetition for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Given a buffer B with a maximum size of M, a list of previously observed classes P initialized as  P = {} with a corresponding list O indicating the number of observations per class c in C, and a  dataset Di from experience ei, the algorithm first checks the present classes Pi in Di and adds them  to P (P ← P ∪ Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Then, for each class c in Pi it increments the number of observations O[c] by  1 if the class was previously seen, otherwise it initializes O[c] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' After updating the number of  observations, FA computes the buffer quota Q for all observed classes by inverting the number of  observations (Q = [  1  O[c]∀c ∈ C]) and normalizes it .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This way, the algorithm offers the less frequent  classes a larger quota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Finally, a procedure ensures the buffer is used to its maximum capacity by  filling unused slots with samples from more frequent classes sorted by their observation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This  is a crucial step since it is possible that an infrequent class which is not present ei will be assigned  with a larger quota than its current number of samples in B, and therefore the buffer will remain  partially empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Figure 4 we show how our method assigns higher ratio of samples for infrequent  classes to overcome the imbalance issue in the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For further analysis and pseudo-code of FA  policy refer to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We present examples of unbalanced scenarios in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  4  EMPIRICAL EVALUATION  We study CIR scenarios by leveraging our generators Gsamp and Gslot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' First, by using Gslot we  provide quantitative results about forgetting in CL strategies when transitioning from CI to DI sce-  narios (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Then, by using Gsamp we focus on long streams with 500 experiences and  study the performance of Replay and Naive (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The long streams give us the opportunity  to study knowledge accumulation over time in the presence of repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We also provide an intu-  itive interpretation of the model dynamics over long streams (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Finally, we show that our  Frequency-Aware Replay is able to exploit the repetitions present in the stream and to surpass the  performance of other replay approaches not specifically designed for CIR scenarios (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  The experiments were conducted using the CIFAR-100 Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2009) and Tiny-ImageNet  LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (1998) datasets with the ResNet-18 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For Gslot, we run experiments for Naive (in-  cremental fine tuning), LwF Li & Hoiem (2018), EWC Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2017), Experience Replay  with reservoir sampling Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2020) (ER-RS) and AGEM Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2018) strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  For Gsamp we run experiments for Naive and ER (CB/RS/FA) strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We set the default buffer  size for CIFAR-100 to 2000, and for Tiny-ImageNet to 4000 in the replay strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We evaluate  all strategies on the Average Test Accuracy (TA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  TRANSITION FROM CLASS-INCREMENTAL TO DOMAIN INCREMENTAL  DI and CI scenarios are heavily studied in the CL literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However, little is known about what  happens to the performance of popular CL strategies when gradually transitioning from one scenario  to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' By changing the value of K in Gslot, we provide a quantitative analysis of such  behavior in CIR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Figure 5 shows the Average Test Accuracy over all classes for different  CL strategies when transitioning from CI (left-most point of each plot) to DI (right-most point of  each plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Replay is one of the most effective strategies in CI scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' As expected, in CIR scenarios the  advantage provided by ER-RS with respect to other CL strategies diminishes as the amount of rep-  etition increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However, in order for the other strategies to match the performance of ER-RS, the  environment needs to provide a large amount of repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  LwF guarantees a consistent boost in the performance, both in CIFAR-100 and Tiny-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  In particular, and quite surprisingly, on Tiny-ImageNet LwF is able to quickly close the gap with  ER-RS and even surpass it as the amount of repetition increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The positive interplay between  distillation and repetition provides an effective way to mitigate forgetting in CIR scenarios, without  the need to explicitly store previous samples in an external memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' EWC showed different sen-  sitivity to the regularization hyper-parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We experimented with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1, 1, 10, 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' While  on MNIST we did not see any difference in performance, on CIFAR-100 and Tiny-ImageNet large  values of λ lead to a dramatic decrease, dropping as low as Naive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We found 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1 to be the best value  5  2  3  5  8  10  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Average Test Accuracy  MNIST  ER-RS  LwF  EWC  A-GEM  Naive  10  30  50  80  100  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='6  CIFAR-100  20  60  100  160  200  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  Tiny-ImageNet  Figure 5: Average Test Accuracy for different values of K in CIR scenarios generated with Gslot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Class-Incremental scenarios are represented by the left-most point of each plot, Domain-Incremental  scenarios by the right-most point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Results averaged over 3 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  50  100  150  200  250  300  350  400  450  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='9  Class Accuracy  Class Status  Present  Missing  Figure 6: Accuracy of a particular class over the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The target class is either present or absent  in the experiences indicated by the blue and orange points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  on both CIFAR-100 and Tiny-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This configuration only provides a low amount of regular-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Overall, the role played by the natural repetition already guarantees a sufficient stability of  the model, which is additionally boosted only in the case of LwF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  IMPACT OF REPETITION IN LONG STREAMS  We investigate the impact of repetition in long streams (N = 500) generated with Gsamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For  the long-stream experiments we also report the missing-classes accuracy (MCA) and seen-classes  accuracy (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' MCA measure the accuracy over the classes that were seen before but are missing  in the current experience, and SCA measure the accuracy over all seen classes up to the current  experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Missing Class Accuracy Increases Over Time  In CI scenarios, a Naive strategy catastrophically  forgets each class as soon as it starts learning on new classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Surprisingly, we found that in CIR  scenarios there is knowledge accumulation over time for all the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Figure 6 shows the accuracy  of a single class over time, highlighting whether the class is present or not in the current experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  At the beginning of the stream missing classes are completely forgotten, which can be noticed by  the instant drop of the accuracy to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However, over time the model accumulate knowledge  and the training process stabilizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' As a result, the accuracy of missing classes tends to increase  over time, suggesting that the network becomes more resistant to forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Notice that this is an  example of continual learning property that is completely ignored when testing on CI scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  This finding prompts the question, ”What is happening to allow knowledge accumulation even for  Naive finetuning?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We investigate this question by analysing the model’s accuracy over time and  the properties of the learned model in the next experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Accuracy Gap Between Naive and Replay Decreases Over Time  To study the impact of long  streams with repetitions we monitor the accuracy gap between ER and Naive fine-tuning by com-  paring their accuracy after each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For the scenario configuration, we set Pf(S) as a  Geometric distribution with a coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='01 and fix the probability of repetition Pr as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2 for  all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For more details and illustrations on distribution types refer to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In such  scenarios, the majority of classes occur for the first time in the first quarter of the stream, and then  repeat with a constant probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2 which makes them appropriate for our experiments since  all classes are observed before the middle of the stream and the repetition probability is low enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  As can be seen in Figure 7, while the accuracy of ER saturates over time, the accuracy of Naive  increases, closing the gap between the two strategies from around 25% in experience 100 to 7% in  6  0  100  200  300  400  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='6  Accuracy  Average Test Accuracy  Strategy  ER-CB  Naive  0  100  200  300  400  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='6  Accuracy  Average Seen Class Accuracy  Strategy  ER-CB  Naive  0  100  200  300  400  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='6  Accuracy  Average Missing Class Accuracy  Strategy  ER-CB  Naive  Figure 7: Average test accuracy and average missing class accuracy plots for long streams streams  with 500 experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  experience 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This supports our hypothesis that neural network’s ability to consolidate knowledge  is significantly influenced by ”natural” repetition in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  The Role of Repetition  The amount of repetition is one of the key aspects of a CIR scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  To find out how strategies perform under different repetition probabilities, we consider a setting  where all components of a scenario are fixed except for Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For this experiment, we set Pf(S)  as geometric distribution with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2 and let Pr change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Figure 8 we demonstrate the seen  class accuracy (SCA) for the Naive and ER-CB strategies in CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' It is clear from the plots,  that the model’s rate of convergence can be significantly affected by the amount of repetition in the  scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Although, it may seem obvious that higher repetition leads to less forgetting, it is not very  intuitive to what extent different strategies may gain from the environment’s repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' While the  Naive strategy gains a lot from increased repetition, the replay strategy saturates after some point  for higher repetitions and the gaps close between different repetition values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  0  100  200  300  400  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='7  Accuracy  ER-CB  p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='0  0  100  200  300  400  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='7  Accuracy  Naive  p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='0  Figure 8: Retained accuracy for different values of p in Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3  MODEL SIMILARITY AND WEIGHT SPACE ANALYSIS  Weight Interpolation  Based on the ”gradual loss drop” observation in missing classes, we study  how the loss surface changes over time if we perturb the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We interpolate between the model  weights from two consecutive checkpoints with an interval of 10 experiences in various segments  of the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Assuming that w∗  t and w∗  t+10 are the obtained solutions for experiences t and t + 10  respectively, we generate eight in-between models wk = α ∗ w∗  t + (1 − α) ∗ w∗  t+50 by by increasing  α from zero to one, and then compute the accuracy of wk for the data of experience t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We show  the interpolation accuracy for various pairs of experiments in different segments of the stream for  the Naive strategy in Figure 9 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In the beginning of the stream, the accuracy of experience t  in each pair drops significantly, while we observe a milder loss drop towards the end of the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  The findings suggest that, towards the end of the stream, even a relatively big perturbation does not  have a large negative effect on the model’s accuracy and the optimal solutions of the consecutive  experiments are connected with a linear low-loss path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Weight changes  Another approach to analyzing the gradual drop of the accuracy is by dissecting  how much, when, and where the weight changes occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' As shown in Figure 9 (right), we can  observe that within the first experiences, there is a significant difference for blocks 0, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This  difference then stalls, showing that as we continue training experiences, the weights of these blocks  stabilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' On the other hand, blocks 3 and 4 show a linear increase in the difference with the number  of experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' An explanation for this phenomenon, is that the first layers of the model capture  knowledge that can be useful for several classes (more general), so it is unnecessary to change them  after several experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' On the other hand, the last blocks are the ones that memorize or learn  more specific patterns, so they adapt to each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='0  Interpolation Weight  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='0  Accuracy  Accuracy over Linear Path  Start  End  50  100  100  150  200  250  300  350  400  450  0  100  200  300  400  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='175  Difference  Difference over the Experience  Blocks  Block 0  Block 1  Block 2  Block 3  Block 4  Figure 9: (left) Interpolation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (right) Weight changes in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The difference used in  (right) is calculated as Dj =  1  |θ0|  �θb  i  ��� (θ0,i−θj,i)  ∥θ0,i∥2  ���, where the weights of experience j are compare  with the initialization θ0 for each block i  CKA Analysis  Finally, we show the CKA Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2019) of the model in the beginning,  middle and the end of the stream with an interval difference of 50 experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' As shown in the  visualizations in Figure 10, the longer the model is trained on more experiences, the less significant  the changes in the representations become especially for the final layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We can see that the diag-  onal of the CKA becomes sharper propagating forward with more experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This indicates that  although the model is trained on different subsets of classes in each experiment, the representations  change less after some point in the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Experience-15  Experience-5  5  15  Experience-35  Experience-25  25  35  Experience-65  Experience-55  55  65  Experience-200  Experience-190  190  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='0  Figure 10: CKA of the model in different parts of the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  FREQUENCY-AWARE REPLAY IN UNBALANCED SCENARIOS  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='5  Accuracy  Strategy  ER-FA  ER-CB  ER-RS  Naive  Figure 11:  Accuracy of Infrequent  Classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  We conduct experiments for bi-modal unbalanced scenar-  ios where classes can have a high frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 or a  low frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We use a fraction factor that de-  termines the amount of infrequent classes in the scenario,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', Fraction=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3 means that 30% of the classes are infre-  quent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Table 2 we compare ER-FA with the Naive, ER-  RS and ER-CB strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The numbers show the MCA  and average Test Accuracy (TA) metrics for each strategy  in the end of the stream averaged over three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Our  strategy outperforms all other scenarios in almost all set-  tings in both CIFAR-100 and TinyImageNet datasets in  terms of TA, and significantly outperforms other methods  in terms of MCA (in the last experience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Moreover, we  illustrate the accuracy of infrequent classes in CIFAR-100 experiments for Fraction=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3 in Figure  11 where ER-FA achieves considerably higher accuracy in the whole stream by assigning larger  quota to infrequent classes without losing its performance on frequent classes (refer to Appendix G  for further illustrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  5  RELATED WORK  Current CL methods are mainly focused on two types of benchmarks namely, Multi Task (MT) and  Single Incremental Task (SIT) Maltoni & Lomonaco (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' MT divides training data into distinct  tasks and labels them during training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' SIT splits a single task into a sequence of  unlabeled experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' SIT can be further divided into Domain-Incremental (DI) where all classes  are seen in each experience, and Class-Incremental (CL) where each experience contains only new  8  DS  Strategy  Fraction= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  Fraction= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3  Fraction= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  MCA  TA  MCA  TA  MCA  TA  C-100  Naive  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  ER-RS  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='9  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='9  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  ER-CB  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  ER-FA  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  TIN  Naive  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  ER-RS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  ER-CB  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3  ER-FA  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='9  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  Table 2: Unbalanced scenario results for the CIFAR-100 (C-100) and TinyImageNet (TIN) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  “Fraction” refers to the fraction of infrequent classes having repetition probability of only 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  (unseen) classes van de Ven & Tolias (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Both DI and CI are extreme cases and are unlikely to  hold in real-world environments Cossu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In a more realistic setting, the role of natural  repetition in CL scenarios was studied in the context of New Instances and Classes (NIC) scenario  Lomonaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2020) and the CRIB benchmark Stojanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' NIC mainly focuses on  small experiences composed of images of the same object, and repetitions in CRIB are adapted to  a certain dataset and protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The Class-Incremental with Repetition (CIR) scenario was initially  formalized in Cossu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2021), however the work lacks a systematic study of CIR scenarios as  the wide range of CIR scenarios makes them difficult to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  To counter the lack of repetition in CI, replay has been extensively used as a CL strategy (Rebuffi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Lopez-Paz & Ranzato, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Belouadah & Popescu, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Douillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In such methods,  natural repetition is artificially simulated by storing past data in an external memory, and replaying  them alongside the scenario stream data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Repetition reduces catastrophic forgetting through implicit  regularization of model’s weights Hayes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In CI benchmarks, replay seems to be the  only working strategy van de Ven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In other words, replay seems to be a necessity when  no natural repetition happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Although replay can be seen as a method to simulate natural repetitions artificially, the two concepts  are fundamentally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Repetition in replay strategies occurs with the same data seen in pre-  vious experiences, which is neither realistic nor biologically plausible Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' On the  other hand, natural repetitions of already seen objects occur in different real-world environments,  and better fit the CIR scenario studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Recently, Lesort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2022) scaled the number  of tasks in a finite world setting (Boult et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Mundt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', 2022) where the model has access to  a random subset of classes in each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The authors proposed naive fine-tuning with masking  techniques to improve retained accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Our work is different in the sense that we compare among  different strategies and study various types of repetitions with two flexible generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  6  DISCUSSION AND CONCLUSION  We defined CIR scenarios which represent CL environments where repetition is naturally present in  the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Although the concept of repetition is quite intuitive, it is not obvious how to realize it in  practice for research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Therefore, we proposed two CIR generators that can be exploited to  address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Through empirical evaluations, we showed that, unlike CI scenarios, knowledge  accumulation happens naturally in CIR streams, even without applying any CL strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This raised  the question of whether the systematic repetition provided by Replay is critical in all CIR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  With several experiments in long streams, we demonstrated that although Replay provides an ad-  vantage in general, even random repetition in the environment can be sufficient to induce knowledge  accumulation given a long enough lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Moreover, we found that existing Replay strategies are exclusively designed for classical CI scenar-  ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Thus, we proposed a novel strategy, ER-FA, to exploit the properties of CIR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' ER-FA  accumulates knowledge even in highly unbalanced stream in terms of class frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' ER-FA out-  performs by a large margin other Replay approaches when monitoring the accuracy for infrequent  classes while preserving accuracy for the frequent ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Overall, ER-FA guarantees a more ro-  bust performance on a wide range of real-world scenarios where classes are not homogeneously  distributed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  9  The framework defined in this work opens new research directions which depart from the existing  ones, mainly focused on the mitigation of forgetting in CI scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We hope that our experiments  and results will promote the study of CIR scenarios and the development of new CL strategies, able  to exploit the inner semantics of repetition, a natural trait of real-world data streams, in many other  clever and imaginative ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content=' Gradient-based learning applied to  document recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Proceedings of the IEEE, 86(11):2278–2324, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Timoth´ee Lesort, Vincenzo Lomonaco, Andrei Stoian, Davide Maltoni, David Filliat, and Natalia  D´ıaz-Rodr´ıguez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Continual learning for robotics: Definition, framework, learning strategies, op-  portunities and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Information fusion, 58:52–68, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Timoth´ee Lesort, Oleksiy Ostapenko, Diganta Misra, Md Rifat Arefin, Pau Rodr´ıguez, Laurent  Charlin, and Irina Rish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Scaling the number of tasks in continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  arXiv preprint  arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='04543, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Zhizhong Li and Derek Hoiem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Learning without forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' IEEE Transactions on Pattern Analysis  and Machine Intelligence, 40(12):2935–2947, Dec 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' ISSN 1939-3539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1109/TPAMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='  Vincenzo Lomonaco, Davide Maltoni, and Lorenzo Pellegrini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Rehearsal-free continual learning  over small non-iid batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In CVPR Workshops, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='  Vincenzo Lomonaco, Lorenzo Pellegrini, Andrea Cossu, Antonio Carta, Gabriele Graffieti, Tyler L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Hayes, Matthias De Lange, Marc Masana, Jary Pomponi, Gido M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' van de Ven, Martin Mundt,  Qi She, Keiland Cooper, Jeremy Forest, Eden Belouadah, Simone Calderara, German I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Parisi,  Fabio Cuzzolin, Andreas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Tolias, Simone Scardapane, Luca Antiga, Subutai Ahmad, Adrian  Popescu, Christopher Kanan, Joost van de Weijer, Tinne Tuytelaars, Davide Bacciu, and Da-  vide Maltoni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Avalanche: An end-to-end library for continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Proceedings of the  IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  3600–3610, June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  David Lopez-Paz and Marc’Aurelio Ranzato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Gradient episodic memory for continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Davide Maltoni and Vincenzo Lomonaco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Continuous learning in single-incremental-task scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Neural Networks, 116:56–73, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Martin Mundt, Iuliia Pliushch, Sagnik Majumder, Yongwon Hong, and Visvanathan Ramesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Uni-  fied probabilistic deep continual learning through generative replay and open set recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Journal of Imaging, 8(4):93, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  German I Parisi, Ronald Kemker, Jose L Part, Christopher Kanan, and Stefan Wermter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Continual  lifelong learning with neural networks: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Neural Networks, 113:54–71, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' icarl:  Incremental classifier and representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Proceedings of the IEEE conference on  Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='  Stefan Stojanov, Samarth Mishra, Ngoc Anh Thai, Nikhil Dhanda, Ahmad Humayun, Chen Yu,  Linda B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Smith, and James M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Rehg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Incremental Object Learning From Contiguous Views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  8777–8786, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Gido M van de Ven and Andreas S Tolias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Three scenarios for continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' arXiv preprint  arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='07734, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  11  Gido M van de Ven, Hava T Siegelmann, and Andreas S Tolias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Brain-inspired replay for continual  learning with artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Nature communications, 11(1):1–14, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Jeffrey S Vitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Random sampling with a reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' ACM Transactions on Mathematical Software  (TOMS), 11(1):37–57, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Yue Wu, Yinpeng Chen, Lijuan Wang, Yuancheng Ye, Zicheng Liu, Yandong Guo, and Yun Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Large scale incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Proceedings of the IEEE/CVF Conference on Computer  Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' 374–382, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  12  A  RELATED WORK (CONTINUED)  Considering benchmark formalization frameworks, De Lange & Tuytelaars (2021) recently pro-  posed a subdivision aimed at framing continual learning setups by categorizing them based on the  batch and observable horizon that the learning agent is able to access at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' With this frame-  work, the authors aim to better formalize the online learning setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' While the concept of observ-  able horizon may be useful in evaluating the significance and local (in time) usefulness of natural  repetition in a training stream, this work does not consider the concept of natural repetition in its  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Recently, Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2022) proposed to introduce blurry task boundaries in class incremental bench-  marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Their proposal is based on previous works Bang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' (2019) that tried  to produce more realistic benchmarks by blurring the class-incremental scenario, which however  resulted in a setup in which no classes are added to new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' They argue that this idea moves the  focus too far away from the class-incremental setup and it is still not quite realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The resulting  setup, named i-Blurry, aims at resolving the aforementioned issues and moving toward a more re-  alistic scenario by partitioning the classes available in the source dataset into two groups: Disjoint  and Blurry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Classes of the disjoint group are gradually added in successive experiences while sam-  ples of classes from the blurry group always appear in all experiences with their numerosity being  controlled through a blur ratio M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The authors show that, based on the degree of disjunction N  and blurry M, this framework can produce class-incremental (no blurring), domain-incremental (no  disjunction), and blurred setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This setup is the one that most moved towards the direction of  introducing repetition in Continual learning benchmarks in a controlled way so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However, the  proposed blurring mechanism is too coarse-grained to simulate a natural repetition of concepts as  the significance of the repetition introduced by blurring relies too much on i) a random (uniform)  sampling of the concepts to be repeated, ii) the static subdivision of classes in the Disjoint and Blurry  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  B  SLOT-BASED GENERATOR  Following the properties of CIR scenarios in Section 2, Gslot generates a subset of CIR streams that  hold the assumptions below in the defined properties:  |Xi ∩ Xj| = 0 : new instances appear in each experiences  �i=N  i=1 Xi = X: all samples are used  |Yi ∩ Yj| ≥ 0, ∀1 ≤ i, j ≤ N where i ̸= j  �i=N  i=1 Yi = Y : all classes are used  X is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  These assumptions allow transitioning through different CIR scenario types between the two ex-  tremes of CI and DI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  ALGORITHM  The overall steps of Gslot are illustrated in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Algorithm 2 we present all steps of Gslot  used to generate arbitrary CIR scenarios given a dataset D, number of experiences N and number  of slots K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The output of the algorithm is a CL stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  TRANSITIONING  Transitioning in Gslot for a scenario with a fixed number of experiences can be done by increasing  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' When K = 1 the generated scenario will be class-incremental and as K get closer to the total  number of classes in D, the scenarios moves towards a domain incremental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In Figure 13 we  show an example of how generated scenarios change by increasing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  13  …  Concepts:  Replicate each concept !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='×#  $  times  and shuffle the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  𝒆𝟏  𝒆𝟐  𝒆𝟑  …  Example: 𝐾 = 4  Slot 1  Slot 2  Slot 3  Slot 4  New Instances  𝒆𝑵  𝓢𝒕𝒓𝒂𝒊𝒏  Figure 12: Illustrations of the overall steps of Gslot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Each shape represents a concept, and the green  color means that new instances of that concept are used in each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Class  Experience  Figure 13: From left to right: transitioning from CI to DI in Gslot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Each class is represented with a  unique color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  C  SAMPLING-BASED GENERATOR  Following the properties of CIR scenarios in Section 2, Gsamp generates a subset of CIR scenarios  that hold all defined properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Additionally, for any stream S = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', eN}, Gsamp defines  a probability distribution for the first occurrence of concepts over S and per-class probabilities for  each concept c ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Gsamp can generate arbitrarily long stream (N ≥ 1) and even from a growing  set of samples X where Y remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  ALGORITHM  The overall steps of the Gslot are shows in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  DISTRIBUTION TYPES  In this section we show some examples of different discrete distributions that can be used for Pf(S)  and Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For Pr we use the unnormalized version of the final distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Distributions used for  Gsamp can be any arbitrary discrete distribution and are not limited to the ones we describe here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  ZIPFIAN  Given the number of elements N and scalar e ≥ 0 , the probability mass function of a Zipfian  distribution over a list of N elements is defined in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' When used for the probability of first  occurrence, the distribution can be defined over the experiences of a stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For example, N can  be considered as the number of experiences and i can indicate the ith experience in the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' By  increasing e, the distribution over the stream will be skewed towards the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In figure 14 we  demonstrate some examples of first occurrence probabilities over a stream of length 10 generated  with Zipf distribution with increasing values of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Many natural distribution follow Zipf distribution  14  Algorithm 1 Slot-Based Generator (Gslot) Pseudo-Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Require: Dataset D = {(xi, yi)}i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=',P with C classes, number of experiences N, experience  size S present in each experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Ensure: K ≤ N  Ensure: C mod N = 0  Ensure: NK mod C = 0  cls-idxs = {}  ▷ Empty dictionary  for y ∈ set({yi}i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=',P ) do  cls-idxs[y] = []  ▷ Empty list init  end for  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' , P do  cls-idxs[yi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='append(i)  end for  slots={}  ▷ Empty dictionary  for y ∈ cls-idxs do  slots[y] = []  ksample = int(len(cls-idxs[y]) /K)  for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' , N×K  C  do  subset-idxs = pop(cls-idxs[y], ksample)  subset-samples = [xidx for idx ∈ subset-idxs]  slots[y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='append(subset-samples)  end for  end for  stream = []  for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' , N do  experience = dataset()  seen-classes = []  for k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=',K do  repeat  y = sample(slots)  until y /∈ seen-classes  seen-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='append(y)  experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='add(pop(slots[y], 1))  end for  stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='append(experience)  end for  return stream  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Probability  e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  0  5  10  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  e = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Figure 14: Zipf distribution with varying values of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  and it can be used to generate highly skewed distributions both for first occurrence and repetition  probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  f(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' e, N) =  1  ie  �N  n=1  1  ne  (1)  15  Algorithm 2 Sampling-Based Generator (Gsamp) Pseudo-Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Require: Dataset D = {(xi, yi)}i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=',P with C classes, number of experiences N, number of  slots K, probability distribution for first occurrence Pf(S), and list of repetition probabilities Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  T = {0}C×N  ▷ Initialize occurrence matrix with zeros  for c ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' C} do  i ∼ Pf(S)  ▷ Samples the first occurrence of class c  T[c, i] = 1  for j ∈ {i, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' N} do  r ∼ U(0, 1)  ▷ U: uniform distribution over [0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0]  if r < Pr[c] then T[c, j] = 1  end if  end for  end for  E = {}  for ei ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' N} do  Ci = RetrieveClasses(ei)  Dei = Sample(D, Ci, S)  ▷ Sample S instances from dataset D for classes Ci  E ← E ∪ Dei  end for  Strain, Stest = GenerateStream(E)  ▷ Generate streams using E  return Strain, Stest  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  POISSON  The PMF for Poisson distribution is given in Equation 2 where µ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Poisson with larger values  of µ can be used for distributions where the probability of occurrence/repetition first rises and then  gradually decreases over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  f(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' µ) = µie−µ  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  (2)  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Probability  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  0  5  10  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Figure 15: Poisson distribution with varying values of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='3  GEOMETRIC  Another useful distribution that can be used for the first occurrence probabilities over a stream is  Geometric distribution with its PMF given in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' This distribution is in particular inter-  esting for transitioning from domain incremental to class incremental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' By setting p = 1, only the  probability of experience i = 0 will be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 and the rest will be zero, and by decreasing p,  the probability will spread over the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In figure 17 we show examples for generated scenarios  with Gsamp with Geometric first occurrence and fixed probability of repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  f(i, p) = (1 − p)i−1p  (3)  16  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Probability  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='01  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  0  5  10  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='7  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Figure 16: Geometric distribution with varying values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Experience  Class  Experience  Class  Experience  Class  Figure 17: Scenarios generated with Geometric first occurrence and probability of repetition equal  to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 for all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The p values for the Geometric distributions from left to right are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2 and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  D  UNBALANCED SCENARIOS  In this section we present a particular type of unbalanced scenarios where a subset of classes in  the stream have a low probability of repetition and the rest repeat very often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We refer to such  scenarios bi-modal scenarios, where each mode refers to a subset of classes with a distinct repetition  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' More specifically, we have a stream of experiences S = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=', eN} where Y S =  �N  1 Yei indicates the set of all available concepts in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In bi-modal scenarios Y = Y if ∪Y fr where  Y if and Y fr are the set of frequent and infrequent concepts respectively, and Y if ∪ Y fr = Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In  Figure 18 we show examples of unbalanced bi-modal scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Experience  Class  Experience  Class  Experience  Class  Figure 18: Unbalanced scenarios with two modes of repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The fractions of infrequent classes  from left to right are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Repetition probabilities for frequent and infre-  quent classes are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='9 accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  E  FREQUENCY-AWARE REPLAY  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  ALGORITHM  In Algorithm 3 we present the steps for updating the buffer in FA storage policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  ANALYSIS: VARYING THE FRACTION OF INFREQUENT CLASSES  In this section, we study the behavior of FA, CB, and RS storage policies by changing the fraction  of infrequent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In our analysis, we consider an unbalanced stream generated with Gsamp  where N = 100 and the probability of repetition for frequent and infrequent classes are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='9 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In such streams, the large probability gap between frequent and infrequent classes  17  Algorithm 3 Frequency-Aware Buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Require: Current Buffer Set B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Maximum buffer size M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' List of Seen Classes C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Number of  Observation per Seen Class O  D = GetExperienceDataset(ei)  P = DetectPresentClasses(D)  C ← C ∪ P  for c ∈ P do  ▷ For each present class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' increment the number of observations  if c ∈ O then O[c]+ = 1  else O[c] = 1  end if  end for  Q = [  1  O[c]∀c ∈ C]  ▷ Calculate quota per class  ˆQ =  Q  |Q|  ▷ Normalize quota values  S = {⌈Q[c] ∗ M⌉∀c ∈ C}  ▷ Calculate buffer slot size for each class  UpdateSlots(S)  ▷ Update assigned slots according to the current state of B  UpdateBuffer(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' S)  return B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' O  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Ratio  Storage Policy  FA (Ours)  CB  RS  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Ratio  Storage Policy  FA (Ours)  CB  RS  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Ratio  Storage Policy  FA (Ours)  CB  RS  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Ratio  Storage Policy  FA (Ours)  CB  RS  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  Ratio  Storage Policy  FA (Ours)  CB  RS  Figure 19:  Ratio of samples for infrequent classes in unbalanced scenarios for the FA,  CB and RS policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Fraction of infrequent classes from top-left to bottom-right are  20%, 40%, 60%, 80%, 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  helps us observe the difference more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' We report the ratio of samples assigned to infrequent  classes in the buffer in the lifetime of the model in the stream for scenarios where the fraction of  infrequent classes is equal to {20%, 40%, 60%, 80%, 100%}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' For this experiment, we set the buffer  size to 500 for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  As demonstrated in Figure 19, when the fraction of infrequent classes is equal to 20%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' only 20%  of classes are infrequent, the ratio is very low for RS policy as it tries to replicate the true distribution  of the stream while CB assigns exactly 20% of the buffer space to the infrequent samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' However,  we can observe that FA starts to assign more samples over time to the infrequent classes over time as  it adapts the buffer slots based on the frequency of repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Moreover, it is evident in the plots that,  by increasing the fraction of infrequent classes, the ratio gap between FA and CB gets smaller as  the quota for CB stays the same while the number of infrequent classes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Eventually, when  the fraction of infrequent classes is equal to 100%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' all classes have the same (low) probably of  repetition, all buffers have exactly the same ratio since all classes are infrequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  In conclusion, FA buffer slots can be very helpful in highly unbalanced streams where a smaller  fraction of classes have a low probability of repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' When the stream moves towards becoming  18  balanced, the FA and CB get closer, and all methods become similar in the extreme case of a fully  balanced stream with similar probability of repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  F  CHANGING Pf(S)  0  100  200  300  400  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='7  Accuracy  ER-CB  p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='0  0  100  200  300  400  500  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='7  Accuracy  Naive  p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='0  Figure 20: Average test accuracy for different values of p in first occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  We conduct experiments to find the differences between situations in which all classes occur early  versus those in which new classes also appear late in the stream in order to analyze the role of first  occurrence type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' How early or late in the stream we observe all classes of a dataset, depends on  the the parameters that control Pf(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In this experiment, we fix the probability of repetition Pr  and change Pf(S)’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' In particular, we opt the geometric distribution for Pf(S) and  choose the values {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='0} for its only parameter 0 < p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Increasing p is inversely  proportional to the spread factor in the first occurrence distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' when p is close to 0 all  classes happen in the first experience and as we move p toward 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='0 the classes start to spread along  the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' Figure 20 shows the CIFAR-100 results for the Naive and ER-CB strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content=' The results  suggest that when the spread factor is low, the model initially has difficulty to learn since there  are more classes in the initial experiments and thus the model has to learn from fewer instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  However, with more experiences, all first occurrence types, reach almost the same SCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  G  ER-FA RESULTS  Results in Figure 21 illustrate the total test accuracy and accuracy of frequent classes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  Although the discrepancy between the accuracies of frequent classes is very small, the total test ac-  curacy can significantly vary due to the difference in the accuracy of infrequent classes as presented  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
+page_content='  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+page_content='6  Accuracy  Strategy  ER-FA  ER-CB  ER-RS  Naive  0  20  40  60  80  100  Experience  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf'}
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+1
+Enabling Augmented Segmentation and Registration
+in Ultrasound-Guided Spinal Surgery via Realistic
+Ultrasound Synthesis from Diagnostic CT Volume
+Ang Li†, Jiayi Han†, Yongjian Zhao, Keyu Li, Li Liu‡
+Abstract—This paper aims to tackle the issues on unavailable
+or insufficient clinical ultrasound (US) data and meaningful
+annotation to enable bone segmentation and registration for US-
+guided spinal surgery. While the US is not a standard paradigm
+for spinal surgery, the scarcity of intra-operative clinical US data
+is an insurmountable bottleneck in training a neural network.
+Moreover, due to the characteristics of US imaging, it is difficult
+to clearly annotate bone surfaces which causes the trained neural
+network missing its attention to the details. Hence, we propose an
+In silico bone US simulation framework that synthesizes realistic
+US images from diagnostic CT volume. Afterward, using these
+simulated bone US we train a lightweight vision transformer
+model that can achieve accurate and on-the-fly bone segmen-
+tation for spinal sonography. In the validation experiments,
+the realistic US simulation was conducted by deriving from
+diagnostic spinal CT volume to facilitate a radiation-free US-
+guided pedicle screw placement procedure. When it is employed
+for training bone segmentation task, the Chamfer distance
+achieves 0.599mm; when it is applied for CT-US registration,
+the associated bone segmentation accuracy achieves 0.93 in
+Dice, and the registration accuracy based on the segmented
+point cloud is 0.13∼3.37mm in a complication-free manner.
+While bone US images exhibit strong echoes at the medium
+interface, it may enable the model indistinguishable between thin
+interfaces and bone surfaces by simply relying on small neighbor-
+hood information. To overcome these shortcomings, we propose
+to utilize a Long-range Contrast Learning Module (LCLM) to
+fully explore the Long-range Contrast between the candidates
+and their surrounding pixels. In the ablation experiments, it is
+verified that the proposed Long-range Contrast Learning module
+is effective in the precise positioning of the US region of interest.
+On top of that, the training data is entirely generated by our
+proposed US simulation framework without fine-tuning based
+on real clinical data, which demonstrates its effectiveness of the
+bone realistic US simulation framework.
+Note to Practitioners—The motivation of this paper is to
+address the issues on unavailable or insufficient bone US images
+and annotation labels. We employ such a data augmentation
+technique to generate realistic simulated bone US and annotation
+associated with the corresponding CT volume. The problems
+of current US data augmentation approaches are mainly the
+inability to generate continuous context-accurate data (neural
+network-based algorithm) and the lack of real-time physical
+simulation capability (traditional acoustics-based algorithm).
+In this paper, we enhance the US simulation derived from
+CT images by supplementing techniques such as extrusion
+simulation, moving space simulation, and image augmentation
+to yield higher quality images, and hence apply these images
+directly to the target task, i.e., train a neural network to
+† These authors contributed equally
+‡ represents the corresponding author. Email: lliu@ee.cuhk.edu.hk
+Ang Li, Yongjian Zhao, Keyu Li and Li Liu is with Chinese University of
+Hong Kong.
+Jiayi Han is with Fudan University.
+guide US-guided spinal surgery. Besides, we demonstrate the
+effectiveness of the realistic US simulation framework and each
+new module of the neural network using ablation experiments. It
+can be concluded that the neural network trained with the data
+generated by US simulation framework promises to enable US-
+guided pediclescrew placement procedures. In the future work,
+we will apply the In silico bone US simulation as a reinforcement
+learning environment and deploy the trained agents to directly
+guide US-guided procedures.
+Index Terms—Realistic US Simulation, Vision Transformer,
+Bone Surface Segmentation, CT-US Registration, US-Guided
+Spinal Navigation
+I. INTRODUCTION
+Segmentation of bone surfaces from intra-operative US data
+followed by CT-US registration is two critical steps for US-
+guided spinal surgery. Recent research has focused on the
+use of deep learning-enabled methods for accurate, robust,
+and real-time segmentation and registration of bone surfaces.
+However, scarcity of data size, due to a lack of standardized
+data and patient privacy concerns, is a major challenge in
+applying deep learning-enabled methods in the intra-operative
+imaging field. This is specifically a challenge due to the fact
+that the US is not a standard imaging modality in spine-related
+surgeries and US-guided spinal surgeries are not common;
+even if spinal sonography can be available for pre-clinical
+collection procedures, annotations would be still a severe
+challenge due to the vast amounts of data. Another limiting
+factor is the manual data acquisition process: sub-optimal
+orientation of the US transducer with respect to the imaged
+spinal anatomy will result in the user-dependent acquisition of
+low-quality bone scans.
+Increasing the size of existing datasets through data aug-
+mentation in order to improve models’ performance is exten-
+sively investigated. Among them, a fundamental approach to
+obtaining a large labeled dataset is In silico realistic simulation
+of spinal US images. The US simulation methods could be
+broadly categorized into three types: acoustic model-based US
+simulation, image-based US simulation, and generation of a
+virtual image using a generative adversarial network (GAN).
+US generation algorithms based on acoustic models are usually
+very slow [1]. Generating US by GAN may require training
+different networks for different organs, meanwhile, obtaining
+corresponding US volume and CT volume pairs aligned in
+the same space is difficult, allowing for the generation of
+large-scale synthetic US images using GAN a challenging
+task. Image-based approaches attempt to utilize a simulated
+arXiv:2301.01940v1  [eess.IV]  5 Jan 2023
+
+2
+Fig. 1. The figure shows examples of generated Ultrasound images which are the junctional surfaces of the two vertebrae, the plane where the vertebral plate
+could be entirely verified, the plane swept along the spinous, and an arbitrary scan plane, respectively.
+US probe to re-sample the original image (such as a CT scan)
+and then consider the image scalar as acoustic parameters of
+organs and then simulate propagation with acoustic properties.
+Nonetheless, different from other imaging methods such as
+CT and MRI, the US is significantly influenced by gas. To
+avoid its influence, the operator presses the probe to remove
+the gas between the probe and the skin, which leads to the
+distortion of tissues in the original images. However, this
+distortion is ignored by the former researchers. Moreover,
+because each tissue has a different reflection rate, the former
+works segment the image and design specific transfer functions
+to each tissue, which is time-consuming. In addition, the
+conventional use of US image generation is only to assist
+in 2d-3d image registration [2] or for training examining
+physicians [3] without further application of synthetic US
+images.
+In this paper, we propose a novel CT-derived realistic US
+synthesis framework incorporating automated image gener-
+ation with sampling methods, as shown in Fig. 3. From
+(a)∼(b), each column represents the real US, corresponding
+CT scan, reflection map, and transmission map, respectively.
+We simulate the distortion resulting by pressing the probe via
+warping the original image and propose an adaptive transfer
+function that could be directly adopted to the whole image
+which eliminates the transfer function designing process for
+each tissue and highly speeds up the US simulation task.
+To fully take the advantage of the proposed simulation and
+conduct real-time US image segmentation, we further propose
+a lightweight vision transformer with Long-range Contrast
+Learning Module (LCLM) which utilizes a designed cascaded
+dilated convolution layers to achieve dense super-large recep-
+tive field which enhances the US image segmentation. Exper-
+iments demonstrate the proposed simulation system achieves
+state-of-the-art performance compared with other approaches
+and benefits the proposed vision transformer for real US image
+segmentation.
+Our contributions are listed as follows:
+1) We propose an In silico bone US simulation framework
+that synthesizes realistic US images from diagnostic CT
+volume.
+2) We develop a lightweight vision transformer model that
+achieves precise and real-time bone segmentation for
+spinal sonography images.
+3) Experiments demonstrate that the proposed In silico
+bone US simulation approach dramatically enhances the
+segmentation performance in comparison with initial CT
+scans, indicating that the proposed data augmentation
+method is capable of pre-training models for real clinical
+spinal sonography.
+II. RELATED WORK
+In this section, the related work on realistic US simulation
+and associated bone segmentation are discussed.
+A. Realistic US Simulation
+Traditional acoustics-based US simulation software was
+pioneered by Jenson et al. [4] in 1996. K-space method for
+fast computation of pulsed photo-acoustic fields was proposed
+by B. T. Cox et al. [5] in 2005 and Treeby et al. [6] optimized
+the US image propagation model on the K-space method and
+then the widely used K-wave tools have been developed. The
+advantage of this conventional acoustic-based algorithm is that
+it can simulate various types of US imaging systems with the
+physical effect of the images as realistic as possible.
+As these algorithms based on physical models of acoustics
+have efficiency problems to generate large datasets, researchers
+started to develop ray-tracing-based approaches. Burger et al.
+[7] developed a US simulation system by segmenting the
+CT dataset into different tissues, and then assigning velocity,
+impedance, scattering factor, and other acoustic properties
+to each tissue; hence, it was used to simulate the sound
+propagation, absorption, and scattering processes. Cong et al.
+[8] proposed a multi-scale enhancement method to augment
+tubular structures to simulate blood flow, and allow for US
+images more realistic. Piorkowski et al. [3] in 2013 applied
+the algorithms of Wein [9] and Kutter [10] to make a Trans-
+esophageal Echocardiography (TEE) Simulator which has a
+tremendously positive effect on training doctors to perform
+TEE examinations. To further accelerate the image generation
+speed, Wang et al. [11] used NVIDIA’s Optix 6.0 ray-tracing
+
+3
+engine to do the Monte Carlo simulation of US with a good
+result compared to GAN and Field-II [12].
+In recent years, generative adversarial network (GAN) mod-
+els have been used extensively in realistic US simulation
+research. It generates realistic US images after learning from a
+large dataset in the US domain. Hu et al. [13] applied a GAN
+model to yield US images for Freehand scanning, the model
+takes the spatial location information as a conditional input,
+and then outputs the US image at the current location. This
+work contributes to producing US data for the corresponding
+location, yet they were unable to create synthetic US images
+for each specific patient. To address this shortcoming in 2018,
+Tom et al. [14] employed cascaded GAN with an image
+segmentation label as conditional input to create more realistic
+US images and to be able to produce synthetic US images for
+different segmentation results. Nonetheless, according to their
+report, it can be concluded that even with precise segmenta-
+tion the GAN-based virtual US system still has difficulty in
+ensuring the edge intensity and shape as the real US in same.
+B. US Segmentation
+Some early study utilized handcrafted features for US
+segmentation, such as active contour [15], [16]. In recent years,
+the most popular model in US segmentation is UNet [17].
+Many works adopt different reinforcements in UNet-based
+models for US segmentation. [18] proposed a multi-task UNet
+which combines classification and segmentation tasks. [19]
+proposed a lightweight UNet model which alternately adopt
+3 × 3 and 1 × 1 convolution layers. They also introduced a
+false output suppression mechanism that combines patch-wise
+classification and segmentation results to eliminate false pos-
+itive. [20] adopts spatial attention on US image segmentation
+task.
+III. METHOD
+A. Realistic US Simulation from CT Volume
+The entire US simulation progress is divided into several
+parts: data sampling, data filtering, US transmitting, and image
+blending.
+a) Image Probe and Press simulation: Since the goal
+of the proposed US simulation system is to train doctors or
+artificial intelligence agents for surgical robots, the moving
+space of probe motion cannot be the entire 3D space. The
+probe in the system can move in the provided scanning space,
+but the probe is a curved surface, it is the probe surface, and
+the scanning space will not be fitted perfectly. The scanning
+space will wrap around and fit the probe if the pressure is
+simulated. However, the CT volume will not deform with the
+surface. Hence, an algorithm needs to be proposed to make
+the CT data deform with the probe surface along the scanning
+space. Using the spring-mass model to simulate deformations
+in scanning space by using the moving least squares algorithm
+to fit the CT deformation is certainly good, but also has
+two demerits. The first is solving the spring-mass simulation
+and the moving least squares equation will consume a lot of
+computational resources. Secondly, the algorithm requires a
+lot of extra work to mark the anchor points on the images.
+Under this situation, we changed the local translation image
+warping algorithm proposed by A. Gustafsson for CT data
+and apply it with the shape of the probe to the image. The
+algorithm only needs to know the shape of the probe and the
+HU value of the CT data of that slice to simulate the motion of
+the tissues. If further acceleration is desired, the weight value
+of the HU parameter can be set to 1, then the UV coordinate
+mapping only needs to be calculated once, without affecting
+the imaging efficiency at all.
+The equation is formulated as Equ. 1
+�
+�
+�
+�
+�
+⃗u = ⃗x
+r2
+max−|⃗x−⃗c|2
+r2max−|⃗x−⃗c|2+D(⃗m − ⃗c)
+D = 100
+f α(hu)|⃗m − ⃗c|2
+.
+(1)
+In the above equation, f controls the ratio of deformation.
+In our case, we want to push the intersection point of the
+center-line of the probe and the top line of the sampled image,
+as the blue dot is shown in Fig.4, to the top of the probe
+which is shown in the figure as the red dot. According to
+the mentioned algorithm, all the tissue between the blue and
+red dots needs to be pressed downwards. Assuming that the
+tissue is a rigid body, the thickness of the tissue does not
+change by pressing with a very big force. In this case, the
+original position at the red dot should be pushed to the top of
+the green curve. Then a green arc that represents the limit of
+tissue movement can be drawn. The squeezing of the tissue
+in the area between the probe arc and the green arc will
+cause high reflections or absorption in this area. Hence, in the
+process of simulating transmission, it is significant to make
+the sound waves attenuate less in this area, otherwise, the
+squeezed tissue generated by the algorithm will completely
+block the transmission of US. As you can see from Fig. 3,
+there are a few bright lines below the probe curve, which are
+the signal of the squeezed tissue and it is identical to the vivo
+US image.
+Fig. 2. This figure illustrates the pressure simulation algorithm. As an enlarged
+part of the dashed box in the figure, the yellow arc represents the shape of
+the probe, and the area between the yellow arc and the green arc is the target
+space being squeezed into.
+b) Restricted Movement Space of Probe: As mentioned
+above, the probe cannot move in space arbitrarily. To automate
+the acquired US images, two types of probe moving regulation
+are designed in this paper using spinal US as an instance.
+Define M as a mesh in 3d space, clipping a 3D sub-mesh Mr
+
+4
+Fig. 3.
+Figure (a) shows the CT images after being processed by the pressure simulation, the skin and muscles are compressed into the probe’s curved
+surface. Figure (b) shows the virtual US image produced with simulated extrusion. Figure (c) shows the reflection map after extrusion. Figure (d) shows that
+the processed image will not affect the propagation.
+Fig. 4. The green part of the figure is the object to be scanned. Figure (a) shows the relationship between the entire feasible space (skin) and the scanned
+object(spine). Figure (b) shows the first type of movement, with the yellow dot representing the tip of the probe and the arrow representing the two-degree
+freedom of movement in the manifold. Figure (c) shows the second type of movement, the red line represents the discrete surface, the dashed arrow represents
+the probe can switch on different curves, and the arrow represents the probe can be moved along the curve direction.
+of interest by the bounding box of the scanning target, the
+movement of the probe is only meaningful in this manifold.
+The first way is to move freely on the grid, for any point p
+on Mr, the direction of movement of p will be curved along
+the grid normal vector. In this case, a very small dx or dy
+will not make a great dz, which would not make the image
+change dramatically and could not acquire a continuous image.
+The second method is to slice Mr into a series of curves
+in 3D space along the forward direction Mr0...Mrn. In each
+polyline, the points can only move forward or backward along
+the tangent direction by a certain distance. It has the merit
+that keeping the components of the probe’s normal vector
+remain consistent across that curve, which is often used for
+automatic circular or flat scanning of a target and ensuring that
+the target being swept is on a line or a point. Besides, both of
+these movement methods can be employed by a reinforcement
+learning-based agent, and the first method is more suitable for
+continuous control, while the second method is more suitable
+for discrete control.
+c) Sound Reflection: The first step in the simulation is
+to map the CT-HU image into an acoustic impedance image,
+with the mapping look-up table references to [21]. In the
+mapping table, each HU value will be mapped to an acoustic
+impedance Z. The image of this acoustic impedance will then
+be processed instead of the CT image. Based on the acoustic
+physical transfer properties, we first introduce the Fresnel
+equation to calculate the reflection. Dividing sound waves
+into those parallel and those perpendicular to the plane, the
+equations are:
+�
+�
+�
+R⊥ = ( Z1cosθi−Z2cosθt
+Z1cosθi+Z2cosθt )2
+R∥ = ( Z2cosθi−Z1cosθt
+Z2cosθi+Z1cosθt )2
+R = 1
+2(R⊥ + R∥)
+.
+(2)
+When sound waves transmitted from one medium to an-
+other, it is not only be reflected but also be refracted. This
+refracted has the same physical properties as the refracted of
+light passing through different media which is given by Snell’s
+law:
+sinθ1
+sinθ2
+= Z2
+Z1
+(3)
+In fact, when a sound wave is refracted or bent, the direction
+will be kept until the next transmission. It is possible to
+simulate multi-level refraction and reflection values using a
+
+5
+recursive ray-tracing algorithm However, since the amplitude
+of the refracted sound waves is small and has little effect on
+the generated image quality, the proposed method does not
+consider its direction after refraction and only superimposes
+the amplitude on the incident sound wave of the next medium.
+Irradiance: Lambert’s cosine theorem is introduced to take
+the irradiance of the sound wave should into consideration.
+This equation means that the more perpendicular the direction
+of the tissue gradient and the sound wave, the stronger the
+sound reflection and the brighter the tissue.
+Ir
+Ii
+= cosθ
+(4)
+d) Sound Attenuation: Attenuation in the propagation of
+US passing through tissue could be broadly divided into re-
+flection, scattering, and absorption, where absorption accounts
+for the majority of it. The paper [22]states that US absorption
+of substances such as bovine and porcine livers accounts for
+90% or even 100% of US propagation attenuation. Even in
+the tissues that exhibit anisotropy such as the bovine brain
+and leg muscle absorption still makes a major contribution
+to the attenuation. Therefore, mapping the CT scan to an
+absorption image is important. The mapping table is obtained
+by interpolating the dataI in the paper [23]. The formula for
+absorption is given by the Lambert-Beer law. Since different
+tissues may not have the same US absorption capacity in
+the same Hu value, in order to get a better result from the
+generated image, an adjustable parameter α is added to the
+original equation. The modified equation is as follows:
+Ia = I010−α∗d∗f∗
+1
+10∗β
+(5)
+Tissues
+Density
+Velocity
+Impedance
+Attenuation
+Skin
+1100
+1631
+1.794
+0.22
+Fat
+916
+1435
+1.352
+0.975
+Muscle
+1041
+1595
+1.647
+1.47
+TABLE I
+ACOUSTICS PROPERTIES OF DIFFERENT TISSUES. WITH DENSITY(KG
+m−3), VELOCITY(ms−1), IMPEDANCE 106(kgm−2s−1),
+ATTENUATION(dBcm−1MHz−n) [23]
+Fig. 5. Illustration of propagation.
+e) Propagation in Discrete Field: The propagation of
+the US is the most time-consuming part of the generation
+process. Particularly, the steps of finding receiver tissues of
+reflection and refraction would take a lot of time in detecting
+intersections that occur frequently.
+At the same time, the direction of the adjacent pixels is
+discrete, whereas the direction of acoustic transmission is
+continuous. So it may happen that the line segments intersect
+while the pixels do not have an intersection. Each pixel among
+the eight neighborhoods should be regarded as the obstacle of
+the wave propagation and the normalized result of the cosine
+value of the direction of propagation and the direction of the
+pixel connection is used as the weight of the obstacle. That
+is, if the connection direction is in the opposite direction to
+the direction of propagation, the weights will be zero and the
+obstacle weight is maximized when the directions are exactly
+the same.
+In order to calculate the amplitude enhancement due to
+reflection and refraction easily, each point also calculates
+its wave propagation to the neighborhood. In this case, a
+negative cosine value of the pixel connection direction and
+the propagation direction will consider as reflection. Cosine
+values less than 0.8 (a hyper-parameter) will be considered
+as contrast enhancement by refraction and scattering. The
+schematic diagram of propagation is shown in 1
+f) Reflection Enhancement:
+When the transducer re-
+ceives the reflected sound waves from the medium gap, a
+bright stripe appears in the US image. Because the reflection
+only occurs when the properties of the medium change, there
+should be only one reflection between the bone surface and
+the muscle tissue, but the US image often shows a very thick
+bright stripe between bone and muscles. hacihaliloglu et al.
+[24] propose that this bright stripe is produced by the thickness
+of the US in the elevational direction. Based on this theory, this
+paper sampled 3d gradient textures in front of and behind the
+current plane (As the blue line in figure 6). These textures are
+multiplied by a weight value α which is inversely proportional
+to the distance and then accumulated into the current generated
+US image. Simulating thick stripes is significant for neural
+network training; if the thick stripes are not present in the
+training data, the output of the model will deviate from the
+ground truth when segmenting the bone surface on the real
+US data.
+In summary, the generated US image is obtained by cal-
+culating the propagation image with the reflection image and
+the absorption image and then blending these three images
+together with the radial noise.
+B. US Image Segmentation
+In this work, we propose a lightweight segmentation frame-
+work for US segmentation, as shown in Fig. 7. We follow
+[25] as the backbone. After each MobileViT block, we add
+an LCLM for further long-range contrast learning. We then
+utilize a feature pyramid network (FPN) to recover the initial
+resolution of the input US image.
+a) MobileViT block: To cover the global information,
+MobileViT block first utilizes a 3×3 convolution to aggregate
+the neighboring features. The pixels are then grouped into
+patches, each patch contains h × w pixels in which h and w
+represent the height and the width of the patch. The ith token
+
+6
+Fig. 6. The thick stripe is indicated by the arrow in Figure a). Figure b) shows how the thickness of the US waves induces the thick stripes artifact. Figure
+c) is a schematic of sampling in a 3d gradient texture. The Blue line indicates the imaging plane. The reflection value between the blue line and the yellow
+line is multiplied and accumulated into the imaging plane. Figure d) shows the simulated thick stripes in the system.
+Fig. 7. The framework of the proposed lightweight vision transformer for US segmentation. For the first two stages, we only utilize basic convolution layers to
+learn local representations. In the last three stages, we implement MobileViT block and LCLM alternately to cover both long-range dependency and long-range
+contrast.
+of each patch transverses its feature from each other by SA.
+When h <= 3 ∧ w <= 3, each pixel can access information
+from all pixels. Details of the MobileViT block could be found
+in [25].
+b) Why is LCLM needed: We find that the region of
+interest (RoI) in US images has similar textures, but is weak in
+amplitude to other tissues. Basic 3×3 convolutions are capable
+of modeling the neighboring texture but fail to learn long-range
+textures. As the local texture of the to-be-segmented region
+is similar to the other tissues, we have to design a module
+that covers the texture within and outside the to-be-segmented
+region.
+In vision transformers, long-range texture modeling is
+achieved by self-attention (SA) of all tokens as in Equ. 6:
+�x = softmax( xWq(xWk)T
+d
+)(xWv),
+(6)
+in which x ∈ RN×d represents the N input tokens with
+dimension d, Wq ∈ Rd×dq, Wk ∈ Rd×dq, and Wv ∈ Rd×dv
+are learnable parameters. We rewrite the self-attention to the
+formula as in Equ. 7:
+�
+�
+�
+�
+�
+�xi = �
+j∈A⟩
+wj(xjWv)
+wj =
+exp(xiWq(xjWk)T /d)
+�
+t
+exp(xiWq(xtWk)T /d)
+.
+(7)
+where Ai represents the set of accessible tokens of token xi.
+As Wq and Wk only attribute in modeling the relationship
+of queries, once Wv is fixed, the space generated by x is
+therefore limited. Meanwhile, as shown in Fig. 8(a), SA
+tends to aggregate “similar” features from the tokens, which
+helps the model to learn long-range dependency. However,
+because of the nature of US images, long-range contrast is
+also important to be learned, but SA fails to do so.
+Convolution operation could be formulated in the same
+formulation as SA, as shown in Equ. 8:
+�xi = �
+j∈Ai
+λj(xjWv) .
+(8)
+In this case, if rank(xjWv) = d, the space generated by x is
+Rd. We randomize 1000 different pairs of Wq and Wv, 1000
+different λ and fixed Wv. The outputs are shown in Fig. 8. As
+demonstrated in the figure, the outputs of convolution fulfill
+the space, while the outputs of SA are gathered.
+
+C+w
+<
+Transducer
+R=
+krk
+k=c-wS
+MViT
+1/2
+1/2
+MViT
+LCLM
+Up
+Up7
+Fig. 8. The possible outputs (dots in red) of self-attention (a) and convolution
+(b) with fixed Wv.
+A recent work [26] also demonstrates that CNNs with
+super large kernels learn feature representations from the
+super large receptive field (sub-global), and achieve compara-
+ble performance with state-of-the-art transformers with fewer
+parameters.
+As aforementioned, we need a module to gather long-range
+textures. As SA fails to learn long-range contrast, and a super-
+large convolution kernel leads to large computational cost,
+we need to design a light yet effective convolution module
+for long-range contrast learning, namely Long-range Contrast
+Learning Module (LCLM).
+c) Long-range contrast learning with dilation convo-
+lution: To cover long-range contrast with fewer parameters,
+a simple way is to utilize dilation convolution. In this work,
+we adopt 3 cascaded dilation convolution layers with dila-
+tion {3,5,11}. Along with one basic convolution layer which
+appears at the end of the MobileViT block, LCLM covers
+up to 41 × 41 pixels densely. We show the receptive field in
+Fig. 9. Each convolution layer is followed by a normalization
+layer and an activation layer. A depth-wise LCLM only needs
+3×3×3×d parameters, which is even smaller than a standard
+convolution layer (3 × 3 × d × d, d >> 3).
+Fig. 9. The receptive field of the pixel in red. The pixels painted not in grey
+are covered by the receptive field.
+IV. EXPERIMENTS
+A. Realistic US Simulation from CT Volume
+It is difficult to provide quantitative evaluation criteria for
+image generation-related works. Qualitatively, the advantages
+of this paper over deep learning-based US image generation
+are patient-specific and more accurate lesion boundaries, and
+when compared with previous papers based on image gener-
+ation, we make the system produce more realistic images by
+simulating squeezing and scattering in soft tissues and bones.
+Nonetheless, the quantitative evaluation is also of significance,
+and since the quality of image generation is hard to be
+evaluated quantitatively, we evaluate the quality of image
+generation from the purpose of the work. Except for training
+doctors to do US examinations, the most important uses of
+the US image generation algorithms are intra-operative reg-
+istration and providing datasets for deep learning. Therefore,
+we put the generated images into various deep learning-based
+segmentation algorithms to verify the usefulness of this image
+generation system for the pre-training of neural networks.
+B. Deep Learning-Based Segmentation
+a) Implementation Details: To validate the proposed
+US simulation method and the lightweight segmentation
+model, we train the proposed model on synthetic US images
+and inference on real US images. We set the batch size to 32,
+utilize the Adam optimizer, and set the learning rate to 1e-4.
+Fig. 10.
+The validation image examples. Figure (a) is the US image of a
+spinal phantom in gel mimic, Figure (b) is the US image of a 3D-printed
+spine in water, and Figure (c) is the US image of a clinically human body.
+b) Comparison with other models: We first compare
+the proposed model with other US segmentation models
+in Tab. II. All performances reported in the table are re-
+implemented by ourselves. Compared with UNet-based mod-
+els, the proposed model is much smaller and more effective.
+Compared with MViT-FPN, the proposed approach achieves
+much better performance.
+To utilize the model, in this case, the segmentation is to
+achieve better registration, therefore we remove the cases with
+obvious segmentation errors to obtain our (selected) result,
+which is used in the subsequent registration. In this case,
+the effect of the network trained with generated data can be
+equivalent to or better than the effect of the SOTA model
+trained by a large number of real US images with an accurately
+labeled. Our label is acquired by ct segmentation, which is
+more challenging for the convergence of lightweight neural
+networks since he can label bone surfaces which is invisible
+or ambiguous in some situation in US.
+c) Ablation study: To demonstrate the effectiveness of
+synthetic US over initial CT scans, long-range dependency,
+and long-range contrast, we validate the aforementioned set-
+tings and show the results in Tab. III. Training the model with
+CT scans leads to a dramatic IoU decrease, which demon-
+strates that the proposed method can effectively synthesize US
+images. Without LCLM, the model fails to learn the long-range
+contrast of US images and results in a performance decrease.
+Without MViT, the IoU slightly drops, which represents that
+
+query
+key
+output
+(b)8
+Method
+Dice
+CD(TP)
+CD(FN)
+#Parm
+FPS
+UNet [17]
+0.574
+1.085mm
+0.590mm
+131.8M
+4.13
+MViT-FPN [25]
+0.294
+4.664mm
+0.824mm
+20.0M
+28.1
+CNLUnet [19]
+0.329
+2.868mm
+1.287mm
+48.6M
+10.9
+Ours
+0.783
+0.599mm
+1.079mm
+20.0M
+26.3
+Ours(Selected)
+0.926
+0.227mm
+0.184mm
+20.0M
+26.3
+TABLE II
+COMPARISON WITH OTHER SEGMENTATION MODELS. NOTE THAT ALL
+PERFORMANCES REPORTED IN THE TABLE ARE RE-IMPLEMENTED BY
+OURSELVES.
+the model benefits from long-range dependency, but is less
+significant compared with long-range contrast.
+Method
+Baseline
+SUS→CT
+w/o MViT
+w/o LCLM
+Dice
+0.783
+0.040
+0.438
+0.294
+CD(TP)
+0.599mm
+6.062mm
+0.910mm
+4.664mm
+CD(FN)
+1.079mm
+3.938mm
+0.674mm
+0.824mm
+TABLE III
+THE ABLATION STUDY OF THE PROPOSED MODEL. “SUS” REPRESENTS
+THE SYNTHETIC US. “SUS→CT”, “W/O MVIT” AND “W/O LCLM”
+REPRESENT TRAINING THE MODEL BY CT SCANS, REMOVING MVIT
+FROM THE BASELINE AND REMOVING LCLM FROM THE BASELINE,
+RESPECTIVELY.
+Fig. 11.
+The result of segmentation. Figure(a), (b), and (c) presents the
+segmented label, the output of the network, and the ground truth respectively.
+C. Validation of US-CT Registration
+a) Implementation Details: The US simulation system
+is proposed to better obtain labeled data from multiple poses,
+patients, and environments, enabling US-based surgical navi-
+gation to be accomplished. In this paper, we validated whether
+the output of the model trained on synthetic data can align
+the preoperative CT and the planned trajectory into the intra-
+operative phase, and the spinal bone surface registration for
+pedicle screw placement is the instance.
+The validation is divided into two parts, the first is to
+validate the accuracy of the rigid body registration, i.e. trans-
+lation and rotation. The second part is to validate whether
+the pre-operatively planned trajectory with the aforementioned
+transforms will cause complications for the surgery or not, i.e.
+whether the intra-operative trajectory will touch vital organs.
+The experiments were performed on three phantoms, a spine
+phantom in water, a 3D-printed human spine in water, and a
+bovine spine in agar gel. The registration steps were divided
+into a coarse alignment and an individual registration for
+each segment. The coarse registration used all point clouds
+to calculate the approximate position and orientation, and the
+final registration used a de-noised Iterative Closest Point(ICP)
+method. The results of the registration are shown in the table
+below.
+Fig. 12. The phantom model for registration validation.
+Fig. 13. The MSE loss of registration in three dimensions.
+Fig. 14.
+The Translation and Rotation of pedicle screw with registration
+matrix. The rotation is presented as an angle in degrees.
+b) Validation of intra-operative model registration: In
+the registration phase, the model inputs the US image outputs
+the segmentation label, and extracts the upper contour of its
+maximum connectivity, hence the point cloud is created with
+the calibration matrix of the US probe. The calibration of the
+US is done by two cross-wire phantom [27], with 0.97mm
+MSE(min square loss) loss. Since the coordinate system of
+
+3
+Translation in three dimension
+X
+1
+y
+0
+7
+-1
+Li
+L2
+L3
+L4
+L5
+TheLumbarsegment3
+V
+1
+Z
+0
+-1
+Angle
+-2
+Li
+L2
+L3
+L4
+L5
+The Lumbar segment9
+each segment does not coincide, the rotation evaluation in
+angular will be represented by the rotation of pedicle screws in
+the next part. To evaluate the error of registration, the point-
+to-point MSE is used, which is also the objective function
+of the ICP algorithm. The result of registration is shown in
+Fig. 15, which shows the ground truth in the intra-operative
+space(white model) and the results of registration with US
+(green model). In summary, the segmentation algorithm trained
+by the generated US image works in the registration with the
+error 0.133 ∼ 3.366mm. The maximum appears in the z-axis
+of L5, which is around 3 mm, which is probably caused by
+the shape of L5 being more different from the shape of other
+lumbar.
+c) Validation of Pedicle Screw Placement Feasibility: To
+verify the registration effectiveness of the system in the target
+tasks, we compared the translation and rotation of the tip of the
+screw with the US point cloud registration and ground truth
+using an expert-planned intra-operative plan for pedicle screw
+placement. The rotation is determined by the angle between
+two pedicle axis vectors in 3D space which is expressed in
+degrees. As shown in Fig. 14, the error in translation for
+the tip of the screw ranges from 0.027 mm to 4.031 mm,
+with the maximum value still occurring on the z-axis of L5,
+which is consistent with the above assessment. The error in the
+rotation is between 3.05 degrees and 4.47 degrees, therefore,
+it can be concluded that the pre-trained model with generated
+US image can perform the alignment with small errors. The
+results of the pedicle screw placement are shown in figure 16,
+which shows this registration does not put the patient at risk
+of complications.
+Fig. 15. The visualization of registration. The green part of the figure is the
+registration result by US segmentation, and the white part is the ground truth.
+Cases derived from 3d printed patient spine, which is (b) in the figure 12.
+Fig. 16. The visualization of registration-based screw placement. The figure
+shows the posture of the screws with standard length and diameter which
+is planned on preoperative CT by an experienced surgeon, compared to the
+intra-operative spine ground truth by the registration matrix acquired by image
+segmentation. Cases derived from the standard spine phantom, which is (c)
+in the figure 12.
+V. CONCLUSION
+In this paper, we propose a US image simulation method
+based on CT images and acoustic properties that automatically
+generates a US simulation environment based on a specific
+patient CT volume, which allows US-based deep learning and
+reinforcement learning algorithms to be trained and validated.
+In addition, we propose a lightweight vision transformer for
+segmenting US images which is a network structure with better
+segmentation accuracy and modal generalization capabilities.
+Experiments show that the model trained using US im-
+ages generated by the realistic US simulation from the CT
+system achieves higher segmentation accuracy and has the
+capability to perform the intraoperative registration process
+without complications compared to the model trained with
+CT images. Compared to other segmentation algorithms, the
+proposed transformer has real-time computational efficiency,
+better segmentation accuracy, and generalization capability,
+which is particularly important in US surgical applications.
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diff --git a/89AzT4oBgHgl3EQf-_4J/content/tmp_files/load_file.txt b/89AzT4oBgHgl3EQf-_4J/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf,len=589
+page_content='1  Enabling Augmented Segmentation and Registration  in Ultrasound-Guided Spinal Surgery via Realistic  Ultrasound Synthesis from Diagnostic CT Volume  Ang Li†, Jiayi Han†, Yongjian Zhao, Keyu Li, Li Liu‡  Abstract—This paper aims to tackle the issues on unavailable  or insufficient clinical ultrasound (US) data and meaningful  annotation to enable bone segmentation and registration for US-  guided spinal surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' While the US is not a standard paradigm  for spinal surgery, the scarcity of intra-operative clinical US data  is an insurmountable bottleneck in training a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Moreover, due to the characteristics of US imaging, it is difficult  to clearly annotate bone surfaces which causes the trained neural  network missing its attention to the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Hence, we propose an  In silico bone US simulation framework that synthesizes realistic  US images from diagnostic CT volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Afterward, using these  simulated bone US we train a lightweight vision transformer  model that can achieve accurate and on-the-fly bone segmen-  tation for spinal sonography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In the validation experiments,  the realistic US simulation was conducted by deriving from  diagnostic spinal CT volume to facilitate a radiation-free US-  guided pedicle screw placement procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' When it is employed  for training bone segmentation task, the Chamfer distance  achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='599mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' when it is applied for CT-US registration,  the associated bone segmentation accuracy achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='93 in  Dice, and the registration accuracy based on the segmented  point cloud is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='13∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='37mm in a complication-free manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  While bone US images exhibit strong echoes at the medium  interface, it may enable the model indistinguishable between thin  interfaces and bone surfaces by simply relying on small neighbor-  hood information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' To overcome these shortcomings, we propose  to utilize a Long-range Contrast Learning Module (LCLM) to  fully explore the Long-range Contrast between the candidates  and their surrounding pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In the ablation experiments, it is  verified that the proposed Long-range Contrast Learning module  is effective in the precise positioning of the US region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  On top of that, the training data is entirely generated by our  proposed US simulation framework without fine-tuning based  on real clinical data, which demonstrates its effectiveness of the  bone realistic US simulation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Note to Practitioners—The motivation of this paper is to  address the issues on unavailable or insufficient bone US images  and annotation labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' We employ such a data augmentation  technique to generate realistic simulated bone US and annotation  associated with the corresponding CT volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The problems  of current US data augmentation approaches are mainly the  inability to generate continuous context-accurate data (neural  network-based algorithm) and the lack of real-time physical  simulation capability (traditional acoustics-based algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  In this paper, we enhance the US simulation derived from  CT images by supplementing techniques such as extrusion  simulation, moving space simulation, and image augmentation  to yield higher quality images, and hence apply these images  directly to the target task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=', train a neural network to  † These authors contributed equally  ‡ represents the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Email: lliu@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='hk  Ang Li, Yongjian Zhao, Keyu Li and Li Liu is with Chinese University of  Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Jiayi Han is with Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  guide US-guided spinal surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Besides, we demonstrate the  effectiveness of the realistic US simulation framework and each  new module of the neural network using ablation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' It  can be concluded that the neural network trained with the data  generated by US simulation framework promises to enable US-  guided pediclescrew placement procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In the future work,  we will apply the In silico bone US simulation as a reinforcement  learning environment and deploy the trained agents to directly  guide US-guided procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Index Terms—Realistic US Simulation, Vision Transformer,  Bone Surface Segmentation, CT-US Registration, US-Guided  Spinal Navigation  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' INTRODUCTION  Segmentation of bone surfaces from intra-operative US data  followed by CT-US registration is two critical steps for US-  guided spinal surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Recent research has focused on the  use of deep learning-enabled methods for accurate, robust,  and real-time segmentation and registration of bone surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  However, scarcity of data size, due to a lack of standardized  data and patient privacy concerns, is a major challenge in  applying deep learning-enabled methods in the intra-operative  imaging field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' This is specifically a challenge due to the fact  that the US is not a standard imaging modality in spine-related  surgeries and US-guided spinal surgeries are not common;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  even if spinal sonography can be available for pre-clinical  collection procedures, annotations would be still a severe  challenge due to the vast amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Another limiting  factor is the manual data acquisition process: sub-optimal  orientation of the US transducer with respect to the imaged  spinal anatomy will result in the user-dependent acquisition of  low-quality bone scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Increasing the size of existing datasets through data aug-  mentation in order to improve models’ performance is exten-  sively investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Among them, a fundamental approach to  obtaining a large labeled dataset is In silico realistic simulation  of spinal US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The US simulation methods could be  broadly categorized into three types: acoustic model-based US  simulation, image-based US simulation, and generation of a  virtual image using a generative adversarial network (GAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  US generation algorithms based on acoustic models are usually  very slow [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Generating US by GAN may require training  different networks for different organs, meanwhile, obtaining  corresponding US volume and CT volume pairs aligned in  the same space is difficult, allowing for the generation of  large-scale synthetic US images using GAN a challenging  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Image-based approaches attempt to utilize a simulated  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='01940v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='IV]  5 Jan 2023  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The figure shows examples of generated Ultrasound images which are the junctional surfaces of the two vertebrae, the plane where the vertebral plate  could be entirely verified, the plane swept along the spinous, and an arbitrary scan plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  US probe to re-sample the original image (such as a CT scan)  and then consider the image scalar as acoustic parameters of  organs and then simulate propagation with acoustic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Nonetheless, different from other imaging methods such as  CT and MRI, the US is significantly influenced by gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' To  avoid its influence, the operator presses the probe to remove  the gas between the probe and the skin, which leads to the  distortion of tissues in the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' However, this  distortion is ignored by the former researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Moreover,  because each tissue has a different reflection rate, the former  works segment the image and design specific transfer functions  to each tissue, which is time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In addition, the  conventional use of US image generation is only to assist  in 2d-3d image registration [2] or for training examining  physicians [3] without further application of synthetic US  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  In this paper, we propose a novel CT-derived realistic US  synthesis framework incorporating automated image gener-  ation with sampling methods, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' From  (a)∼(b), each column represents the real US, corresponding  CT scan, reflection map, and transmission map, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  We simulate the distortion resulting by pressing the probe via  warping the original image and propose an adaptive transfer  function that could be directly adopted to the whole image  which eliminates the transfer function designing process for  each tissue and highly speeds up the US simulation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  To fully take the advantage of the proposed simulation and  conduct real-time US image segmentation, we further propose  a lightweight vision transformer with Long-range Contrast  Learning Module (LCLM) which utilizes a designed cascaded  dilated convolution layers to achieve dense super-large recep-  tive field which enhances the US image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Exper-  iments demonstrate the proposed simulation system achieves  state-of-the-art performance compared with other approaches  and benefits the proposed vision transformer for real US image  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Our contributions are listed as follows:  1) We propose an In silico bone US simulation framework  that synthesizes realistic US images from diagnostic CT  volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  2) We develop a lightweight vision transformer model that  achieves precise and real-time bone segmentation for  spinal sonography images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  3) Experiments demonstrate that the proposed In silico  bone US simulation approach dramatically enhances the  segmentation performance in comparison with initial CT  scans, indicating that the proposed data augmentation  method is capable of pre-training models for real clinical  spinal sonography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' RELATED WORK  In this section, the related work on realistic US simulation  and associated bone segmentation are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Realistic US Simulation  Traditional acoustics-based US simulation software was  pioneered by Jenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [4] in 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' K-space method for  fast computation of pulsed photo-acoustic fields was proposed  by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Cox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [5] in 2005 and Treeby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [6] optimized  the US image propagation model on the K-space method and  then the widely used K-wave tools have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The  advantage of this conventional acoustic-based algorithm is that  it can simulate various types of US imaging systems with the  physical effect of the images as realistic as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  As these algorithms based on physical models of acoustics  have efficiency problems to generate large datasets, researchers  started to develop ray-tracing-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Burger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  [7] developed a US simulation system by segmenting the  CT dataset into different tissues, and then assigning velocity,  impedance, scattering factor, and other acoustic properties  to each tissue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' hence, it was used to simulate the sound  propagation, absorption, and scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Cong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  [8] proposed a multi-scale enhancement method to augment  tubular structures to simulate blood flow, and allow for US  images more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Piorkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [3] in 2013 applied  the algorithms of Wein [9] and Kutter [10] to make a Trans-  esophageal Echocardiography (TEE) Simulator which has a  tremendously positive effect on training doctors to perform  TEE examinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' To further accelerate the image generation  speed, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [11] used NVIDIA’s Optix 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='0 ray-tracing  3  engine to do the Monte Carlo simulation of US with a good  result compared to GAN and Field-II [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  In recent years, generative adversarial network (GAN) mod-  els have been used extensively in realistic US simulation  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' It generates realistic US images after learning from a  large dataset in the US domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [13] applied a GAN  model to yield US images for Freehand scanning, the model  takes the spatial location information as a conditional input,  and then outputs the US image at the current location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' This  work contributes to producing US data for the corresponding  location, yet they were unable to create synthetic US images  for each specific patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' To address this shortcoming in 2018,  Tom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [14] employed cascaded GAN with an image  segmentation label as conditional input to create more realistic  US images and to be able to produce synthetic US images for  different segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Nonetheless, according to their  report, it can be concluded that even with precise segmenta-  tion the GAN-based virtual US system still has difficulty in  ensuring the edge intensity and shape as the real US in same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' US Segmentation  Some early study utilized handcrafted features for US  segmentation, such as active contour [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In recent years,  the most popular model in US segmentation is UNet [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Many works adopt different reinforcements in UNet-based  models for US segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [18] proposed a multi-task UNet  which combines classification and segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [19]  proposed a lightweight UNet model which alternately adopt  3 × 3 and 1 × 1 convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' They also introduced a  false output suppression mechanism that combines patch-wise  classification and segmentation results to eliminate false pos-  itive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' [20] adopts spatial attention on US image segmentation  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' METHOD  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Realistic US Simulation from CT Volume  The entire US simulation progress is divided into several  parts: data sampling, data filtering, US transmitting, and image  blending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  a) Image Probe and Press simulation: Since the goal  of the proposed US simulation system is to train doctors or  artificial intelligence agents for surgical robots, the moving  space of probe motion cannot be the entire 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The  probe in the system can move in the provided scanning space,  but the probe is a curved surface, it is the probe surface, and  the scanning space will not be fitted perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The scanning  space will wrap around and fit the probe if the pressure is  simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' However, the CT volume will not deform with the  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Hence, an algorithm needs to be proposed to make  the CT data deform with the probe surface along the scanning  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Using the spring-mass model to simulate deformations  in scanning space by using the moving least squares algorithm  to fit the CT deformation is certainly good, but also has  two demerits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The first is solving the spring-mass simulation  and the moving least squares equation will consume a lot of  computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Secondly, the algorithm requires a  lot of extra work to mark the anchor points on the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Under this situation, we changed the local translation image  warping algorithm proposed by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Gustafsson for CT data  and apply it with the shape of the probe to the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The  algorithm only needs to know the shape of the probe and the  HU value of the CT data of that slice to simulate the motion of  the tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' If further acceleration is desired, the weight value  of the HU parameter can be set to 1, then the UV coordinate  mapping only needs to be calculated once, without affecting  the imaging efficiency at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  The equation is formulated as Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 1  �  �  �  �  �  ⃗u = ⃗x  r2  max−|⃗x−⃗c|2  r2max−|⃗x−⃗c|2+D(⃗m − ⃗c)  D = 100  f α(hu)|⃗m − ⃗c|2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  (1)  In the above equation, f controls the ratio of deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  In our case, we want to push the intersection point of the  center-line of the probe and the top line of the sampled image,  as the blue dot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='4, to the top of the probe  which is shown in the figure as the red dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' According to  the mentioned algorithm, all the tissue between the blue and  red dots needs to be pressed downwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Assuming that the  tissue is a rigid body, the thickness of the tissue does not  change by pressing with a very big force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In this case, the  original position at the red dot should be pushed to the top of  the green curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Then a green arc that represents the limit of  tissue movement can be drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The squeezing of the tissue  in the area between the probe arc and the green arc will  cause high reflections or absorption in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Hence, in the  process of simulating transmission, it is significant to make  the sound waves attenuate less in this area, otherwise, the  squeezed tissue generated by the algorithm will completely  block the transmission of US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' As you can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 3,  there are a few bright lines below the probe curve, which are  the signal of the squeezed tissue and it is identical to the vivo  US image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' This figure illustrates the pressure simulation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' As an enlarged  part of the dashed box in the figure, the yellow arc represents the shape of  the probe, and the area between the yellow arc and the green arc is the target  space being squeezed into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  b) Restricted Movement Space of Probe: As mentioned  above, the probe cannot move in space arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' To automate  the acquired US images, two types of probe moving regulation  are designed in this paper using spinal US as an instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Define M as a mesh in 3d space, clipping a 3D sub-mesh Mr  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Figure (a) shows the CT images after being processed by the pressure simulation, the skin and muscles are compressed into the probe’s curved  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure (b) shows the virtual US image produced with simulated extrusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure (c) shows the reflection map after extrusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure (d) shows that  the processed image will not affect the propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The green part of the figure is the object to be scanned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure (a) shows the relationship between the entire feasible space (skin) and the scanned  object(spine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure (b) shows the first type of movement, with the yellow dot representing the tip of the probe and the arrow representing the two-degree  freedom of movement in the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure (c) shows the second type of movement, the red line represents the discrete surface, the dashed arrow represents  the probe can switch on different curves, and the arrow represents the probe can be moved along the curve direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  of interest by the bounding box of the scanning target, the  movement of the probe is only meaningful in this manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  The first way is to move freely on the grid, for any point p  on Mr, the direction of movement of p will be curved along  the grid normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In this case, a very small dx or dy  will not make a great dz, which would not make the image  change dramatically and could not acquire a continuous image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  The second method is to slice Mr into a series of curves  in 3D space along the forward direction Mr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='Mrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In each  polyline, the points can only move forward or backward along  the tangent direction by a certain distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' It has the merit  that keeping the components of the probe’s normal vector  remain consistent across that curve, which is often used for  automatic circular or flat scanning of a target and ensuring that  the target being swept is on a line or a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Besides, both of  these movement methods can be employed by a reinforcement  learning-based agent, and the first method is more suitable for  continuous control, while the second method is more suitable  for discrete control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  c) Sound Reflection: The first step in the simulation is  to map the CT-HU image into an acoustic impedance image,  with the mapping look-up table references to [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In the  mapping table, each HU value will be mapped to an acoustic  impedance Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The image of this acoustic impedance will then  be processed instead of the CT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Based on the acoustic  physical transfer properties, we first introduce the Fresnel  equation to calculate the reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Dividing sound waves  into those parallel and those perpendicular to the plane, the  equations are:  �  �  �  R⊥ = ( Z1cosθi−Z2cosθt  Z1cosθi+Z2cosθt )2  R∥ = ( Z2cosθi−Z1cosθt  Z2cosθi+Z1cosθt )2  R = 1  2(R⊥ + R∥)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  (2)  When sound waves transmitted from one medium to an-  other, it is not only be reflected but also be refracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' This  refracted has the same physical properties as the refracted of  light passing through different media which is given by Snell’s  law:  sinθ1  sinθ2  = Z2  Z1  (3)  In fact, when a sound wave is refracted or bent, the direction  will be kept until the next transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' It is possible to  simulate multi-level refraction and reflection values using a  5  recursive ray-tracing algorithm However, since the amplitude  of the refracted sound waves is small and has little effect on  the generated image quality, the proposed method does not  consider its direction after refraction and only superimposes  the amplitude on the incident sound wave of the next medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Irradiance: Lambert’s cosine theorem is introduced to take  the irradiance of the sound wave should into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  This equation means that the more perpendicular the direction  of the tissue gradient and the sound wave, the stronger the  sound reflection and the brighter the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Ir  Ii  = cosθ  (4)  d) Sound Attenuation: Attenuation in the propagation of  US passing through tissue could be broadly divided into re-  flection, scattering, and absorption, where absorption accounts  for the majority of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The paper [22]states that US absorption  of substances such as bovine and porcine livers accounts for  90% or even 100% of US propagation attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Even in  the tissues that exhibit anisotropy such as the bovine brain  and leg muscle absorption still makes a major contribution  to the attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Therefore, mapping the CT scan to an  absorption image is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The mapping table is obtained  by interpolating the dataI in the paper [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The formula for  absorption is given by the Lambert-Beer law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Since different  tissues may not have the same US absorption capacity in  the same Hu value, in order to get a better result from the  generated image, an adjustable parameter α is added to the  original equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The modified equation is as follows:  Ia = I010−α∗d∗f∗  1  10∗β  (5)  Tissues  Density  Velocity  Impedance  Attenuation  Skin  1100  1631  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='794  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='22  Fat  916  1435  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='352  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='975  Muscle  1041  1595  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='647  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='47  TABLE I  ACOUSTICS PROPERTIES OF DIFFERENT TISSUES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' WITH DENSITY(KG  m−3), VELOCITY(ms−1), IMPEDANCE 106(kgm−2s−1),  ATTENUATION(dBcm−1MHz−n) [23]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Illustration of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  e) Propagation in Discrete Field: The propagation of  the US is the most time-consuming part of the generation  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Particularly, the steps of finding receiver tissues of  reflection and refraction would take a lot of time in detecting  intersections that occur frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  At the same time, the direction of the adjacent pixels is  discrete, whereas the direction of acoustic transmission is  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' So it may happen that the line segments intersect  while the pixels do not have an intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Each pixel among  the eight neighborhoods should be regarded as the obstacle of  the wave propagation and the normalized result of the cosine  value of the direction of propagation and the direction of the  pixel connection is used as the weight of the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' That  is, if the connection direction is in the opposite direction to  the direction of propagation, the weights will be zero and the  obstacle weight is maximized when the directions are exactly  the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  In order to calculate the amplitude enhancement due to  reflection and refraction easily, each point also calculates  its wave propagation to the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In this case, a  negative cosine value of the pixel connection direction and  the propagation direction will consider as reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Cosine  values less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='8 (a hyper-parameter) will be considered  as contrast enhancement by refraction and scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The  schematic diagram of propagation is shown in 1  f) Reflection Enhancement:  When the transducer re-  ceives the reflected sound waves from the medium gap, a  bright stripe appears in the US image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Because the reflection  only occurs when the properties of the medium change, there  should be only one reflection between the bone surface and  the muscle tissue, but the US image often shows a very thick  bright stripe between bone and muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' hacihaliloglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  [24] propose that this bright stripe is produced by the thickness  of the US in the elevational direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Based on this theory, this  paper sampled 3d gradient textures in front of and behind the  current plane (As the blue line in figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' These textures are  multiplied by a weight value α which is inversely proportional  to the distance and then accumulated into the current generated  US image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Simulating thick stripes is significant for neural  network training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' if the thick stripes are not present in the  training data, the output of the model will deviate from the  ground truth when segmenting the bone surface on the real  US data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  In summary, the generated US image is obtained by cal-  culating the propagation image with the reflection image and  the absorption image and then blending these three images  together with the radial noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' US Image Segmentation  In this work, we propose a lightweight segmentation frame-  work for US segmentation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' We follow  [25] as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' After each MobileViT block, we add  an LCLM for further long-range contrast learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' We then  utilize a feature pyramid network (FPN) to recover the initial  resolution of the input US image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  a) MobileViT block: To cover the global information,  MobileViT block first utilizes a 3×3 convolution to aggregate  the neighboring features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The pixels are then grouped into  patches, each patch contains h × w pixels in which h and w  represent the height and the width of the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The ith token  6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The thick stripe is indicated by the arrow in Figure a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure b) shows how the thickness of the US waves induces the thick stripes artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure  c) is a schematic of sampling in a 3d gradient texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The Blue line indicates the imaging plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The reflection value between the blue line and the yellow  line is multiplied and accumulated into the imaging plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure d) shows the simulated thick stripes in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The framework of the proposed lightweight vision transformer for US segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' For the first two stages, we only utilize basic convolution layers to  learn local representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In the last three stages, we implement MobileViT block and LCLM alternately to cover both long-range dependency and long-range  contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  of each patch transverses its feature from each other by SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  When h <= 3 ∧ w <= 3, each pixel can access information  from all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Details of the MobileViT block could be found  in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  b) Why is LCLM needed: We find that the region of  interest (RoI) in US images has similar textures, but is weak in  amplitude to other tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Basic 3×3 convolutions are capable  of modeling the neighboring texture but fail to learn long-range  textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' As the local texture of the to-be-segmented region  is similar to the other tissues, we have to design a module  that covers the texture within and outside the to-be-segmented  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  In vision transformers, long-range texture modeling is  achieved by self-attention (SA) of all tokens as in Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 6:  �x = softmax( xWq(xWk)T  d  )(xWv),  (6)  in which x ∈ RN×d represents the N input tokens with  dimension d, Wq ∈ Rd×dq, Wk ∈ Rd×dq, and Wv ∈ Rd×dv  are learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' We rewrite the self-attention to the  formula as in Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 7:  �  �  �  �  �  �xi = �  j∈A⟩  wj(xjWv)  wj =  exp(xiWq(xjWk)T /d)  �  t  exp(xiWq(xtWk)T /d)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  (7)  where Ai represents the set of accessible tokens of token xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  As Wq and Wk only attribute in modeling the relationship  of queries, once Wv is fixed, the space generated by x is  therefore limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Meanwhile, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 8(a), SA  tends to aggregate “similar” features from the tokens, which  helps the model to learn long-range dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' However,  because of the nature of US images, long-range contrast is  also important to be learned, but SA fails to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Convolution operation could be formulated in the same  formulation as SA, as shown in Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 8:  �xi = �  j∈Ai  λj(xjWv) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  (8)  In this case, if rank(xjWv) = d, the space generated by x is  Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' We randomize 1000 different pairs of Wq and Wv, 1000  different λ and fixed Wv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The outputs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' As  demonstrated in the figure, the outputs of convolution fulfill  the space, while the outputs of SA are gathered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  C+w  <  Transducer  R=  krk  k=c-wS  MViT  1/2  1/2  MViT  LCLM  Up  Up7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The possible outputs (dots in red) of self-attention (a) and convolution  (b) with fixed Wv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  A recent work [26] also demonstrates that CNNs with  super large kernels learn feature representations from the  super large receptive field (sub-global), and achieve compara-  ble performance with state-of-the-art transformers with fewer  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  As aforementioned, we need a module to gather long-range  textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' As SA fails to learn long-range contrast, and a super-  large convolution kernel leads to large computational cost,  we need to design a light yet effective convolution module  for long-range contrast learning, namely Long-range Contrast  Learning Module (LCLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  c) Long-range contrast learning with dilation convo-  lution: To cover long-range contrast with fewer parameters,  a simple way is to utilize dilation convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In this work,  we adopt 3 cascaded dilation convolution layers with dila-  tion {3,5,11}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Along with one basic convolution layer which  appears at the end of the MobileViT block, LCLM covers  up to 41 × 41 pixels densely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' We show the receptive field in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Each convolution layer is followed by a normalization  layer and an activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' A depth-wise LCLM only needs  3×3×3×d parameters, which is even smaller than a standard  convolution layer (3 × 3 × d × d, d >> 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The receptive field of the pixel in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The pixels painted not in grey  are covered by the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Realistic US Simulation from CT Volume  It is difficult to provide quantitative evaluation criteria for  image generation-related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Qualitatively, the advantages  of this paper over deep learning-based US image generation  are patient-specific and more accurate lesion boundaries, and  when compared with previous papers based on image gener-  ation, we make the system produce more realistic images by  simulating squeezing and scattering in soft tissues and bones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Nonetheless, the quantitative evaluation is also of significance,  and since the quality of image generation is hard to be  evaluated quantitatively, we evaluate the quality of image  generation from the purpose of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Except for training  doctors to do US examinations, the most important uses of  the US image generation algorithms are intra-operative reg-  istration and providing datasets for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Therefore,  we put the generated images into various deep learning-based  segmentation algorithms to verify the usefulness of this image  generation system for the pre-training of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Deep Learning-Based Segmentation  a) Implementation Details: To validate the proposed  US simulation method and the lightweight segmentation  model, we train the proposed model on synthetic US images  and inference on real US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' We set the batch size to 32,  utilize the Adam optimizer, and set the learning rate to 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  The validation image examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure (a) is the US image of a  spinal phantom in gel mimic, Figure (b) is the US image of a 3D-printed  spine in water, and Figure (c) is the US image of a clinically human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  b) Comparison with other models: We first compare  the proposed model with other US segmentation models  in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' All performances reported in the table are re-  implemented by ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Compared with UNet-based mod-  els, the proposed model is much smaller and more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Compared with MViT-FPN, the proposed approach achieves  much better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  To utilize the model, in this case, the segmentation is to  achieve better registration, therefore we remove the cases with  obvious segmentation errors to obtain our (selected) result,  which is used in the subsequent registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In this case,  the effect of the network trained with generated data can be  equivalent to or better than the effect of the SOTA model  trained by a large number of real US images with an accurately  labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Our label is acquired by ct segmentation, which is  more challenging for the convergence of lightweight neural  networks since he can label bone surfaces which is invisible  or ambiguous in some situation in US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  c) Ablation study: To demonstrate the effectiveness of  synthetic US over initial CT scans, long-range dependency,  and long-range contrast, we validate the aforementioned set-  tings and show the results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Training the model with  CT scans leads to a dramatic IoU decrease, which demon-  strates that the proposed method can effectively synthesize US  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Without LCLM, the model fails to learn the long-range  contrast of US images and results in a performance decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Without MViT, the IoU slightly drops, which represents that  query  key  output  (b)8  Method  Dice  CD(TP)  CD(FN)  #Parm  FPS  UNet [17]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='574  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='085mm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='590mm  131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='8M  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='13  MViT-FPN [25]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='294  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='664mm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='824mm  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='0M  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='1  CNLUnet [19]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='329  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='868mm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='287mm  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='6M  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='9  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='783  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='599mm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='079mm  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='0M  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='3  Ours(Selected)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='926  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='227mm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='184mm  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='0M  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='3  TABLE II  COMPARISON WITH OTHER SEGMENTATION MODELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' NOTE THAT ALL  PERFORMANCES REPORTED IN THE TABLE ARE RE-IMPLEMENTED BY  OURSELVES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  the model benefits from long-range dependency, but is less  significant compared with long-range contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Method  Baseline  SUS→CT  w/o MViT  w/o LCLM  Dice  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='783  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='438  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='294  CD(TP)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='599mm  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='062mm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='910mm  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='664mm  CD(FN)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='079mm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='938mm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='674mm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='824mm  TABLE III  THE ABLATION STUDY OF THE PROPOSED MODEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' “SUS” REPRESENTS  THE SYNTHETIC US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' “SUS→CT”, “W/O MVIT” AND “W/O LCLM”  REPRESENT TRAINING THE MODEL BY CT SCANS, REMOVING MVIT  FROM THE BASELINE AND REMOVING LCLM FROM THE BASELINE,  RESPECTIVELY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  The result of segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Figure(a), (b), and (c) presents the  segmented label, the output of the network, and the ground truth respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Validation of US-CT Registration  a) Implementation Details: The US simulation system  is proposed to better obtain labeled data from multiple poses,  patients, and environments, enabling US-based surgical navi-  gation to be accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In this paper, we validated whether  the output of the model trained on synthetic data can align  the preoperative CT and the planned trajectory into the intra-  operative phase, and the spinal bone surface registration for  pedicle screw placement is the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  The validation is divided into two parts, the first is to  validate the accuracy of the rigid body registration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' trans-  lation and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The second part is to validate whether  the pre-operatively planned trajectory with the aforementioned  transforms will cause complications for the surgery or not, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  whether the intra-operative trajectory will touch vital organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  The experiments were performed on three phantoms, a spine  phantom in water, a 3D-printed human spine in water, and a  bovine spine in agar gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The registration steps were divided  into a coarse alignment and an individual registration for  each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The coarse registration used all point clouds  to calculate the approximate position and orientation, and the  final registration used a de-noised Iterative Closest Point(ICP)  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The results of the registration are shown in the table  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The phantom model for registration validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The MSE loss of registration in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  The Translation and Rotation of pedicle screw with registration  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The rotation is presented as an angle in degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  b) Validation of intra-operative model registration: In  the registration phase, the model inputs the US image outputs  the segmentation label, and extracts the upper contour of its  maximum connectivity, hence the point cloud is created with  the calibration matrix of the US probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The calibration of the  US is done by two cross-wire phantom [27], with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='97mm  MSE(min square loss) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Since the coordinate system of  3  Translation in three dimension  X  1  y  0  7  1  Li  L2  L3  L4  L5  TheLumbarsegment3  V  1  Z  0  1  Angle  2  Li  L2  L3  L4  L5  The Lumbar segment9  each segment does not coincide, the rotation evaluation in  angular will be represented by the rotation of pedicle screws in  the next part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' To evaluate the error of registration, the point-  to-point MSE is used, which is also the objective function  of the ICP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The result of registration is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 15, which shows the ground truth in the intra-operative  space(white model) and the results of registration with US  (green model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' In summary, the segmentation algorithm trained  by the generated US image works in the registration with the  error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='133 ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='366mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The maximum appears in the z-axis  of L5, which is around 3 mm, which is probably caused by  the shape of L5 being more different from the shape of other  lumbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  c) Validation of Pedicle Screw Placement Feasibility: To  verify the registration effectiveness of the system in the target  tasks, we compared the translation and rotation of the tip of the  screw with the US point cloud registration and ground truth  using an expert-planned intra-operative plan for pedicle screw  placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The rotation is determined by the angle between  two pedicle axis vectors in 3D space which is expressed in  degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 14, the error in translation for  the tip of the screw ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='027 mm to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='031 mm,  with the maximum value still occurring on the z-axis of L5,  which is consistent with the above assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The error in the  rotation is between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='05 degrees and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='47 degrees, therefore,  it can be concluded that the pre-trained model with generated  US image can perform the alignment with small errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The  results of the pedicle screw placement are shown in figure 16,  which shows this registration does not put the patient at risk  of complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The visualization of registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The green part of the figure is the  registration result by US segmentation, and the white part is the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Cases derived from 3d printed patient spine, which is (b) in the figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The visualization of registration-based screw placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' The figure  shows the posture of the screws with standard length and diameter which  is planned on preoperative CT by an experienced surgeon, compared to the  intra-operative spine ground truth by the registration matrix acquired by image  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Cases derived from the standard spine phantom, which is (c)  in the figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' CONCLUSION  In this paper, we propose a US image simulation method  based on CT images and acoustic properties that automatically  generates a US simulation environment based on a specific  patient CT volume, which allows US-based deep learning and  reinforcement learning algorithms to be trained and validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  In addition, we propose a lightweight vision transformer for  segmenting US images which is a network structure with better  segmentation accuracy and modal generalization capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content='  Experiments show that the model trained using US im-  ages generated by the realistic US simulation from the CT  system achieves higher segmentation accuracy and has the  capability to perform the intraoperative registration process  without complications compared to the model trained with  CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
+page_content=' Compared to other segmentation algorithms, the  proposed transformer has real-time computational efficiency,  better segmentation accuracy, and generalization capability,  which is particularly important in US surgical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQf-_4J/content/2301.01940v1.pdf'}
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diff --git a/99E0T4oBgHgl3EQfwwHR/content/tmp_files/2301.02638v1.pdf.txt b/99E0T4oBgHgl3EQfwwHR/content/tmp_files/2301.02638v1.pdf.txt
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index 0000000000000000000000000000000000000000..fe80717ed1b9f2322628048b087bec835637a6a7
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@@ -0,0 +1,1669 @@
+Asynchronous Wait-Free Runtime Verification and
+Enforcement of Linearizability
+ARMANDO CASTAÑEDA, Instituto de Matemáticas, Universdad Nacional Autónoma de México, México
+GILDE VALERIA RODRÍGUEZ, Posgrado en Ciencia e Ingeniería de la Computación, Universidad
+Nacional Autónoma de México, México
+This paper studies the problem of asynchronous wait-free runtime verification of linearizability for concurrent
+shared memory implementations, where one seeks for an asynchronous wait-free concurrent shared memory
+algorithm for verifying at runtime that the current execution of a given concurrent implementation is
+linearizable. It proposes an interactive model for distributed runtime verification of correctness conditions, and
+shows that it is impossible to runtime verify linearizability for some common sequential objects such as queues,
+stacks, sets, priority queues, counters and the consensus problem, regardless of the consensus power of base
+objects. Then, the paper argues that actually a stronger version of the problem can be solved, if linearizability
+is indirectly verified. Namely, it shows that (1) linearizability of a class of concurrent implementations can be
+distributed runtime strongly verified using only read/write base objects (i.e. without the need of consensus),
+and (2) any implementation can be transformed to its counterpart in the class using only read/write objects too.
+As far as we know, this is the first distributed runtime verification algorithm for any correctness condition that
+is fully asynchronous and fault-tolerant. As a by-product, a simple and generic methodology for the design
+of self-enforced linearizable implementations is obtained. This type implementations produce outputs that
+are guaranteed linearizable, and are able to produce a certificate of it, which allows the design of concurrent
+systems in a modular manner with accountable and forensic guarantees. We are not aware of prior concurrent
+implementations in the literature with such properties. These results hold not only for linearizability but for a
+correctness condition that includes generalizations of it such as set-linearizability and interval-linearizability.
+Additional Key Words and Phrases: Concurrent algorithms, Distributed runtime verification, Enforcement,
+Fault-tolerance, Linearizability, Lock-freedom, Monitoring, Shared memory, Verification, Wait-freedom
+1
+INTRODUCTION
+Linearizability and its challenges. Linearizability [60] is the de facto correctness condition for
+asynchronous shared memory concurrent implementations of objects defined through sequential
+specifications. Intuitively, an implementation is linearizable if each operation happens atomically at
+a single moment of time between its invocation and response. Linearizability is so popular in part
+because of its properties: it never forces the use of locks (it is non-blocking) and allows the design of
+systems in a modular manner (it is composable) [60, 87]. Designing linearizable implementations is a
+simple task due to Herlihy’s Universal Construction [57], although the resulting solutions typically
+do not scale well in practice. In contrast, designing linearizable and scalable implementations is
+a challenging task, as it requires the use of fine grained synchronization mechanisms in order
+to exploit the parallelism in concurrent systems, which typically derives in an large number of
+scenarios and subtle corner cases that need to be considered in correctness proofs [59, 72, 78]. It is
+not uncommon to discover bugs in implementations that were thought to be linearizable.
+The importance of linearizability, and more broadly of correct concurrent software, naturally
+calls for formal verification techniques. Over the past years, a variety of techniques for verifying
+linearizability have been developed, using different approaches and providing different levels of
+guarantees. See for example survey [26]. Despite of all efforts, verifying correctness of linearizable
+implementations remains difficult. Model checking is feasible only for small cases (i.e. for a bounded
+number of processes, and/or invocations to operations), and finding linearization points, simulations,
+invariants, etcetera, is hard, sometimes requiring a great amount of expertise of the user. The
+problem has been also studied from a theoretical perspective. It is known that deciding whether
+1
+arXiv:2301.02638v1  [cs.DC]  6 Jan 2023
+
+Castañeda and Rodríguez
+an implementation is linearizable might be EXPSPACE-complete or even undecidable [15], while
+deciding if a given finite execution is linearizable might be NP-complete [49, 76].
+Distributed runtime verification and its challenges. Runtime verification is a dynamic, lightweight,
+yet rigorous, formal method that complements static verification techniques with a more practical
+approach. It only seeks to verify that the current execution of a system is correct, and maybe
+prevent an incorrect action or enforce a correct behaviour otherwise. The system under inspection
+can be of any type, from hardware to software, centralized or distributed. We refer the reader
+to [6, 35, 54, 68] for a detailed exposition of the field.
+Broadly speaking, in runtime verification, two, non necessarily disjoint tasks need to be accom-
+plished [14]: (1) the design of a communication interface that captures the current execution of
+the underlaying system under inspection, and (2) the design of a monitoring system that verifies
+whether the captured execution is correct with respect to some correctness criteria. A big source of
+difficulty is that the underlaying system and the communication interface are typically decoupled.
+Namely, the underlaying system is designed, implemented and deployed without considering that
+in the future a runtime verification mechanism might be integrated to it, hence it might not export
+enough data of the current execution from which a communication interface and a monitoring
+system can be built later on. The situation gets more difficult when the underlaying system is
+distributed as no process of the system “knows” what the current execution is, each process has only
+a partial view of what the execution could be. The problem is even worse if we seek for a distributed,
+asynchronous and fault-tolerant communication interface, as the processes might not even have the
+ability to agree on the partial view of a process of the underlaying system (e.g. [36, 69]). In such
+scenario, we have several computational entities that exchange information (typically by means of
+a shared memory or a network), subject to delays and failures, yet they have to make consistent
+decisions; namely, we have a distributed computing problem. Designing distributed communication
+interfaces that are asynchronous and fault-tolerant is a challenging problem in runtime verification
+(see [14, 27, 32, 43, 70, 82]). In fact, there are only proposal for runtime verification (of a number
+of properties) with distributed communication interfaces that are failure-free and synchronous
+(e.g. [4, 83]), fault-tolerant with timing assumptions (e.g. [8, 10, 11, 13, 28, 33, 41, 42, 63, 80]), or asyn-
+chronous failure-free (e.g. [20, 37, 39, 85, 88]). That is, the known distributed runtime verification
+algorithms are not fully asynchronous and fault-tolerant. As far as we know, runtime verification of
+linearizability has only been studied in [29, 30], with centralized communication interfaces.
+Asynchronous wait-free runtime verification of linearizability. We are interested in runtime verifi-
+cation of linearizability where the underlaying system A is an asynchronous concurrent implemen-
+tation of an object, and the communication interface C is an asynchronous wait-free shared memory
+concurrent algorithm, where wait-free means that all processes but one can crash. Moreover, we
+focus on the case where the underlaying system A is a black-box, i.e., we do not have access to its
+specification/pseudocode, and hence the only way to obtain information of the current execution is
+by analyzing the sequence of invocations and responses each process obtains from it.
+Why is this setting interesting? While arguably A and C being asynchronous and A being a
+black-box, model the challenging scenario described above, where A and C are decoupled, C being
+asynchronous and wait-free guarantees that the “quality of service” of A is preserved. For example,
+if A is non-blocking but C is blocking (maybe because it lock-based), the system that results of
+integrating A and C (and the corresponding monitoring system) will be blocking, i.e. non fault-
+tolerant, hence loosing A’s progress property. Similarly, A might be designed as an asynchronous
+implementation for sake efficiency, hence if C is synchronous or semi-synchronous, A’s timing
+property is lost. Indeed, a target in runtime verification is to design communication interfaces and
+2
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+monitoring systems that are “as less intrusive as possible”; ideally they should not interfere with
+the behavior of the underlaying system (e.g. [6, 12, 25]).
+p1
+p2
+Push(1) : true
+Pop() : 1
+p1
+p2
+Push(1) : true
+Pop() : 1
+Fig. 1. Two executions of a stack where the two processes have the same partial views, i.e. sequence of
+invocations and responses; while the execution at the top is linearizable, the execution at the bottom is not.
+Real-time, unaccessible to the processes, is what ultimately defines the executions.
+Under this circumstances, is it possible to capture the actual execution of A in order to verify
+correctness, i.e. linearizability? The answer is clearly no. After the seminal work of Lamport [64], we
+know that it is impossible to determine the order of non-causally related events in fully asynchronous
+distributed systems; and invocations to and responses from A are local events, which are non-
+causally related. Thus, the partial view of a process of A’s actual execution is just the sequence
+with its invocation and responses, and for the processes is just impossible to access the real-time
+order of these local events, which ultimately defines A’s execution (see Figure 1). This reasoning
+suggests that very little should be possible when considering linearizability, as in the end this
+correctness condition totally depends on the real-time order. However, there are ways to runtime
+verify linearizability, as we show here.
+Results. We propose an interactive model to study the problem of distributed runtime verification
+of linearizability; the model actually serves for studying any correctness condition. In the model,
+an asynchronous concurrent implementation A that is presumably linearizable interacts with a
+client C (modelling a communication interface), which, as already said, is an asynchronous wait-free
+shared memory concurrent algorithm. The client C is required to invoke operations of A, receive
+the corresponding responses, and somehow compute relevant information of the current execution
+of A in order to decide whether it is linearizable or not; the interaction between A and C is infinite.
+The client C runtime verifies a correctness property P, if it is sound and complete, i.e., it reports
+ERROR if and only if A’s current execution does not satisfy P.
+Once the interactive model and the distributed runtime verification problem are clearly stated, a
+simple indistinguishability argument shows that it is impossible to runtime verify linearizability
+for some common objects such as queues, stacks, sets, priority queues, counters and even the
+consensus problem, regardless of the consensus number [57] of the base objects used in C. Thus,
+the problem of distributed runtime verification of linearizability is beyond consensus.
+Then, somewhat surprisingly, we show that a stronger version of the problem can be solved, and
+without the need of consensus, if linearizability is indirectly verified. Namely, we identify a class of
+concurrent implementations, which we call DRV, such that linearizability can be runtime verified;
+furthermore, a stronger version of the problem can be solved where any client C is required to
+be complete and strongly sound, which means that it might report ERROR when A’s execution is
+linearizable (i.e. a false negative), as long as it “discovers” an execution of A that is not linearizable
+(hence concluding that A is not linearizable). Intuitively, any implementation in the class DRV
+3
+
+Castañeda and Rodríguez
+produces an “sketch” of its current execution. It turns out that such sketches are good enough to
+runtime strongly verify linearizability, for the class DRV, and using only read/write base objects,
+which have consensus number one [57] (hence uncapable of solving consensus among two or
+more processes). Linearizability can then be indirectly strongly verified because any concurrent
+implementation A can be transformed into an implementation A∗ in DRV, using only read/write
+base objects, such that A and A∗ have the same progress properties, and A is linearizable if and
+only if A∗ is linearizable (both w.r.t. the same sequential object). Moreover, these results holds
+for a correctness condition GenLin that we define here, which includes linearizability, as well as
+other correctness conditions such as set-linearizability [75] and interval-linearizability [18], both
+generalizations of linearizability for concurrent objects with no sequential specifications [19].
+Finally, we show the usefulness of the proposed interactive model by obtaining concrete dis-
+tributed runtime verification algorithms from an interactive client C. For linearizability, we obtain
+a simple methodology to derive self-enforced linearizable implementations, intuitively whose re-
+sponses are guaranteed linearizable (i.e. verified), and moreover the implementations are able to
+produce a certificate of it. Thus, self-enforced linearizable implementations allow the design of
+systems in a modular manner with accountable and forensic guarantees. We are not aware of prior
+concurrent implementations in the literature with such properties. The methodology takes any
+implementation A and produces an implementation B using A and read/write objects such that A
+and B have the same progress conditions and A is linearizable if and only if B is self-enforced lin-
+earizable (both w.r.t. the same sequential object). Furthermore, a concurrent asynchronous wait-free
+runtime verification algorithm for linearizability (i.e. a communication interface and a monitoring
+system) can be easily obtained from B. As far as we know, this is the first distributed runtime
+verification algorithm for any correctness condition that is at the same time fully asynchronous
+and fault-tolerant. Remarkably, all proposed algorithms are generic and simple to implement, hence
+prone to being program synthesized, a property of interests in runtime verification.
+Structure of the paper. Once Section 2 states the model of computation, the interactive model for
+distributed runtime verification and the problem of distributed runtime verification are introduced
+in Section 3. Then, Section 4 recalls the definition of linearizability and Section 5 shows the
+impossibility for runtime verification of linearizability for some objects. A high-level perspective of
+how the impossibility result is evaded and the definition of the strong version of the problem appear
+in Section 6. Sections 7 and 8 implement the high-level strategy described in Section 6. Section 9
+explains some extension of our results. Finally, Section 10 discusses related work, explaining
+differences with our results, and Section 11 concludes the paper with a final discussion.
+2
+MODEL OF COMPUTATION
+We consider a standard concurrent shared memory system (e.g. [59, 60]) with 𝑛 ≥ 2 asynchronous
+processes, 𝑝1, . . . , 𝑝𝑛, which may crash at any time during an execution. All but one process can crash
+in any execution. The index of process 𝑝𝑖 is 𝑖. Processes communicate with each other by invoking
+atomic operations of shared base objects that reside in the shared memory: either simple Read/Write
+operations, or more complex and powerful Read-Modify-Write operations, such as Fetch&Inc or
+Compare&Swap. Base objects are atomic, hence we consider a sequentially consistent [65] shared
+memory. Shared base objects are denoted with uppercase letters; local variables used by a process
+for performing its local computations are denoted with lowercase letter subscripted with the index
+of the process.
+For ease of exposition, it is assumed that base objects are of unbounded size; Section 11 however
+explains how this unrealistic assumption can be removed from the proposed algorithms. We consider
+the possibility that processes have perfectly synchronized local clocks. Each process can read its
+4
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+local clock in a local computation step, information that then can be written in the shared memory.
+We stress that the time that elapses between reading a local clock and writing it in the shared
+memory is unpredictable, as the system is asynchronous. This assumption is not relevant in our
+algorithms, however it makes our impossibility result stronger.
+An implementation of a concurrent object O (e.g. a queue or a stack), specified in some way (more
+details in Section 4), is a distributed algorithm A consisting of 𝑛 local state machines 𝐴1, . . . ,𝐴𝑛,
+some of them possibly non-deterministic. Local machine 𝐴𝑖 specifies which operations on base
+objects and local computations 𝑝𝑖 executes in order to return a response when it invokes a high-level
+operation of O (sometimes simply called operation). Each of these base objects operations and local
+computations is a step. Invocations and responses are local computations as well but we do not
+refer to them as steps.
+An execution of A is a possibly infinite sequence of steps, plus invocations and responses to
+high-level operations of the concurrent object O, with the following well-formedness properties:
+(1) Each process is sequential, namely, it first invokes a high-level operation, and only when it
+has a corresponding response, it can invoke another high-level operation.
+(2) A process takes steps only between an invocation and a response.
+(3) For any invocation of an operation op𝑖 by a process 𝑝𝑖, denoted 𝑖𝑛𝑣𝑖 (op𝑖), the steps of 𝑝𝑖
+between that invocation and its corresponding response (if there is one), denoted 𝑟𝑒𝑠𝑖 (op𝑖),
+are steps specified by 𝐴𝑖 when 𝑝𝑖 invokes op𝑖.
+It is assumed that after a process completes an operation, it non-deterministically picks the operation
+it executes next.
+For simplicity, and without loos of generality, we assume that every concurrent object provides
+a single high-level operation, called Apply, that receives as input op, a description of the actual
+operation that is invoked, which includes the inputs to the actual operation. We also assume, again
+without loss of generality, that Apply is invoked with a given input op only once (a fictitious input
+value to the actual operation can be added such that all inputs are different).
+Typically, invocation and responses of an implementation A also include A as part of their
+description. We however drop that information from invocations and responses because it facilitates
+our discussion, although some ambiguity is introduced.
+An implementation A can use other implementation B in order to produce responses, namely,
+the processes can invoke high-level operation of B. Hence, when a process invokes an operation of
+B, it continues executing the steps specified by A only after it receives the corresponding response
+from B. Consider any execution 𝐸 of A. We let 𝐸|B denote the sequence of steps, invocations and
+responses of B in 𝐸. Then, 𝐸|B is a well-formed execution of B. In what follows, unless stated
+otherwise, when we talk about operations in 𝐸, we mean only the operations of A, excluding the
+nested calls to operations of B.
+A high-level operation in an execution is complete if both its invocation and response appear in
+the execution. An operation is pending if only its invocation appears in the execution. A process
+is correct in an infinite execution of an implementation if it takes infinitely many steps. When
+considering infinite executions, we focus on those that are fair: for every correct process and every
+𝐾 ≥ 1, there is a finite prefix with 𝐾 steps of that process. An implementation A is wait-free
+if in every infinite execution, every correct process completes infinitely many operations [57].
+An implementation A is lock-free if in every infinite execution, infinitely many operations are
+complete [60]. Thus, a wait-free implementation is lock-free but not necessarily vice versa. We
+consider only implementations that are at least lock-free. The notions of wait-freedom and lock-
+freedom naturally extend to specific operations or fragments of pseudocode.
+5
+
+Castañeda and Rodríguez
+The step complexity of an implementation is the maximum number of base operations a process
+needs to take to produce a response.
+Sometimes it will be convenient to think an implementation A as a black-box whose specification
+cannot be accessed, and hence the only information that can be obtained from it are the executions
+it produces without steps. We call such executions without steps histories, namely, sequences of
+invocations and responses satisfying the first two well-formedness properties stated above. As
+in [60], we define an abstract implementation as a set of well-formed histories. By abuse of notation,
+for any execution 𝐸 of an implementation A, we let 𝐸 itself denote the history obtained from 𝐸
+(i.e. the sequence obtained by removing from 𝐸 all its steps), and let A denote itself the abstract
+implementation with all histories of A (i.e., the histories obtained from all its executions). This
+abuse of notation will facilitate the discussion, at the cost of introducing some ambiguity.
+The consensus number of a shared object O is the maximum number of processes that can solve
+the consensus problem, using any number of instances of O in addition to any number of Read/Write
+base objects [57]. Consensus numbers induce the consensus hierarchy where objects are classified
+according their consensus numbers. The simple Read/Write operations stand at the bottom of
+the hierarchy, with consensus number one and the lowest coordination power. At the top of the
+hierarchy we find operations with infinite consensus number, like Compare&Swap, that provide
+the maximum possible coordination power.
+3
+AN INTERACTIVE MODEL FOR DISTRIBUTED RUNTIME VERIFICATION
+Let us fix a concurrent object O, specified in some way. Let A be a lock-free implementation of O.
+Intuitively, a correctness condition is a mechanism to separate the correct implementations of O
+from the incorrect ones. Basically, it is a predicate PO that all finite executions of A need to satisfy
+for A being declared correct, with respect to PO. It is known that if PO is linealizability, deciding
+whether A is linearizable might be EXPSPACE-complete or even undecidable [15], depending
+on the object O. Deciding if a given finite history is linearizable is decidable, but it might be NP-
+complete [49, 76], although for some objects this question can be decided in polynomial time [16, 31].
+From now on, we will assume that each process can locally test if a given finite history satisfies PO.
+Shared Variables:
+Shared memory 𝑀
+Operation Verify(A) is
+(01) 𝑠𝑒𝑡𝑖 ← ∅
+(02) while true do
+(03)
+op𝑖 ← non-deterministically chosen high-level operation that is not in 𝑠𝑒𝑡𝑖
+(04)
+𝑠𝑒𝑡𝑖 ← 𝑠𝑒𝑡𝑖 ∪ {op𝑖 }
+(05)
+Encode in 𝑀 the invocation to Apply(op𝑖) of A
+(06)
+Invoke operation Apply(op𝑖) of A
+(07)
+𝑟𝑒𝑠𝑝𝑖 ← response from operation Apply(op𝑖) of A
+(08)
+Encode the response 𝑟𝑒𝑠𝑝𝑖 in 𝑀
+(09)
+𝑒𝑥𝑒𝑐𝑖 ← description of the current execution of A in 𝑀
+(10)
+if ¬ P𝑂 (𝑒𝑥𝑒𝑐𝑖) then
+(11)
+report (ERROR,𝑒𝑥𝑒𝑐𝑖)
+(12)
+end if
+(13) end while
+end Verify
+Fig. 2. Generic structure of a verifier VO for correctness condition PO (code of process 𝑝𝑖).
+Let us suppose that A is presumably correct with respect to PO, but maybe this fact has not been
+formally proven. Let us also suppose the existence of a client C (i.e. a concurrent algorithm) that
+solves some distributed problem using A, namely, the processes in C invoke high-level operations
+6
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+of A. We would like to design an intermediate layer VO between C and A that, from time to
+time, verifies that the current execution of A is correct, namely, it satisfies the predicate PO, and
+reports ERROR otherwise. If A is indeed correct, then we would like the client C not to be able to
+distinguish whether it is interacting with A or VO, and hence we require that VO is asynchronous
+and wait-free so that the properties of A are preserved, for example its progress properties.
+We model the intermediate layer just described as a distributed algorithm VO, called verifier, that
+interacts with A. The generic structure of the interaction appears in Figure 2, where A is a black-
+box, and hence the only way VO can obtain information of A is by invoking high-level operations
+of it; namely, the processes in VO only invoke operations of A and only receive responses. Thus,
+the verifier VO interacts with an abstract object A, from which receives one of its histories in
+every execution of VO.
+During the interaction, each process invokes a series of non-deterministically chosen high-level
+operations of A (modeling the operations of A that any client C might invoke). For simplicity,
+every process tests if the history so far satisfies PO after each of its high-level operations of A.
+If the predicate is not satisfied, the process reports ERROR together with a witness for A, i.e. a
+history of A that does not satisfy PO; in any case, the interaction continues. (In a practical setting
+the interaction would stop and ERROR and the witness would be returned to the client C; for
+simplicity, in our model the interaction continues.) Naturally, the processes in VO need to exchange
+information in order to obtain a description of the current history of A (which ultimately creates
+causal relations between A’s invocations and responses). Thus, each process might store some
+information in the shared memory before and after each of its invocations to and responses from A.
+Since A is a black-box in the generic verifier in Figure 2, VO cannot be designed for a specific
+presumably correct implementation for O, it must work for any possibly abstract concurrent
+implementation (even if it is correct with respect to an object O′ ≠ O). This requirement is modeled
+as conceptually VO taking A as its input to the computation.
+We will restrict our attention to verifiers where the segments of the code in Lines 03–05 and
+Lines 08–12 are wait-free; we say that such a verifier is wait-free. The step complexity of a verifier is
+the maximum number of base operations (i.e. excluding the invocation and response in Lines 06
+and 07) a process needs to take in order to complete one iteration of the while loop.
+In the definition of the distributed runtime verification problem below, for simplicity, it is assumed
+that no process crashes, and hence all executions of a verifier are infinite, since we are assuming
+that A is at least lock-free. The assumption makes the problem easier to state. The possible crashes
+that can occur in the system are captured by asynchrony and the fact that correctness is tested
+at finite prefixes of a given infinite execution. The soundness and completeness properties that
+specify the problem have been considered in the past (see for example [27, 73, 74]); the definition
+basically adapts them to fit in our interactive setting.
+Definition 3.1 (Distributed Runtime Verification). Let O be a concurrent object specified
+in some way, and consider a correctness condition PO for O. We say that a wait-free verifier VO
+distributed runtime verifies PO if the following two requirements are fulfilled in every infinite execution
+𝐸 of VO with an arbitrary input (abstract) implementation A:
+(1) Soundness: If for every finite prefix 𝐸′ of 𝐸, 𝐸′|A satisfies PO, then no process reports ERROR.
+(2) Completeness. If 𝐸 has a finite prefix 𝐸′ such that 𝐸′|A does not satisfy PO, then at least one
+process reports ERROR together with a witness for A.
+We say that PO is distributed runtime verifiable if there is a wait-free verifier that distributed
+runtime verifies PO.
+The previous definition is flexible, it is not difficult to modify it to cover the cases where A is
+blocking [59] (i.e. it internally uses locks) or obstruction-free [59] (namely, progress is guaranteed
+7
+
+Castañeda and Rodríguez
+only when a process runs solo), or correctness conditions for one-shot distributed problems such
+as tasks [58], where each process invokes one high-level operation. The main difference is that in
+these cases, the interaction might be only finite.
+Below, for sake of compactness, sometime we will simply say verify/verifiable instead of dis-
+tributed runtime verify/verifiable.
+To conclude the section, we observe that the intermediate layer in the situation described above
+can be easily obtained from a verifier VO. Let V𝑂,A denote the implementation whose single
+high-level operation Apply(op𝑖) executes Lines 05 to 09 with a fixed implementation A, and returns
+𝑟𝑒𝑠𝑝𝑖 if P𝑂 (𝑒𝑥𝑒𝑐𝑖) holds, and returns (ERROR,𝑒𝑥𝑒𝑐𝑖) otherwise. Basically, V𝑂,A “wraps” A, and
+thus the client C would use VA,O instead of A without noticing the difference, if A is indeed
+correct with respect to PO. This claim will be formalized in Section 8 for linearizability and a
+correctness condition generalizing it.
+4
+LINEARIZABILITY
+Linearizability [60] is the de facto standard correctness condition for concurrent implementations
+of objects defined with sequential specifications. It extends the concept of atomicity introduced by
+Lamport [66, 67] to any sequential object. Intuitively, a history of an implementation is linearizable
+if its operations can be ordered sequentially, without reordering non-overlapping operations, so
+that their responses satisfy the sequential specification of the implemented object. Figure 3 depicts
+examples of linearizable and non-linearizable histories of a stack implementation.
+Definition 4.1 (Seqential Specifications). A sequential specification of a concurrent object O
+is a state machine specified through a (possibly partial) transition function 𝛿. Given a state 𝑞 and an
+invocation 𝑖𝑛𝑣𝑖 (op𝑖) of process 𝑝𝑖, 𝛿(𝑞,𝑖𝑛𝑣𝑖 (op𝑖)) returns the tuple (𝑞′,𝑟𝑒𝑠𝑖 (op𝑖)) (or a set of tuples if
+the machine is non-deterministic) indicating that the machine moves to state 𝑞′ and the response to
+op𝑖 is 𝑟𝑒𝑠𝑖 (op𝑖). The sequences of invocation-response tuples, ⟨𝑖𝑛𝑣𝑖 (op𝑖) : 𝑟𝑒𝑠𝑖 (op𝑖)⟩, produced by the
+state machine are its sequential histories.
+p1
+p2
+Push(1) : true
+p3
+Push(2) : true
+Pop() : 2
+Pop() : 1
+p1
+p2
+Push(1) : OK
+p3
+Push(2) : true
+Pop() : empty
+Pop() : 1
+Fig. 3. Two 3-process histories of a stack implementations are depicted. Time goes from left to right, and an
+operation is denoted with a double-ended arrow. For clarity, each operation Apply(op) is simply denoted op.
+The execution at the top is linearizable with respect to the usual sequential specification of a stack; a
+linearization of it is: ⟨Push(2) : true⟩⟨Push(1) : true⟩⟨Pop() : 1⟩⟨Pop() : 2⟩. The execution at the bottom is
+not linearizable because the stack cannot be empty when the operation ⟨Pop() : empty⟩ starts.
+For sake of clarity, a tuple ⟨𝑖𝑛𝑣𝑖 (op𝑖) : 𝑟𝑒𝑠𝑖 (op𝑖)⟩ is simply denoted op𝑖. Also, subscripts of
+invocations and responses are omitted.
+8
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+An history 𝐸′ is an extension of a finite history 𝐸, if 𝐸′ can be obtained from 𝐸 by appending zero
+or more responses for some of 𝐸’s pending operations.
+For any history 𝐸 and any process 𝑝𝑖, 𝐸|𝑝𝑖 denotes the sequence of invocations and responses of
+𝑝𝑖 in 𝐸. Two histories 𝐸 and 𝐹 are equivalent if 𝐸|𝑝𝑖 = 𝐹 |𝑝𝑖, for every process 𝑝𝑖.
+For any finite history 𝐸 of an implementation A, 𝑐𝑜𝑚𝑝(𝐸) denotes the history obtained by
+removing from 𝐸 all invocations of pending operations; note that 𝑐𝑜𝑚𝑝(𝐸) is well-formed. To
+formalize linearizability, we define a partial order <𝐸 on the complete operations of any history 𝐸:
+op <𝐸 op′ if and only if 𝑟𝑒𝑠(op) precedes 𝑖𝑛𝑣(op′) in 𝐸. Two complete operations are concurrent if
+they are incomparable by <𝐸. The history is sequential if <𝐸 is a total order.
+We consider the definition of linearizability in [87], which is a slight variant of the original
+definition [60] that fixes some corner cases.
+Definition 4.2 (Linearizability). Let O be any concurrent object. A finite history 𝐸 is linearizable
+with respect to O if there is an extension 𝐸′ of 𝐸 and a sequential history 𝑆 of O such that
+(1) 𝑐𝑜𝑚𝑝(𝐸′) and 𝑆 are equivalent and
+(2) <𝑐𝑜𝑚𝑝 (𝐸′) ⊆ <𝑆.
+The sequential history 𝑆 is said to be a linearization of 𝐸. We say that an implementation A is
+linearizable with respect to O, if each of its finite histories is linearizable with respect to O.
+5
+LINEARIZABILITY FOR SOME OBJECTS IS NOT RUNTIME VERIFIABLE
+This section shows that linearizability for some common objects such queues, stacks, priority
+queues, counters, and even the fundamental consensus problem, is not distributed runtime verifiable,
+regardless of the coordination power of the base objects the processes use to communicate with.
+The following simple impossibility proof captures informal arguments that have been used in the
+past (e.g. [27, 47]) to argue that distributed runtime verification of some correctness conditions in
+asynchronous distributed systems is impossible.
+Theorem 5.1 (Impossibility of Distributed Runtime Verification). Linearizability for queues,
+stacks, sets, priority queues, counters and the consensus problem (defined as a sequential object) is not
+distributed runtime verifiable, even if a verifier uses base objects with consensus number infinity such
+as Compare&Swap.
+Proof. We focus on the case of the queues as all other cases are very similar. By contradiction,
+suppose that there is a wait-free verifier Vqueue that verifies linearizability for queues. Consider the
+following non-linearizable implementation A: every Enqueue operation returns true, and every
+Dequeue operation returns empty, for every process 𝑝𝑖 ≠ 𝑝1, and for 𝑝1, it returns 1 in its first
+operation, and returns empty in every subsequent operation.
+We will exhibit two executions 𝐸 and 𝐹 of Vqueue with input A and argue that Vqueue cannot
+simultaneously satisfy both the soundness and completeness requirements of the distributed
+runtime verification problem. We use the generic structure in Figure 2 to describe the executions.
+Execution 𝐸 is the next:
+(1) In its first iteration of the while loop, process 𝑝1 picks 𝑜𝑝1 = Dequeue() in Line 03, executes
+the local step in Line 04 and the base operations corresponding to Line 05.
+(2) In its first iteration of the while loop, process 𝑝2 picks 𝑜𝑝2 = Enqueue(1) n Line 03, executes
+the local step in Line 04 and the base operations corresponding to Line 05.
+(3) Process 𝑝1 executes Lines 06 and 07 of its first iteration of the while loop. Thus it obtains
+response 𝑟𝑒𝑠𝑝1 = 1 for its high-level operation 𝑜𝑝1 = Dequeue() .
+(4) Process 𝑝2 executes Lines 06 and 07 of its first iterations of the while loop. Thus it obtains
+response 𝑟𝑒𝑠𝑝2 = true for its high-level operation 𝑜𝑝2 = Enqueue(1) .
+9
+
+Castañeda and Rodríguez
+(5) Process 𝑝1 executes Lines 08 to 12 of its first iteration of the while loop.
+(6) Process 𝑝2 executes Lines 08 to 12 of its first iteration of the while loop.
+(7) For each 𝑘 = 1, 2, . . . , ∞ (in this order), 𝑝(𝑘 mod 𝑛)+1 executes a whole iteration of the while
+loop where it picks 𝑜𝑝(𝑘 mod 𝑛)+1 = Dequeue() in Line 03 (and hence the response in Line 07
+is 𝑟𝑒𝑠𝑝(𝑘 mod 𝑛)+1 = empty).
+Execution 𝐹 is similarly constructed, with the exception that the steps 3 and 4 of the previous
+construction appear in the opposite order. In other words, in 𝐸, the first high-level operations of 𝑝1
+is executed first and then the first high-level 𝑜𝑝2 is executed, whereas in 𝐹, the high-level operations
+are executed in the opposite order. Observe that the history of A obtained from every finite prefix
+of 𝐹 is linearizable; in contrast, the history of A obtained from every finite prefix of 𝐸 containing
+at least the first operation of 𝑝1, is not.
+By a simple induction, it can be shown that the local state of any process 𝑝𝑖 after its 𝑘-th step
+is the same in both executions 𝐸 and 𝐹. This means that the executions are indistinguishable to
+all processes, and hence in both executions they make the same sequence of decisions in Lines 08
+to 10. If no process reports ERROR in 𝐸 and 𝐹, then Vqueue does not fulfills completeness due to 𝐸,
+and if at least one process reports ERROR in 𝐸 and 𝐹, then Vqueue does not fulfills soundness due
+to 𝐹. Therefore, Vqueue cannot exist.
+For the other objects, the argument is nearly the same. For example, for the case of the stack, Pop
+operations replace Dequeue and Push operations replace Enqueue. For the case of the consensus,
+we define an object with a single Decide operation that can be invoked several times, and the first
+operations sets its value as the decision.
+□
+The previous proof can easily be extended to several variations of linearizability whose aim
+is modeling relaxed versions of sequential objects or objects with no sequential specification.
+Examples of such variations are quasi-linearizabilty [2], 𝑘-stuttering [56], set-linearizability [75],
+interval-linearizability [18] and intermediate value linearizability [79]. The reason is that all these
+relaxations include the sequential executions with the “exact” sequential behavior of the object that
+is relaxed, which suffices for the previous proof to hold. Furthermore, nearly the same proof shows
+the impossibility of runtime verification for sequential consistency [65], the only difference in the
+argument is that now there is finite prefix of the execution 𝐸|A that is not sequentially consistent,
+concretely, the one that only has the first operation of 𝑝1; the rest of the proof is the same.
+6
+EVADING THE IMPOSSIBILITY RESULT: THE STRATEGY AT A HIGH-LEVEL
+The rest of the paper is devoted to show that it is possible to circumvent the impossibility in
+Theorem 5.1. Roughly speaking, it will argue that linearizability of any implementation A with
+respect to any sequential object O, can be indirectly verified through a class of implementations,
+called DRV. Somewhat surprisingly the class DRV can be verified under a stronger definition of
+the distributed runtime verification problem. This section gives a high-level perspective of the
+strategy followed in the next sections, and introduces the stronger version of the distributed runtime
+verification problem that will be studied.
+The impossibility in Theorem 5.1 basically comes from the inability of the processes to capture
+the actual history of an arbitrary implementation A. Since the processes are asynchronous, a delay
+of arbitrary length can happen between the steps in Lines 05 and 06, and between the steps in
+Lines 07 and 08, in the generic verifier in Figure 2. Thus, basically the processes are only able to
+capture a history of A where the operations might be “stretched”. Figure 4 exemplifies this situation.
+Let 𝐸 denote the actual history of A and 𝐸′ denote the history of A captured by the processes.
+Hence history 𝐸′ only “sketches” the actual history 𝐸. Actually, 𝐸′ might be less restrictive to be
+linearized (with respect to some sequential object) as some operations that do not overlap in 𝐸
+10
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+p1
+p2
+enq(1) : true
+deq() : 1
+Detected history E′:
+Actual history E of A:
+p1
+p2
+enq(1) : true
+deq() : 1
+Fig. 4. Due to asynchrony, processes might detect histories where the operations of A are “stretched”. This
+phenomenon is exemplified with two histories of a queue implementation where an operation Apply(op)
+is simply denoted op. In the history at the top, both histories, the actual one and the detected one, are
+linearizable, while in the history at the bottom, the actual history is not linearizable, however the detected
+history is linearizable due to a long delay between the event that announce the operation of A that 𝑝1 is
+going to call next and the actual moment when the operation is called.
+might actually overlap in 𝐸′ (see the history at the bottom in Figure 4). Thus we have:
+𝐸 is linearizable =⇒ 𝐸′ is linearizable.
+Arguably, this is the best the processes can do for capturing the actual execution. For the time being,
+let us suppose that the processes somehow can compute 𝐸′ and each process is able to read it from
+the shared memory. With the help of 𝐸′, the processes can easily satisfy soundness, each process
+simply needs to locally test whether 𝐸′ is linearizable; however completeness cannot be satisfied: if
+𝐸 is not linearizable, 𝐸′ might or might not be linearizable (see again the history at the bottom in
+Figure 4). This discussion suggests a weaker version of the distributed runtime verification problem,
+requiring soundness and a weaker version of completeness where processes are allowed to output
+false positives, i.e., they do not report ERROR although the actual history of A is not linearizable.
+In fact, this weaker version of the problem can be formally defined and shown to be solvable [62].
+This result is not completely satisfactory because, after all, the main motivation of any verification
+technique is to detect incorrect solutions. But we can do better, as explained next.
+Detected history E′:
+Actual history E of A:
+p1
+p2
+enq(1) : true
+deq() : 1
+Actual history E∗ of A∗:
+p1
+p2
+enq(1) : true
+deq() : 1
+Fig. 5. The operations of any history of the implementation A∗ might “shrink” in the detected history and in
+the actual history of A. Two histories of A∗ where A is a queue implementation are shown; an operation
+Apply(op) is simply denoted op. In the history at the top, the actual history of A∗ is linearizable but the
+detected history is not. In contrast, in the history at the bottom, the actual history of A∗ is not linearizable,
+which implies that the detected history is not linearizable either.
+11
+
+Castañeda and Rodríguez
+We can look at the limitation above from a positive perspective, since in some sense the detected
+history 𝐸′ might “fix” the incorrect history 𝐸 of A. We can take the mechanism that computes 𝐸′
+and use it to produce a new implementation, denoted A∗, where processes output the responses
+obtained from A, somehow together with the captured history 𝐸′. In a sense, the idea in A∗ is
+to make asynchrony an “ally” instead of an “enemy”: in the case 𝐸, the actual execution of A, is
+not linearizable, if the delays are short, then 𝐸′ will not be linearizable, hence the mistake can be
+detected; and if delays are long, then 𝐸′ will be linearizable, hence the mistake cannot be detected
+but 𝐸′ “fixes” 𝐸. Thus, A can be indirectly verified through A∗ because, as it will be shown, A is
+linearizable if and only if A∗ is linearizable (both w.r.t. the same sequential object).
+In every execution 𝐸∗ of A∗, we now have that 𝐸′ “sketches” both the actual history 𝐸∗ of A∗ and
+the actual history 𝐸 of A, with the difference that the operations in 𝐸′ and 𝐸 now might “shrink”,
+compared to 𝐸∗. This situation is schematized in Figure 5. We thus have the following:
+𝐸 is linearizable =⇒ 𝐸′ is linearizable =⇒ 𝐸∗ is linearizable.
+From the sequence of implications we obtain that if 𝐸∗ is not linearizable then its sketch 𝐸′ is not
+linearizable either. The processes now can easily fulfill completeness for the derived implementation
+A∗ by locally analyzing 𝐸′. Therefore, the processes are able to “catch” any possibly non-linearizable
+history of A∗. For soundness, however it can be the case that 𝐸′ is not linearizable although 𝐸∗ is
+indeed linearizable (see the history at the top in Figure 5); thus the processes might output false
+negatives. It turns out that 𝐸′ is a history of A∗ and then it is a witness for A∗ in those cases.
+Hence if processes output a false negative, it is for a reason: the current execution of A∗ might be
+linearizable but the processes have “discovered” that A∗ is not linearizable, and they have a history
+that witnesses that fact. Moreover, no process can distinguish between 𝐸∗ and 𝐸′ (i.e. the histories
+are equivalent), hence it is actually possible that the actual execution of A∗ is 𝐸′ instead of 𝐸∗,
+which motivates the processes to take the “preventive” action of reporting ERROR. Therefore, the
+processes can fulfill a stronger version of soundness where false negatives are allowed, as long as
+they come with witnesses.
+The previous discussion motivates the next stronger version of the runtime verification problem,
+which will be shown to be solvable for implementations such as A∗; DRV will denote the class
+with the implementations like A∗.
+Definition 6.1 (Distributed Runtime Strong Verification). Let O be a concurrent object
+specified in some way, and consider a correctness condition PO for O. We say that a wait-free verifier VO
+distributed runtime strongly verifies PO if the following two requirements are fulfilled in every infinite
+execution 𝐸 of VO with an arbitrary input (abstract) implementation A∗:
+(1) Strong Soundness: If for every finite prefix 𝐸′ of 𝐸, 𝐸′|A∗ satisfies PO, then either no process
+reports ERROR, or at least one process reports ERROR together with a witness for A∗.
+(2) Completeness. If 𝐸 has a finite prefix 𝐸′ such that 𝐸′|A∗ does not satisfy PO, then at least one
+process reports ERROR together with a witness for A∗ in 𝐸.
+We say that PO is distributed runtime strongly verifiable if there is a wait-free verifier that
+distributed runtime strongly verifies PO.
+We conclude this section mentioning that a slight varian of the proof of Theorem 5.1 shows that
+the stronger version of the problem in the definition above is impossible for the general class of
+implementations (see Appendix A).
+12
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+7
+THE CLASS OF DISTRIBUTED RUNTIME STRONGLY VERIFIABLE
+IMPLEMENTATIONS (DRV)
+This section introduces the class DRV and shows its main properties. Roughly speaking, it will
+show that (1) every implementation can be easily transformed into its counterpart in DRV and (2)
+every implementation in DRV provides an “sketch” of the current execution, additionally to the
+outputs of the object it implements. As we will see, the sketches are the key property that makes
+any of these implementations strongly verifiable. Furthermore, the same result holds not only for
+linearizability, but for a class of objects, with its companion correctness condition, that generalize
+linearizability. After defining the generalized class of objects, the section defines the class DRV and
+shows the main properties of the implementations.
+7.1
+Generalizing linearizability
+In the rest of the paper, similarly to [60], an abstract object, called just object for simplicity, is defined
+as a set of well-formed finite histories. The associated correctness condition is the membership
+predicate. Namely, a finite history of an implementation is correct with respect to the object if the
+history belongs to the set specifying the object. Then, an implementation is correct with respect to
+the object if each of its finite histories belong the set specifying the object.
+In order to generalize linearizability, we define a partial order on the set of operations, complete
+and pending, of any history 𝐸 of an implementation; the relation is denoted ≺𝐸. For any two
+operations op and op′ in 𝐸, we have that op ≺𝐸 op′ if and only if 𝑟𝑒𝑠(op) precedes 𝑖𝑛𝑣(op′) in 𝐸.
+Thus, the only difference with <𝐸 is that ≺𝐸 also relates pending operations.
+Definition 7.1 (Similarity between histories). A finite history 𝐸 is similar to a finite history 𝐹
+if there is a history 𝐸′ such that:
+(1) 𝐸′ can be obtained from 𝐸 by appending responses to some pending operations and removing
+invocations of some pending operations,
+(2) 𝐸′ and 𝐹 are equivalent, and
+(3) ≺𝐸′ ⊆ ≺𝐹.
+Definition 7.2 (The class GenLin). The class GenLin contains every abstract object that is closed
+by prefixes and similarity. Namely, if the set specifying the object contains 𝐹, then it also contains:
+(1) every prefix 𝐸 of 𝐹, and
+(2) every history 𝐸 that is similar to 𝐹.
+We now show that linearizability belongs to the class GenLin.
+Lemma 7.1 (Linearizability is in GenLin). Let 𝐹 be any finite history that is linearizable with
+respect to some sequential object O (i.e. a state machine). Then,
+(1) every prefix 𝐸 of 𝐹 is linearizable with respect to O, and
+(2) every history 𝐸 that is similar to 𝐹 is linearizable with respect to O.
+Therefore, GenLin contains the object (set) with all finite histories that are linearizable with respect
+to O.
+Proof. The prefix closure proof in [51][Theorem 4] assumes a definition of linearizability that
+is slightly different than the one we use here; that proof however also holds in our case. For
+completeness, we present the proof. Let 𝐹 = 𝐸𝐸′. Consider any linearization 𝑆 of 𝐹. Then, there is
+an extension 𝐹 ′ of 𝐹 such that 𝑐𝑜𝑚𝑝(𝐹 ′) and 𝑆 are equivalent and <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆. Let 𝑆 = 𝑆𝐸𝑆𝐸′,
+where 𝑆𝐸 is the shortest prefix of 𝑆 with all complete operations in 𝐸. We argue that 𝑆𝐸 is a
+linearization of 𝐸. Let us observe first that 𝑆𝐸 does not have an operation whose invocation appears
+13
+
+Castañeda and Rodríguez
+in 𝐸′: (1) by definition of 𝑆𝐸, the last operation op of 𝑆𝐸 is complete in 𝐸, and (2) if 𝑆𝐸 has such an
+operation op′, then op′ <𝑆𝐸 op, which contradicts that <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆 bacause we certainly have
+op <𝑐𝑜𝑚𝑝 (𝐹 ′) op′. Now, let 𝐸′′ be the extension of 𝐸 obtained by appending all response in 𝑆𝐸 to
+pending operations in 𝐸. It is not hard to see that 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑆𝐸 are equivalent: if there is a
+𝑝𝑖 such that 𝑐𝑜𝑚𝑝(𝐸′′)|𝑝𝑖 ≠ 𝑆𝐸|𝑝𝑖, then simply 𝑐𝑜𝑚𝑝(𝐹 ′) and 𝑆 are not equivalent. We now argue
+that <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑆𝐸, from which we conclude that indeed 𝑆𝐸 is a linearization of 𝐸. Consider any
+op <𝑐𝑜𝑚𝑝 (𝐸′′) op′. Since 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑆𝐸 are equivalent, both op and op′ appear in 𝑆𝐸. Moreover,
+𝑆𝐸 is a sequential history, and hence <𝑆𝐸 must relate op and op′. If both 𝑜𝑝 and 𝑜𝑝′ are complete
+in 𝐸, then op <𝐸 op′, hence op <𝐹 op′, and consequently op <𝑐𝑜𝑚𝑝 (𝐹 ′) op′, and as <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆,
+op <𝑆 op′, which implies op <𝑆𝐸 op′. Consider now the case where at least one of op and op′ are
+pending in 𝐸. Note that if op is pending in 𝐸, then a response to it is appended in 𝐸′′, and hence it
+cannot be the case that op <𝑐𝑜𝑚𝑝 (𝐸′′) op′. Thus, op is complete in 𝐸, and op′ is pending in 𝐸. In 𝐹 ′,
+either a response to op′ is appended, or no responses to it is appended because op′ is complete in 𝐹.
+In any case, we have op <𝑐𝑜𝑚𝑝 (𝐹 ′) op′, and hence op <𝑆 op′, as <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆, which ultimately
+implies that op <𝑆𝐸 op′. Thus, we conclude that <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑆𝐸, and hence 𝑆𝐸 is a linearization
+of 𝐸.
+For the second claim, consider any history 𝐸′ obtained from 𝐸 by appending responses to some
+pending operations and removing invocations of some pending operations, with 𝐸′ and 𝐹 being
+equivalent and ≺𝐸′ ⊆ ≺𝐹. Let 𝐼 and 𝑅𝐸 denote the sets with the invocations and responses removed
+and appended, respectively, to obtain 𝐸′. Consider any linearization 𝑆 of 𝐹. Then, there is an
+extension 𝐹 ′ of 𝐹 such that 𝑐𝑜𝑚𝑝(𝐹 ′) and 𝑆 are equivalent and <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆. Let 𝑅𝐹 denote the
+set with the responses appended to obtain 𝐹 ′. Let 𝐸′′ be any extension of 𝐸 obtained by appending
+to it the responses in 𝑅𝐸 ∪ 𝑅𝐹. We prove that 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑆 are equivalent and <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑆,
+from which follows that 𝑆 is a linearization of 𝐸. As 𝐸′ and 𝐹 are equivalent, it is easy to see
+that 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑐𝑜𝑚𝑝(𝐹 ′) are equivalent (just note that the invocations in 𝐼 do not appear in
+𝑐𝑜𝑚𝑝(𝐹 ′)), and hence 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑆 are equivalent. Now, consider any op <𝑐𝑜𝑚𝑝 (𝐸′′) op′. Observe
+that it cannot be the case that the response of op is in 𝑅𝐸 ∪ 𝑅𝐹. Then, op is completed in 𝐸; however
+op′ might be pending or complete in 𝐸. Note that op <𝑐𝑜𝑚𝑝 (𝐸′′) op′ implies op ≺𝐸′ op′, and hence
+op ≺𝐹 op′, as ≺𝐸′ ⊆ ≺𝐹, from which follows op <𝑐𝑜𝑚𝑝 (𝐹 ′) op′. We thus have <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑐𝑜𝑚𝑝 (𝐹 ′),
+and then <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑆 because <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆. Therefore, 𝑆 is a linearization of 𝐸.
+□
+It can similarly be shown that variants of linearizability such as set-linearizability [75] and
+interval-linearizability [18], among others, are in the class GenLin. The reason is that these variants
+differ from linearizability only in the properties of the linearization 𝑆 of a given execution. In some
+cases, 𝑆 can “deviate” from sequential executions of state machine O, or it might be the case that 𝑆
+is not necessarily a sequential execution, where several operations can occur simultaneously at the
+same time, or even an operation can overlap several operations.
+7.2
+The class DRV
+Let A be any implementation. We already have seen that it is impossible to verify whether A
+is linearizable with respect to some sequential objects. As anticipated, we will see however that
+A can be indirectly verified through an implementation A∗ that can be easily obtained from A.
+The implementation A∗ appears in Figure 6, which assumes in Line 05 that the content of all
+entries of a shared array can be atomically read with a Snapshot operation [1]. It is known that the
+atomic Snapshot operation can be wait-free linearizable implemented using only Read/Write base
+operations, e.g. [1, 61]. Thus, the step in Line 05 can be assumed to be wait-free and atomic, due to
+the modular properties of linearizability [60, 87].
+14
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+Shared Variable:
+𝑁 [1, 2, . . . ,𝑛] = shared array of Read/Write base objects, each initialized to ∅
+Local Persistent Variable:
+𝑠𝑒𝑡𝑖 = a set initialized to ∅
+Operation Apply(op𝑖) is
+(01) 𝑠𝑒𝑡𝑖 ← 𝑠𝑒𝑡𝑖 ∪ {(𝑝𝑖, op𝑖) }
+(02) 𝑁 [𝑖].Write(𝑠𝑒𝑡𝑖)
+(03) Invoke operation Apply(op𝑖) of A
+(04) 𝑦𝑖 ← response from operation Apply(op𝑖) of A
+(05) 𝑠𝑖 ← 𝑁 .Snapshot()
+(06) 𝜆𝑖 ← �
+𝑘∈{1,2,...,𝑛} 𝑠𝑖 [𝑘]
+(07) return (𝑦𝑖, 𝜆𝑖)
+end Verify
+Fig. 6. From A to A∗ ∈ DRV (code of process 𝑝𝑖).
+In A∗, every process 𝑝𝑖 simply announces in a shared memory the next high-level operation op𝑖
+it wants to execute, then obtains a response for op𝑖 using A, atomically reads all operations that
+have been announced so far in the shared memory, storing them all together in a set 𝜆𝑖, and finally
+returns the set together with the response obtained from A. The set 𝜆𝑖 is called the view of (𝑝𝑖, op𝑖).
+As we will see, this simple mechanism, the views, succinctly encode an “sketch” of A’s and A∗’s
+current histories, and this sketch is what makes A∗ strongly verifiable.
+Definition 7.3 (The class DRV). DRV denotes the the class of concurrent implementations
+obtained through the construction in Figure 6.
+7.3
+Analyzing A∗
+A∗ preserves A’s properties. We first show that A∗ preserves progress and correctness properties
+of A. For the discussion in the rest of the section, we disregard the views in the responses of A∗
+(i.e. the sets 𝜆𝑖), unless stated otherwise. Recall that the execution itself of an algorithm denotes the
+history obtained from it.
+Lemma 7.2 (Correctness of A∗). Consider any implementation A and any object O in the class
+GenLin. Then, A is correct with respect to O if and only if A∗ is correct with respect to O (disregarding
+the views in the responses of A∗). Furthermore, A∗ possesses the same progress condition as A, and
+its step complexity is the step complexity of A plus 𝑂(𝑛).
+Proof. First, it is easy to see the segment of code in Lines 01 to 02 is wait-free, as well as the
+segment in Lines 05 to 07. Thus, if A is lock-free (resp. wait-free) then A∗ lock-free (resp. wait-free).
+As for step complexity, it executes one Write before invoking A, and one Snapshot after, which can
+be implemented in 𝑂(𝑛) step using the algorithm in [61]. For correctness, we prove each direction
+separately.
+⇒ Let 𝐸 be any finite execution of A∗. We show that 𝐸 is correct, namely, the history obtained
+from it (denoted 𝐸 as well) belongs to O. Since 𝐸|A ∈ O and O is closed by similarity, both
+by assumption, it suffices to prove that 𝐸 is similar to history 𝐸|A. Consider the history
+𝐸′ obtained from 𝐸 by: (1) appending to 𝐸 the response in 𝐸|A to every operations that is
+pending in 𝐸 but complete in 𝐸|A, and (2) removing every invocation of a pending operation
+in 𝐸 that does not appear in 𝐸|A. From the definition of 𝐸′ and the pseudocode in Figure 6, it
+can be easily verified that 𝐸′ and 𝐸|A are equivalent. Additionally, it holds that ≺𝐸′ ⊆ ≺𝐸 |A: if
+op ≺𝐸′ op′, then we must have that 𝑟𝑒𝑠(op) precedes 𝑖𝑛𝑣(op′) in 𝐸|A because in A∗ operation
+calls to A are nested in operation calls of A∗, and hence op ≺𝐸 |A op′.
+15
+
+Castañeda and Rodríguez
+⇐ Let 𝐸 be any finite history of A. We argue that 𝐸 ∈ O. The asynchrony in the model
+guarantees the existence of the following execution 𝐸′ of A∗:
+(1) for every process 𝑝𝑖, for each of its operations Apply(op𝑖), the corresponding invocation and
+the steps from Lines 01 to 02 appear all together right before the invocation to Apply(op𝑖)
+of A in Line 02, and
+(2) for every process 𝑝𝑖, for each of its operations Apply(op𝑖), the corresponding response and
+the steps from Lines 05 to 07 appear all together right after the response from Apply(op𝑖)
+of A in Line 04 (if there is one), and
+(3) 𝐸 and 𝐸′ are the same history.
+Basically, 𝐸′ is obtained from 𝐸 by adding steps of A∗ right before and after the invocations
+and responses in 𝐸. By assumption, A∗ is correct with respect to O, and hence 𝐸′ ∈ O, from
+which follows that 𝐸 ∈ O, as 𝐸 and 𝐸′ are the same history.
+□
+p1
+p2
+enq(1) : true
+deq() : 1
+Current execution A∗:
+Current execution A:
+Fig. 7. Due to asynchrony, A∗ is able to “fix” some histories of A. The figure depicts a history of A∗ where
+the history of A is not linearizable with respect to the queue, however, the history of A∗ is linearizable. Each
+operation Apply(op) is simply denoted op and the views of the operations of A∗ are not shown.
+The implementation A∗ can be alternatively understood as a mechanism that “fixes” some
+incorrect histories of A, as the example in Figure 7 shows. Nevertheless, Lemma 7.2 implies that
+A∗ cannot fix all incorrect histories of A. For those histories that it is not able to fix, the views
+provide a mechanism to detect they are incorrect, which will be crucial to fulfill the completeness
+requirement of the distributed runtime strong verification problem.
+Tight executions. Let A∗ be any implementation in the class DRV, and consider any finite
+execution 𝐸 of it. We say that 𝐸 is tight if: (a) each of its pending operations has its Write step
+in Line 02 but no Snapshot step in Line 05, (b) the invocation and local step in Line 01 of every
+operation, complete or pending, appear in a sequence right before its Write step in Line 02, and (c)
+the local steps in Lines 06 and 07 and response of every complete operation appear in a sequence
+right after its Snapshot step in Line 05. Basically, in a tight execution, the beginning and end of an
+operation are identified with the Write and Snapshot steps in Lines 02 and 05, respectively.
+Any finite execution 𝐸 of A∗ can be “transformed” into a tight execution𝑇 (𝐸) of A∗ as described
+next: (a) remove the invocation and local step in Line 01 of every pending operation with no Write
+step in Line 02, (b) for each of the remaining operations, complete or pending, move forward the
+invocation and local step in Line 01 in a sequence right before its Write step in Line 02, (c) for each
+of the remaining operations with its Snapshot step in Line 05, first move backwards the local steps
+in Lines 06 and 07 and response (if there are any) in a sequence right after its Snapshot step in
+Line 05, and then insert (if necessary) the missing local steps in Lines 06 and 07 and response to
+complete the sequence that makes the operation complete. We say that 𝑇 (𝐸) is the tight execution
+associated to 𝐸. Observe that indeed 𝑇 (𝐸) is an execution of A∗ because all invocations, responses
+and steps that are moved, forward of backward, or inserted are local to processes, and hence it
+is immaterial when they occur, or if they occur. Moreover, in 𝐸 and 𝑇 (𝐸), the operations obtain
+16
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+the same responses from A and compute same views as the order of invocations to and responses
+from A are not modified to obtain 𝑇 (𝐸), neither the order of Write and Snapshot steps. Intuitively,
+the difference between 𝐸 and 𝑇 (𝐸) is that operations in 𝑇 (𝐸) span a possibly “shorter” interval of
+time.
+Lemma 7.3 (Tight executions and actual executions). Let 𝐸 be any finite execution of A∗.
+Then, 𝑇 (𝐸) is an execution of A∗, and for any object O in GenLin,
+𝐸|A ∈ O =⇒ 𝑇 (𝐸) ∈ O =⇒ 𝐸 ∈ O.
+Proof. As already argued, 𝑇 (𝐸) is an execution of A∗. We will show that (history) 𝑇 (𝐸) is
+similar to (history) 𝐸|A and (history) 𝐸 is similar to (history) 𝑇 (𝐸). These two facts will prove the
+implications because O is closed by similarity, as it belongs to GenLin.
+We argue first that 𝐸 is similar to𝑇 (𝐸). Let 𝐸′ be the history obtained from 𝐸 by: (1) removing the
+invocation of every pending operation that is removed from 𝐸 to obtain 𝑇 (𝐸), and (2) appending
+the response in 𝑇 (𝐸) of every pending operation in 𝐸 whose response is inserted to obtain 𝑇 (𝐸).
+Basically, 𝐸′ is obtained following steps (a) and (c) in the construction from 𝐸 to 𝑇 (𝐸). It is nor
+hard to see that 𝐸 and 𝑇 (𝐸) are equivalent. Consider now any op ≺𝐸′ op′. Hence 𝑟𝑒𝑠(op) precedes
+𝑖𝑛𝑣(op′) in 𝐸′. Observe that it cannot be that 𝑟𝑒𝑠(op) is appended to 𝐸 to obtain 𝐸′, and 𝑇 (𝐸), and
+hence op is complete in 𝐸; op′ is complete or pending. We must have that op ≺𝑇 (𝐸) op′ becase to
+obtain 𝑇 (𝐸), responses of complete operations might only moved backward and invocations of
+complete or pending operations might only moved forward. Therefore we have ≺𝐸 ⊆ ≺𝑇 (𝐸), from
+which follows that 𝐸 is similar to 𝑇 (𝐸).
+We show now that 𝑇 (𝐸) is similar to 𝐸|A. Consider the history 𝐸′ obtained from 𝑇 (𝐸) by: (1)
+removing every invocation of a pending operation in 𝑇 (𝐸) that does not appear in 𝐸|A (any such
+operation executes its Write step in Line 02 but does not executes the invocation in Line 03), and (2)
+appending the response in 𝐸|A to every operations that is pending in 𝑇 (𝐸) but complete in 𝐸|A
+(any such operation operation executes its response in Line 04, but does not execute its Snapshot
+step in Line 05). It is not difficult to verify that 𝐸′ and 𝐸|A are equivalent. It is also true that
+≺𝐸′ ⊆ ≺𝐸 |A: if op ≺𝐸′ op′, then we must have that 𝑟𝑒𝑠(op) precedes 𝑖𝑛𝑣(op′) in 𝐸|A because in A∗
+operation calls to A are nested between the Write and Snapshot steps in Lines 02 and 05, from
+which follows that op ≺𝐸 |A op′. Therefore, 𝑇 (𝐸) is similar to 𝐸|A.
+□
+Lemma 7.3 suggests a way to strongly verify the implementations of DRV through their tight
+executions; the idea is actually very simple. Suppose that somehow processes are able to compute
+𝑇 (𝐸) of any execution 𝐸 of A∗ ∈ DRV. First, Lemma 7.3 shows that 𝑇 (𝐸) is a history of A∗. If 𝐸 is
+not correct, then the lemma implies that 𝑇 (𝐸) is not correct either, and hence it is a witness for A∗
+that can be reported to satisfy completeness; and if 𝐸 is correct, then 𝑇 (𝐸) might be correct or not,
+but in either case strong soundness can be satisfied because if 𝑇 (𝐸) is not correct, it is a witness
+for A∗ that can be reported.
+In the rest of the section, we argue that the views encode the tight execution associated to the
+actual execution of A∗, which opens the possibility to implement the simple idea just described.
+From views to tight executions and vice versa. Let 𝐸 be any finite execution of A∗. In the associated
+tight execution𝑇 (𝐸), let us replace each invocation of 𝑝𝑖 to operation Apply(op𝑖), with the invocation
+pair (𝑝𝑖, op𝑖), and replace each response from operation Apply(op𝑖) to 𝑝𝑖 (if there is one) with the
+set with all invocation pairs (𝑝𝑗, op𝑗) that precedes the response. Figure 8 depicts an example of
+the replacement. Since invocations and responses in 𝑇 (𝐸) are associated to the Write and Snapshot
+steps in Lines 05 and 02, respectively, the view returned by any operation in 𝑇 (𝐸) is exactly the set
+just defined. We show that a “sketch” of the history 𝑇 (𝐸) can be directly obtained from the views
+17
+
+Castañeda and Rodríguez
+of operations in 𝐸. First, from the properties of the Snapshot [1] and the pseudocode of A∗, we
+obtain the following:
+Remark 7.1. Consider the views 𝜆𝑖 and 𝜆𝑗 in the responses of any pair of completed operations
+Apply(op𝑖) and Apply(op𝑗) by 𝑝𝑖 and 𝑝𝑗 (possibly with 𝑝𝑖 = 𝑝𝑗) in any execution of A∗. The next
+properties are satisfied.
+(1) Self-inclusion: (𝑝𝑖, op𝑖) ∈ 𝜆𝑖.
+(2) Containment comparability: 𝜆𝑖 ⊆ 𝜆𝑗 ∨ 𝜆𝑗 ⊆ 𝜆𝑖.
+(3) Process sequentiality: if 𝑝𝑖 = 𝑝𝑗 ∧ op𝑖 ≠ op𝑗, then (𝑝𝑖, op𝑖) ∉ 𝜆𝑗 ∨ (𝑝𝑗, op𝑗) ∉ 𝜆𝑖.
+For any tight execution 𝐸 of A∗, let 𝜆𝐸 will denote the set with all 4-tuples (𝑝𝑖, op𝑖,𝑦𝑖, 𝜆𝑖) such
+that (𝑦𝑖, 𝜆𝑖) is the response of operation Apply(op𝑖) of 𝑝𝑖 in 𝐸 (see Figure 8). We now explain that a
+well-formed history 𝑋 (𝜆𝐸) can be obtained from 𝜆𝐸, and explain in what sense 𝑋 (𝜆𝐸) is a sketch
+of 𝐸. The construction that follows is from [18].
+By Remark 7.1 (2), all distinct views that appear in 𝜆𝐸 can be ordered in strictly containment
+ascending order: 𝜎1 ⊂ 𝜎2 ⊂ . . . ⊂ 𝜎𝑚. Let 𝜎0 denote ∅. For each 𝑘 = 1, 2, . . . ,𝑚 (in ascending order),
+𝑋 (𝜆𝐸) is iteratively obtained following the next two steps in order (see Figure 8 for an example):
+(1) For each invocation pair (𝑝𝑖, op𝑖) ∈ 𝜎𝑘 \ 𝜎𝑘−1, the invocation to operation Apply(op𝑖) by 𝑝𝑖 is
+appended to 𝑋 (𝜆𝐸); the invocations are appended in any arbitrary order.
+(2) For each (𝑝𝑖, op𝑖,𝑦𝑖, 𝜆𝑖) ∈ 𝜆𝐸 with 𝜆𝑖 = 𝜎𝑘, the response from operation Apply(op𝑖) with
+output (𝑦𝑖, 𝜆𝑖) by 𝑝𝑖 is appended to 𝑋 (𝜆𝐸); the responses are appended in any arbitrary order.
+p1
+p2
+Apply(op1) : a
+p3
+Apply(op′
+1) : b
+Apply(op2) : c
+Apply(op3) : d
+(p1, op1)
+(p1, op′
+1)
+(p1, op1)
+(p2, op2)
+(p1, op1)
+(p1, op′
+1)
+(p2, op2)
+(p3, op3)
+(p1, op1)
+(p1, op′
+1)
+(p2, op2)
+(p3, op3)
+λE = {(p1, op1, a, view), (p1, op′
+1, b, view′), (p3, op3, d, view′′)}
+view = {(p1, op1)}
+view′ = {(p1, op1), (p1, op′
+1), (p2, op2)}
+view′′ = {(p1, op1), (p1, op′
+1), (p2, op2), (p3, op3)}
+Fig. 8. The figure shows the history of a tight execution 𝐸 of A∗. The invocation pair and the view of each
+operation is depicted. The set 𝜆𝐸 is depicted too. It is easy to check that the construction from 𝜆𝐸 produces a
+history 𝑋 (𝜆𝐸) that is equivalent to 𝐸 with ≺𝐸 = ≺𝑋 (𝜆𝐸), and hence 𝐸 and 𝑋 (𝜆𝐸) are similar to each other.
+In each of the steps of the construction above, either a set of invocations or responses are placed
+in some arbitrary sequential order. For any of these orders, the resulting history has the same
+relation ≺ over pending and complete operations, by construction, and hence all possible histories
+obtained in this way are similar to one another. Thus, in fact, 𝑋 (𝜆𝐸) denotes an equivalence class of
+histories.1 By similarity-closure of GenLin, we have:
+1In the parlance of [18], 𝑋 (𝜆𝐸) is an interval-sequential history, i.e., an alternating sequence of non-empty sets with only
+either invocations or responses, starting with a set of invocations. Interval-sequential histories are used in [18] to define
+interval-linearizability, a generalization of linearizability where, roughly speaking, an operation is allowed to overlap other
+operations in an interval-linearization of an execution.
+18
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+Claim 7.1. For every object O in GenLin and every tight execution 𝐸 of A∗, either all histories
+of 𝑋 (𝜆𝐸) are in O or none of them is in O.
+By abuse of notation, we let 𝑋 (𝜆𝐸) denote any history of the equivalence class, unless stated
+otherwise.
+The duality between histories and sets of views has been investigated in [18], where it is shown
+that the construction above is a bijection between (equivalence classes of) well-formed finite
+histories and sets of 4-tuples (𝑝𝑖, op𝑖,𝑦𝑖, 𝜆𝑖) whose views satisfy the properties in Remark 7.1. The
+views can be understood as a static mechanism that captures the dynamic real-time order of
+operations in a history. Lemma 7.1 in [18] directly implies that 𝑋 (𝜆𝐸) is indeed an accurate sketch
+of 𝐸 in the following sense:
+Lemma 7.4 (Views are sketches of tight executions). For any tight execution 𝐸 of A∗,
+𝐸 and 𝑋 (𝜆𝐸) are equivalent with ≺𝐸 = ≺𝑋 (𝜆𝐸), hence the histories are similar to one another.
+As every O ∈ GenLin is closed by similarity, we have:
+Corollary 7.1. For any tight execution 𝐸 of A∗ and every object O in GenLin,
+𝐸 ∈ O ⇐⇒ 𝑋 (𝜆𝐸) ∈ O.
+Claim 7.2. If 𝐻 is a history of an implementation B (not necessarily in the class DRV), then any
+history 𝐻 ′ that is equivalent to 𝐻 with ≺𝐻 = ≺𝐻′ is a history of B as well.
+Proof. Since 𝐻 is a history of B, there is an execution 𝐸 of B such that 𝐻 is the history obtained
+from 𝐸. History 𝐻 has the form 𝐼1𝑅1𝐼2𝑅2 . . ., where each 𝐼𝑥 (resp. 𝑅𝑥) is a non-empty sequence
+of invocations (resp. responses); note that the invocations in 𝐼𝑥 (resp. responses in 𝑅𝑥) does not
+necessarily appear in a continuos sequence in 𝐸. The main observation to prove the claim is that for
+every history 𝐻 ′ such that is equivalent to 𝐻 with ≺𝐻 = ≺𝐻′, we must have that 𝐻 ′ = 𝐼 ′
+1𝑅′
+1𝐼 ′
+2𝑅′
+2 . . .
+with 𝐼 ′
+𝑥 (resp 𝑅′
+𝑥) being a permutation of 𝐼𝑥 (resp. 𝑅𝑥). Then, consider the execution 𝐸′ obtained
+from 𝐸 as follows: (1) for each 𝐼𝑥, first move all invocations in 𝐼𝑥 to the position of the first invocation
+in 𝐼𝑥, and then permute them according to 𝐼 ′
+𝑥, and similarly (2) for each 𝑅𝑥, first move all responses
+in 𝑅𝑥 to the position of the last response in 𝑅𝑥, and then permute them according to 𝑅′
+𝑥. Observe
+that 𝐸′ is an execution of B because only invocation and responses of 𝐸, which are local steps, are
+modified to obtain 𝐸′, and it is immaterial when these local steps actually occur, as long as the
+specification of B is satisfied, as it happens in 𝐸′. By construction, 𝐻 ′ is the history obtained from
+𝐸′, and therefore, 𝐻 ′ is a history of B.
+□
+Finally, Lemma 7.4 and Claim 7.2 imply:
+Corollary 7.2. For any tight execution 𝐸 of A∗, 𝑋 (𝜆𝐸) is a history of A∗.
+7.4
+A note on the class DRV and refined task solvability
+The views mechanism was introduced in [18] (implicitly defined in [44] too) in order to extend the
+task specification formalism [58] such that it is able to capture linearizable sequential long-lived
+objects. The resulting formalism is called multi-shot refined tasks, where processes are required to
+produce outputs and implicitly produce views. It turns out that multi-shot refined tasks are strictly
+more expressive than linearizability, and are equally expressive as interval-linearizability, also
+introduced in [18]: for every interval-sequential object, there is an equivalent multi-shot refined
+task, and vice versa. From this perspective, the implementations in DRV solve the corresponding
+equivalent multi-shot refined tasks, producing explicit views, which is what makes them runtime
+verifiable.
+19
+
+Castañeda and Rodríguez
+8
+THE CLASS DRV IS DISTRIBUTED RUNTIME STRONGLY VERIFIABLE
+A wait-free strong verifier for DRV. Lemmas 7.3 and 7.4 in the previous section are the basis of the
+wait-free strong verifier VO in Figure 9. The idea of the verifier is simple. For any finite execution 𝐸
+of the verifier, the views in 𝜆𝑇 (𝐸 |A∗) sketch the tight execution 𝑇 (𝐸|A∗) associated to the current
+execution 𝐸|A∗ of A∗ (Lemma 7.4), and tight executions suffice to fulfill strong soundness and
+completeness (Lemma 7.3). Thus, in VO the processes simply exchange their views using a Write
+and a Snapshot (Lines 07 to 09) and then each process locally tests if the execution it reads from
+the shared memory is correct (Lines 10 to 12).
+Shared Variables:
+𝑀 [1, 2, . . . ,𝑛] = shared array of Read/Write base objects, each initialized to ∅
+Operation Verify(A∗ ∈ DRV) is
+(01) 𝑟𝑒𝑠𝑖 ← ∅
+(02) while true do
+(03)
+𝑜𝑝𝑖 ← non-deterministically chosen operation that does not appear in 𝑟𝑒𝑠𝑖
+(04)
+Invoke operation Apply(𝑜𝑝𝑖) of A∗
+(05)
+(𝑦𝑖, 𝜆𝑖) ← response from operation Apply(𝑜𝑝𝑖) of A∗
+(06)
+𝑟𝑒𝑠𝑖 ← 𝑟𝑒𝑠𝑖 ∪ {(𝑝𝑖,𝑜𝑝𝑖, 𝑦𝑖, 𝜆𝑖) }
+(07)
+𝑀 [𝑖].Write(𝑟𝑒𝑠𝑖)
+(08)
+𝑠𝑖 ← 𝑀.Snapshot()
+(09)
+𝜏𝑖 ← �
+𝑘∈{1,2,...,𝑛} 𝑠𝑖 [𝑘]
+(10)
+if 𝑋 (𝜏𝑖) ∉ O then
+(11)
+report (ERROR,𝑋 (𝜏𝑖))
+(12)
+end if
+(13) end while
+end Verify
+Fig. 9. A wait-free strong verifier VO for any object O in GenLin, and any implementation of the class DRV
+(code of process 𝑝𝑖).
+Although VO relies on a simple idea, proving it correct is not simple at all. The main reason is
+that in its Snapshot step in Line 08, a process might obtain only a proper subset of the set 𝜆𝑇 (𝐸 |A∗),
+and hence the history 𝑋 (𝜏𝑖) used in the test in Line 10 might not be exactly𝑇 (𝐸|A∗). This situation
+can happen due to asynchrony because some processes might have already obtained a response
+from A∗ in 𝐸|A∗ but no written yet their responses in the 𝑀. To deal with this issue, Lemmas 8.1
+and 8.2 below show useful properties of the histories computed in Line 10.
+As mentioned above, Snapshot can be wait-free implemented using only Read/Write base objects,
+and moreover, there are implementations with 𝑂(𝑛) step complexity [61]. Thus, clearly VO is a
+wait-free, and uses only Read/Write base objects, and every iteration of the while loop takes 𝑂(𝑛)
+steps. Thus we have:
+Claim 8.1. The verifier VO in Figure 9 is wait-free, uses only Read/Write base objects with step
+complexity 𝑂(𝑛).
+To prove the correctness of VO, we analyze only its infinite executions in which the sequence of
+local computation steps in Lines 09 to 12 of a process 𝑝𝑖 appear all together right after the previous
+Snapshot step of 𝑝𝑖 in Line 08 (which is part of the same loop iteration). Restricting our attention
+to these executions facilitates the discussion and proves the verifier to be correct in all cases, as
+it is immaterial when these steps actually occur. This restriction can also be seen from a slightly
+different perspective: given any infinite execution of the verifier, the local steps in Lines 09 to 12
+of a process 𝑝𝑖 can be moved backwards to be right next to the previous Snapshot step of 𝑝𝑖 in
+Line 08, and the processes still make the same decisions in the modified execution.
+20
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+Lemma 8.1. Let 𝐸 be any infinite execution of the wait-free verifier VO in Figure 9 with an arbitrary
+input A∗ ∈ DRV. Consider any finite prefix 𝐸′ of 𝐸 whose last sequence of steps correspond to the
+steps in Lines 08 to 12 of a process 𝑝𝑖, and let 𝜏 ′ denote the content of 𝜏𝑖 of 𝑝𝑖 at the end of 𝐸′. Then:
+(1) 𝑋 (𝜏 ′) is a history of A∗;
+(2) for every object O in GenLin, 𝑇 (𝐸′|A∗) ∈ O =⇒ 𝑋 (𝜏 ′) ∈ O.
+Proof. By Lemma 7.3,𝑇 (𝐸′|A∗) is an execution of A∗. By analyzing𝑇 (𝐸′|A∗), we will conclude
+that 𝑋 (𝜏 ′) is a history of A∗. By the definition of 𝐸′, it follows that 𝜏 ′ contains all 4-tuples that
+appear in 𝑀 at the end of 𝐸′. Due to asynchrony, it is possible that not all 4-tuples in 𝜆𝑇 (𝐸′|A∗)
+appear in 𝑀 at the end of 𝐸′; the reason is that it can be the case that in 𝐸′ a process executes its
+Snapshot in Line 05 of A∗, and hence the 4-tuple of the corresponding operation is in 𝑇 (𝐸′|A∗),
+by definition of tight executions, but does not execute its Write step in Line 07 of VO. Hence, at the
+end of 𝐸′, the response of the last operation in 𝑇 (𝐸′|A∗) of a process in might be “missing” in 𝜏 ′.
+We thus have that 𝜏 ′ ⊆ 𝜆𝑇 (𝐸′|A∗); moreover, 𝜆𝑇 (𝐸′|A∗) \ 𝜏 ′ has at most one 4-tuple for each process.
+We will modify 𝑇 (𝐸′|A∗) to obtain an execution 𝐹 of A∗ whose history is precisely 𝑋 (𝜏 ′). As
+already said, each 4-tuple that appears in 𝜆𝑇 (𝐸′|A∗) \𝜏 ′ corresponds to an operation that is complete
+in 𝑇 (𝐸′|A∗) and that operation is the last one of the corresponding process in the execution. Let 𝐹
+be the execution obtained from 𝑇 (𝐸′|A∗) as follows: for each 4-tuple in 𝜆𝑇 (𝐸′|A∗) \ 𝜏 ′, remove the
+steps in Lines 05 to 07 of A∗ (see Figure 6) of the operation corresponding to the 4-tuple. Clearly, 𝐹
+is an execution of A∗ as only the last steps and response of the last operation of some processes
+are removed from 𝑇 (𝐸′|A∗). Moreover, it is a tight execution as each of its pending operations
+does not execute the Snapshot step in Line 05 of A∗. By construction, we have that 𝜏 ′ = 𝜆𝐹, from
+which follows that 𝑋 (𝜏 ′) and 𝑋 (𝜆𝐹) are equivalent with ≺𝑋 (𝜏′) = ≺𝑋 (𝜆𝐹 ). Finally, 𝐹 and 𝑋 (𝜆𝐹) are
+equivalent with ≺𝐹 = ≺𝑋 (𝜆𝐹 ), by Lemma 7.4, and hence 𝑋 (𝜏 ′) is a history of A∗, by Claim 7.2 and
+as 𝐹 is an execution of A∗.
+From the previous discussion, it is easy to see that 𝐹 is similar to 𝑇 (𝐸′|A∗). Let 𝐹 ′ be the history
+obtained from 𝐹 by appending the responses the responses in the 4-tuples of 𝜆𝑇 (𝐸′|A∗) \ 𝜏 ′. It
+is not difficult to see that 𝐹 ′ and 𝑇 (𝐸′|A∗) are equivalent with ≺𝐹 ′ = ≺𝑇 (𝐸′|A∗). Since any object
+O ∈ GenLin is closed by similarity, we conclude that𝑇 (𝐸′|A∗) ∈ O =⇒ 𝐹 ∈ O. From the discussion
+above, we know that 𝐹 and 𝑋 (𝜏 ′) are similar to one another, and hence 𝐹 ∈ O ⇐⇒ 𝑋 (𝜏 ′), and
+therefore 𝑇 (𝐸′|A∗) ∈ O =⇒ 𝑋 (𝜏 ′) ∈ O.
+□
+Lemma 8.2. Let 𝐸 be any infinite execution of the wait-free verifier VO in Figure 9 with an arbitrary
+input A∗ ∈ DRV. Consider any finite prefix 𝐸′ of 𝐸. There is a finite prefix 𝐹 of 𝐸 such that for every
+finite prefix 𝐸′′ of 𝐸 that has 𝐹 as one its prefixes and whose last sequence of steps correspond to the
+steps in Lines 08 to 12 of a process 𝑝𝑖, it holds that 𝑇 (𝐸′|A∗) is similar to a prefix of 𝑋 (𝜏 ′′), where 𝜏 ′′
+denotes the content of 𝜏𝑖 of 𝑝𝑖 at the end of 𝐸′′.
+Proof. Lemma 7.4 implies that we can concentrate on 𝑋 (𝜆𝑇 (𝐸′|A∗)) to reason about 𝑇 (𝐸′|A∗).
+Observe that due to asynchrony, it is possible that not all 4-tuples in 𝜆𝑇 (𝐸′|A∗) appear in 𝑀 at the
+end of 𝐸′; the reason is that it is possible that in 𝐸′ a process executes its Snapshot in Line 05 of A∗
+(see Figure 6), and hence the 4-tuple of the corresponding operation is in 𝜆𝑇 (𝐸′|A∗), by definition
+of tight executions, but does not execute its Write step in Line 07 of VO. Thus, at the end of 𝐸′,
+the response of the last operation in 𝜆𝑇 (𝐸′|A∗) of a process might be “missing” in 𝑀. Since 𝐸 is
+infinite and by assumption fair (see Section 2), there is a finite prefix 𝐹 of it in which all 4-tuples in
+𝜆𝑇 (𝐸′|A∗) appear in 𝑀 at the end of 𝐹; note that at the of 𝐹, 𝑀 might contain 4-tuples of operations
+that do not appear in 𝜆𝑇 (𝐸′|A∗). We claim that 𝐹 is the prefix of 𝐸 with the desired property.
+Let 𝐸′′ be any finite prefix of 𝐸 that has 𝐹 as one its prefixes and whose last sequence of steps
+correspond to the steps in Lines 08 to 12 of a process 𝑝𝑖, and let 𝜏 ′′ denotes the content of 𝜏𝑖 of
+21
+
+Castañeda and Rodríguez
+𝑝𝑖 at the end of 𝐸′′. It is not difficult to verify that 𝐸′ is a prefix of 𝐹. Using similar arguments as
+above, it can be argued that 𝜏 ′′ ⊆ 𝜆𝑇 (𝐸′′|A∗). Moreover, the election of 𝐹 and the definition of tight
+execution give that 𝜆𝑇 (𝐸′|A∗) ⊆ 𝜏 ′′. It directly follows from the definition of tight execution that if
+an operation is pending in 𝑇 (𝐸′|A∗), then in 𝐸′ it does not execute its Snapshot step in Line 05
+of A∗ (see Figure 6). We thus have that the view of any 4-tuple in 𝜏 ′′ \𝜆𝑇 (𝐸′|A∗) contains the largest
+view among the views in the 4-tuples in 𝜆𝑇 (𝐸′|A∗).
+We will now reason how 𝑋 (𝜆𝑇 (𝐸′|A∗)) and 𝑋 (𝜏 ′′) are constructed from 𝜆𝑇 (𝐸′|A∗) and 𝜆𝜏′′, respec-
+tively. Let 𝜎′
+1 ⊂ 𝜎′
+2 ⊂ . . . ⊂ 𝜎′
+𝑘′ and 𝜎′′
+1 ⊂ 𝜎′′
+2 ⊂ . . . ⊂ 𝜎′′
+𝑘′′ be respectively the distinct views in
+𝜆𝑇 (𝐸′|A∗) and 𝜏 ′′, ordered in ascending order by containement. For the reasons above exposed, we
+have that (1) 𝑘′ ≤ 𝑘′′, (2) 𝜎′
+ℓ = 𝜎′′
+ℓ , for 1 ≤ ℓ ≤ 𝑘′, and (3) 𝜎′
+𝑘′ ⊂ 𝜎′′
+ℓ , for 𝑘′ < ℓ ≤ 𝑘′′. Therefore,
+the construction of 𝑋 (𝜆𝑇 (𝐸′|A∗)) and 𝑋 (𝜏 ′′) from 𝜆𝑇 (𝐸′|A∗) and 𝜏 ′′ use the same first 𝑘′ views. Let
+𝑆 be the subset of 𝜏 ′′ with all 4-tuples whose views are subset of 𝜎′
+𝑘′. Note that 𝜆𝑇 (𝐸′|A∗) ⊆ 𝑆;
+intuitively, the difference between 𝑆 and 𝜆𝑇 (𝐸′|A∗) is that 𝜆𝑇 (𝐸′|A∗) might be missing the response
+of the last operation of some processes and the views of these operation are contained by 𝜎′
+𝑘′.
+Hence, 𝑆 \ 𝜆𝑇 (𝐸′|A∗) has at most one 4-tuple for each process. Let 𝐻 be the shortest prefix of
+𝑋 (𝜆𝑇 (𝐸′|A∗)) containing all operations whose views are subset of 𝜎′
+𝑘′. From the definition of 𝑆 and
+the construction of 𝑋 (𝑆), it directly follows that 𝐻 and 𝑋 (𝑆) are similar to each other. To conclude
+the proof, we argue that 𝑋 (𝜆𝑇 (𝐸′|A∗)) is similar to 𝐻. Let 𝑋 ′ be the history obtained by appending
+to 𝑋 (𝜆𝑇 (𝐸′|A∗)) the responses in the 4-tuples of 𝑆 \ 𝜆𝑇 (𝐸′|A∗). From the discussion above, it can be
+concluded that 𝑋 ′ and 𝐻 are equivalent and ≺𝑋 ′ = ≺𝐻. Therefore, 𝑋 (𝜆𝑇 (𝐸′|A∗)) is similar to 𝐻, and
+hence 𝑇 (𝐸′|A∗) is similar to 𝐻 too.
+□
+We now are finally ready to prove that GenLin is strongly verifiable for the class DRV of
+implementations.
+Theorem 8.1 (Distributed runtime strong verifiability of GenLin for the class DRV).
+Let O be any object in the class GenLin. The verifier VO in Figure 9 is a wait-free strong verifier for
+the correctness of O for the class DRV of implementations. Furthermore, VO satisfies the following
+properties.
+(1) Efficiency. It uses only Read/Write base objects with step complexity 𝑂(𝑛).
+(2) Soundness for correct executions of A. In each infinite execution 𝐸 of it with input A∗, if
+for every finite prefix 𝐸′ of 𝐸, it holds that 𝐸′|A ∈ O (i.e. the history of A, the underlaying
+implementation of A∗, is correct), then no process reports ERROR in 𝐸.
+(3) Stability. For every infinite execution 𝐸 of it in which at least one process reports ERROR, there
+is a finite prefix of it such that it is reported ERROR in every new loop iteration starting after
+the prefix.
+Proof. As it is shown in Claim 8.1, the verifier VO is wait-free, uses only Read/Write base
+objects with step complexity 𝑂(𝑛), and hence it satisfies the efficiency property.
+We now argue that VO is a strong verifier for the correctness of O, namely, it satisfies strong
+soundness and completeness; we also argue that it satisfies soundness for correct executions of A
+and stability. Consider any infinite execution 𝐸 of VO with an arbitrary input implementation
+A∗ ∈ DRV.
+• Strong soundness. Suppose that there is a finite prefix 𝐸′ of 𝐸, whose last sequence of steps
+correspond to the steps in Lines 08 to 12 of a process 𝑝𝑖, and 𝑝𝑖 reports ERROR at the end
+of 𝐸′. Clearly, 𝑝𝑖 reports ERROR because 𝑋 (𝜏 ′) ∉ O, where 𝜏 ′ denotes the content of 𝜏𝑖 of 𝑝𝑖
+at the end of 𝐸′; hence 𝑝𝑖 reports the history 𝑋 (𝜏 ′). By Lemma 8.1 (1), 𝑋 (𝜏 ′) is a history of
+A∗, and hence is a witness for A∗. Thus, strong soundness is satisfied.
+22
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+• Soundness for correct executions of A. Consider any finite prefix 𝐸′′ of 𝐸, whose last sequence
+of steps correspond to the steps in Lines 08 to 12 of a process 𝑝𝑖. By Lemmas 7.3 and 8.1 (2),
+𝐸′′|A ∈ O =⇒ 𝑇 (𝐸′′|A∗) ∈ O =⇒ 𝑋 (𝜏 ′′) ∈ O, where 𝜏 ′′ denotes the content of 𝜏𝑖 of 𝑝𝑖 at
+the end of 𝐸′′. Thus, 𝑝𝑖 does not report ERROR at the end of 𝐸′′.
+• Completeness and stability. Consider any finite prefix 𝐸′ of 𝐸 such that 𝐸′|A∗ ∉ O. By
+Lemma 7.3, 𝐸′|A∗ ∉ O =⇒ 𝑇 (𝐸′|A∗) ∉ O. Lemma 8.2 guarantees the existence of a finite
+prefix 𝐹 of 𝐸 such that for every finite prefix 𝐸′′ of 𝐸 that has 𝐹 as one its prefixes and
+whose last sequence of steps correspond to the steps in Lines 08 to 12 of a process 𝑝𝑖, it
+holds that 𝑇 (𝐸′|A∗) is similar to a prefix 𝐻 of 𝑋 (𝜏 ′′), where 𝜏 ′′ denotes the content of 𝜏𝑖
+of 𝑝𝑖 at the end of 𝐸′′. Since O is closed by similarity, we have 𝑇 (𝐸′|A∗) ∉ O =⇒ 𝐻 ∉ O,
+and since it is closed by prefixes, 𝐻 ∉ O =⇒ 𝑋 (𝜏 ′′) ∉ O. Thus we conclude that 𝑝𝑖 reports
+(ERROR,𝑋 (𝜏 ′′)) at the end of 𝐸′′. By Lemma 8.1 (1), 𝑋 (𝜏 ′′) is a history of A∗, and hence is
+a witness for A∗. Therefore, completeness is satisfied. Observe that stability is satisfied too
+as the analysis holds for every such 𝐸′′ having 𝐹 as a prefix.
+□
+Runtime self-enforced correct implementations for GenLin. A by-product of Theorem 8.1 is a
+simple generic methodology for developing runtime self-enforced correct implementations for any
+object in GenLin. Roughly speaking, a self-enforced implementation produces verfied histories up
+to a moment, which might never happen, where only ERROR is reported.
+Consider the strong verifier VO in Figure 9 and let A be any implementation. In Theorem 8.2
+below, we consider the implementation V𝑂,A∗ obtained from VO and A∗ as defined at the end of
+Section 3. Recall that V𝑂,A∗ has a single high-level operation Apply(op𝑖) that executes Lines 04
+to 09 of VO with A∗, and returns 𝑟𝑒𝑠𝑝𝑖 if 𝑋 (𝜏𝑖) ∈ O, and returns (ERROR,𝑋 (𝜏𝑖)) otherwise. By
+Lemmas 7.3 and 8.1 (2), any witness 𝑋 (𝜏𝑖) for A∗ implies that the current execution of A is not
+correct. Thus, although 𝑋 (𝜏𝑖) is not necessarily a witness for A, it might be helpful for debugging A.
+Furthermore, at any time V𝑂,A∗ is able to produce a certificate of the current computation, i.e., a
+history that might not be the current one but it is indistinguishable to it. Therefore, our runtime self-
+enforced implementations allow the design of concurrent systems with accountable and forensic
+guarantees.
+Theorem 8.2 (Runtime self-enforced correct implementations for GenLin). Let O be any
+object in GenLin and A be any implementation. Consider the implementation VA∗,O obtained through
+the strong verifier VO in Figure 9. Then,
+(1) VA∗,O and A have the same progress condition, and
+(2) if A is correct with respect to O, then VA∗,O is correct with respect to O; otherwise every finite
+execution of VA∗,O is correct with respect to O up to a prefix (which does not necessarily exist)
+where every new operation returns ERROR together with a witness for A∗.
+Proof. First, since VO is wait-free, by Theorem 8.1, and A and A∗ have the same progress
+condition, by Lemma 7.2, we have that VA∗,O and A have the same progress condition.
+Observe that, by the definition of VA∗,O, every infinite execution 𝐸 of V𝑂 (A∗) is naturally
+mapped to a unique infinite execution 𝐸′ of VA∗,O, where the histories of A and A∗ are exactly
+the same in both executions, and every process passes through essentially the same sequence of
+local states. Basically, 𝐸′ is obtained from 𝐸 by respectively replacing the beginning and end of a
+loop iteration with the invocation and response of the operation invoked in the iteration.
+Consider the case where A is correct with respect to O, and let 𝐸 be any infinite execution
+of VA∗,O. The soundness for correct executions of A property of VO (see Theorem 8.1) implies
+that no process reports ERROR in 𝐸. Thus, every operation returns the same response in 𝐸|A and 𝐸.
+23
+
+Castañeda and Rodríguez
+Consider any finite prefix 𝐸′ of 𝐸. Using a similar reasoning as in previous proofs, it can be shown
+that 𝐸′ is similar to 𝐸′|A, and hence 𝐸′|A ∈ O =⇒ 𝐸′ ∈ O, as O is closed by similarity. Therefore,
+VA∗,O is correct with respect to O, as A is correct with respect to O.
+Suppose now that A is not correct with respect to O, and let 𝐸 be any infinite execution of VA∗,O.
+Consider any finite prefix 𝐸′ of 𝐸. The argument in the previous paragraph shows that if 𝐸′|A ∈ O,
+then no process reports ERROR in 𝐸′, and hence 𝐸′ ∈ O. Thus consider the case 𝐸′|A ∉ O.
+The completeness and stability properties of VO (see Theorem 8.1) implies that eventually all
+new operations in 𝐸 report ERROR together with a witness 𝑋 for A∗. To conclude the proof,
+for the sake of contradiction, suppose that no process reports ERROR in 𝐸′ but 𝐸′ ∉ O. Using a
+similar reasoning as in previous proofs, it can be shown that 𝐸′ is similar to 𝑇 (𝐸′|A∗), and hence
+𝐸′ ∉ O =⇒ 𝑇 (𝐸′|A∗) ∉ O, as O is closed by similarity. Since no process reports ERROR in 𝐸′, all
+invocation an responses that appear in it have been written in the shared memory 𝑀 at the end
+of 𝐸′. Thus the last process that takes its Snapshot in Line 08 of VO reads the views of all these
+operations, and hence the history it computes in Line 08 is 𝑇 (𝐸′|A∗), from which follows that it
+reports ERROR in 𝐸′, as we already saw that 𝑇 (𝐸′|A∗) ∉ O. We have reached a contradiction.
+□
+At first glance, one might think that the implementation VA∗,O in Theorem 8.2 is somehow
+verifying A, hence contradicting the impossibility in Theorem 5.1; there is no contradiction however.
+As A∗ is able to “fix” some executions of A (see Section 7), it is possible that in an execution of
+VA∗,O the history of A is not correct, but no process reports ERROR, which happens because A∗
+is able to fix that execution of A.
+9
+EXTENSIONS
+Base objects of bounded size. The implementation A∗ (Figure 6) and the verifier VO (Figure 9)
+use shared object of unbounded size. This unrealistic assumption can be removed by representing
+sets as linked lists. For A∗, each entry in the shared memory 𝑁 contains the first node of a single
+linked list with the items in the set of the process associated to that entry. Every time a process adds
+an item to its set, it creates a new node, links it to the first node of the list (which then becomes
+the second node) and then writes the new node in its entry of the array. A snapshot operation
+in A∗ now returns a vector of positions in the linked lists, hence the elements in the sets in the
+snapshot are the nodes that are accesible from the positions. The sets in VO can be represented in
+the same way.
+Decoupling clients and verifiers, and lightweight verification. Constructing 𝑋 (𝜏𝑖) in Line 10 of VO
+(Figure 9) can be locally computed in polynomial time in the number of operations in 𝜏𝑖 (see the
+construction in Section 7.3). Thus the local test in that line is computed in the time it takes to test if
+𝑋 (𝜏𝑖) is in O, plus a polynomial overhead. This polynomial overhead is desirable for linearizability
+since it is known that, for some sequential objects, linearizability of an execution can be decided in
+polynomial time [16, 31].
+In the verifier VO (Figure 9), the clients (i.e. the processes that use A∗ (Figure 6) to solve some
+distributed problem) are in charge of verifying the current execution of A∗. This might slow down
+the client’s computation, which in general is undesirable. We can deal with this issue by designating
+a group of processes, the verifiers, to exclusively solve the verification task. That is, while the
+clients use A∗ and extract the sketch of the current execution (i.e. the communication interface),
+the verifiers test whether the sketch is correct or not (i.e. the monitoring system). Furthermore,
+using the snapshot implementations in [7, 90], each client only incurs in a constant or logarithmic
+step overhead in every invocation to a high-level operation of A∗. More concretely, by using any
+of those wait-free snapshots implementations, each client in A∗ returns a reference of a snapshot
+of 𝑁 instead of its view (obtained through a snpashot). Such snapshot references can be obtained
+24
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+in constant [90] or logarithmic [7] number of steps. Once a process (a verifier in our case) has
+a snapshot reference, it can obtain the actual snapshot (and hence the corresponding view) in
+𝑂(𝑛 log𝑛) steps [7], or in a number of steps proportional to the number of updates in 𝑀 after the
+reference was taken [90]. Thus, in VO, each client writes its snapshot reference in 𝑀 and continues
+with the next loop, i.e., it does not verify the sketch in 𝑀. The verifiers periodically read the sketch
+so far in 𝑀 of VO, and test if it is correct, namely, they execute in an infinite loop lines 08 to 12
+of VO. The resulting runtime verified system satisfies strong soundness and completeness, as long
+as not all verifiers crash in an execution.
+Verifying obstruction-free and blocking implementations, and task solvability. The proposed inter-
+active model can be modified to include obstruction-free [59] or blocking [59] implementations. The
+difference is that the interaction between the implementation and a verifier might be finite because
+the implementation might block; thus the required properties, strong soundness and completeness,
+need to be satisfied even if the interaction is finite.
+Similarly, task solvability can be strongly runtime verified. A task is a one-shot distributed
+problem [58]. It is known that any task can be equivalently modelled and a one-shot interval-
+sequential interval-linearizable object [18], which belongs to GenLin, since GenLin includes interval-
+linearizability. Again, the difference is that the interaction with this object is finite. Moreover,
+since every process invokes exactly one high-level operation, now a process can declare that the
+computation is correct if it reads from the shared memory the views of all processes in the system
+and the corresponding history is correct.
+Extension to other models of computation. The proposed implementations assume a fixed number
+of processes that can participate in the computation. The implementations can however be adapted
+to shared memory models with an unbounded number of processes, i.e., where there is no prior
+knowledge on the number of processes that participate. We only have to use the wait-free snapshot
+implementation in [45] for this kind of models.
+Due to the shared memory simulation algorithm in [5], the same strong verifiability results hold
+for fully asynchronous message-passing systems where less than half processes can crash in any
+execution. The reason is simply that A∗ (Figure 6) and the verifier VO (Figure 9) can be simulated
+in such message-passing systems.
+10
+RELATED WORK
+Runtime verification. Runtime verification is an active field of research with important advance-
+ments in the last two decades. It has been employed in academia and industry for testing, verification
+and debugging before system deployment, and to ensure reliability, safety, robustness and security
+after deployment. Runtime verification has been used mostly to analyze software, however it
+has also been applied to other types of systems, e.g. hardware, hybrid and embedded systems,
+cyber-physical systems, distributed and concurrent systems, financial transaction systems, and
+more. For a detailed exposition of the field, we refer the reader to the recent textbook [6] and
+surveys [35, 54, 68].
+Distributed runtime verification. Distributed runtime verification of distributed systems is con-
+sidered and emergent and important topic, that poses several challenges that are yet to be solved
+(see [14, 27, 32, 43, 70, 82]). Designing distributed, asynchronous, fault-tolerant communication
+interfaces are regarded as a difficult problem. As far as we know, there are no runtime verification al-
+gorithms in the literature (for any correctness property) with a distributed communication interface
+that is at the same time fully asynchronous and fault-tolerant. To runtime verify different properties,
+25
+
+Castañeda and Rodríguez
+there have been proposed distributed runtime verification algorithms that are failure-free synchro-
+nous message-passing (e.g. [4, 83]), fault-tolerant message-passing where processes have access to
+clocks that are synchronized at some level (e.g. [8, 10, 11, 28, 33, 63, 80]), and fully asynchronous
+message-passing or shared memory with failure-free processes (e.g. [20, 37, 39, 85, 88]).
+Distributed fault-tolerant runtime verification. Our approach is close to the series of papers [13,
+41, 42] initiated by Fraigniaud, Rajsbaum and Travers, who pioneered the study of distributed
+fault-tolerant verification [40]. In them, it is studied a shared memory concurrent system that
+solves a series of tasks [58], and the aim is to runtime verify that the outputs for each task are
+correct, i.e. they satisfy the inputs/outputs relation specifying the task. To do so, an asynchronous
+wait-free read/write shared memory algorithm runs every time a task in the series is solved, and
+it is assumed that the verification algorithm terminates before the processes solve the next task.
+Thus, the distributed runtime verification algorithm proposed in those papers is distributed and
+fault-tolerant but it is not fully asynchronous. In sharp contrast, we consider a fully asynchrony and
+wait-free shared memory system, where some processes might be verifying the current execution
+while at the same time others are executing a high-level operation. GenLin includes tasks, hence
+task solvability is covered by our results. The main concern in [13, 41, 42] is to understand how
+many (input,output) pairs of the other processes (opinions in the parlance of those papers) a process
+must know in order to make a verdict. Our approach here is slightly different, as a process makes
+no verdict (or implicitly makes a “so far so good” verdict) as long as the computation looks correct
+from its perspective, and in the worts case when a process “sees” the pairs of all other processes,
+it can decide whether the computation is correct or not. Another crucial difference is that the
+wait-free interactive verifier proposed here can detect validity violations in a finer way, as real-time
+relations of the actual execution are taken into account, whereas in [13, 41, 42] only (input,output)
+pairs are used in the verification algorithm. For example, by observing only (input,output) pairs,
+for consensus it is impossible to detect when a process ran solo and decided a value distinct from
+its input, which violates validity. That scenario, in contrast, can be detected by our verifier through
+the views mechanism of the class DRV.
+Runtime verification of concurrent algorithms. For concurrent shared memory algorithms, runtime
+verification has been mostly used to detect data races, serializability violations (also called atomicity
+violations) and deadlocks; less studied properties are order instruction violations, missed signals,
+starvation, and high-level correctness conditions such as sequential consistency and linearizability
+(see [27, 70, 77]). Typically, in these works, asynchronous failure-free processes are assumed, and
+achieving a distributed communication interface is not a primary target. For detecting data races and
+serializability violations, several algorithms have been proposed; some algorithms use techniques
+based on the assumption that the concurrent algorithm under inspection uses locks (e.g. [84]),
+and others use some form of vector clocks to capture the happens-before relation, i.e. relations
+of causally-related events (e.g. [39]), or a mixture of both (e.g. [37]). It seems to us that none of
+these techniques can be adapted to our setting because: (1) we focus on lock-free implementations
+and (2) linearizability totally depends on the real-time order of non-causally related events. In
+general, a main difficulty is capturing the actual execution of a concurrent algorithm, as explained
+in [27, Section 4]. A simple solution is to serialize events using a lock (e.g. [71]); we find this type
+of solutions undesirable because, first, it might change the progress condition of the algorithm
+under inspection (as explained in the Introduction), and second, the lock creates a bottleneck.
+Other algorithms rely on bytecode-level added instructions in order to capture the execution
+(e.g. [55]), and moreover there are dynamic analysis frameworks working at a bycodelevel that
+provide information of the current execution for performing dynamic analysis (e.g. [38]). A problem
+with these techniques is that the moment when an event happens and the moment when the event
+26
+
+Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability
+is registered are not the same (i.e. they do not occur atomically, as in our interactive model), and
+hence the actual execution might not be captured (which is the main argument in the proof of the
+impossibility in Theorem 5.1).
+Runtime verification of linearizabilty. As far as we know, runtime verification of linearizability has
+only been studied in [29, 30], with centralized communication interfaces. Those papers study I/O
+refinement, which generalizes linearizability for objects without sequential specifications. GenLin
+includes objects without sequential specifications too as it includes set-linearizability and interval-
+linearizability [18, 19, 75]. In [29, 30], specific code is added to a white-box implementation in order
+to record the execution in a log (a sequence of events) that later is verified by a single process
+(hence the runtime verification algorithm is neither distributed nor fault-tolerant). Events must be
+atomically recorded in the log, which necessarily requires consensus or the use of locks [59, 72, 78].
+High-level operations are divided in mutators and observers. For mutators, the user has to add
+code that records in the log when the operation takes effect (i.e. its linearization point), and for
+observers, invocations and responses are recorded separately. It is known that there are linearizable
+implementations whose linearization points are not fixed (see [59, 72, 78]), hence the approach
+in [29, 30] is not general. Finally, it is not explained in [29, 30] what the relation is between the
+actual execution of the algorithm under inspection and the execution recorded in the log.
+Runtime enforcement. Runtime enforcement is an extension of runtime verification whose aim
+is to evaluate the current execution of a system under inspection, and halt the system whenever
+it deviates from a desired property (see survey [34]). Runtime enforcement initiated with the
+security automata formalism of Schneider [86]. Our interactive model for distributed runtime
+verification can be understood as as distributed version of Schneider’s security automata. As far as
+we know, so far there have been proposed only a few distributed runtime enforcement algorithms
+(e.g. [9, 46, 48, 53, 81]).
+Accountability. In general, accountability requires correct processes to irrevocably detect safety
+violations. Note that false positives are not allowed: once a violation is detected, the detection cannot
+be revoked. The concept of accountability in the context of distributed computing was introduced
+in [52]. Motivated by blockchain technologies, there have been recently proposed accountable
+algorithms for consensus [17, 21, 22, 89] and general tasks [23]. All these works consider semi-
+synchronous message-passing systems with malicious Byzantine failures, and the main target is to
+irrevocably detect Byzantine processes. Here we consider concurrent systems with benign crash
+failures, hence processes never deviate from its specification. In this scenario the safety violation
+one can detect are invalid outputs, as our self-linearizable implementations do.
+11
+FINAL DISCUSSION
+This paper studied the problem of distributed runtime verification of linearizability in asynchronous
+wait-free shared memory systems, through a novel interactive model for runtime verification of
+correctness conditions. Distributed runtime verification of linearizability is not an agreement
+problem: regardless of the consensus number of the base objects used for verification, the problem
+is impossible for common sequential objects such as queues, stacks, sets, priority queues, counters
+and the consensus problem. However, a stronger version of the problem can be solved for a class DRV
+of concurrent implementations, and without the need of consensus. Moreover, the possibility result
+holds for a correctness condition GenLin that includes linearizability and generalizations of it such as
+set-linearizability [75] and interval-linearizability [18, 19], the latter known to be expressive enough
+to model tasks [58] and any concurrent object satisfying some reasonable assumptions [18, 50].
+GenLin contains any object that is closed by prefixes and similarity, the latter being a property
+27
+
+Castañeda and Rodríguez
+identified here. Any concurrent implementation can be transformed into its counterpart in DRV,
+and there is a wait-free verifier that satisfies strong soundness and completeness, for the class DRV
+and any object in GenLin. A crucial building block in the transformation to implementations in
+DRV is that of the views mechanism for capturing the real-time in executions [18]. Read/write
+objects suffice to solve strong runtime verification, hence consensus is not needed.
+A simple and generic methodology for designing self-enforced correct GenLin implementations
+was obtained. Given any concurrent implementation for some GenLin object, one can produce a
+self-enforced correct concurrent implementation with the same progress properties, and for the
+same object such that all outputs are guaranteed correct (i.e. verified), or the implementation blocks,
+reporting error to every new invocation. These implementation are able to produce a certificate of
+the current computation at any time, hence allowing the design of systems in a modular manner with
+accountable and forensic guarantees. We are not aware of previous concurrent implementations in
+the literature with such properties. Furthermore, a concurrent asynchronous wait-free runtime
+verification algorithm for linearizability can be easily obtained from any of our self-linearizable
+implementations. As far as we know, this is the first distributed runtime verification algorithm for
+any correctness condition that is at the same time fully asynchronous and fault-tolerant.
+We believe several directions are worth to be explored. A natural direction is to study other
+correctness conditions in our interactive model, such as sequential consistency [65] (which is
+impossible to verify, as argued at the end of Section 5) or causal consistency [3], or data races in
+white-box concurrent implementations. Runtime verification of hyperproperties [24] is interesting
+as well. The main algorithmic technique used in our solutions is that of the views mechanism for
+sketching the current execution. In general it is interesting to explore if the mechanism helps to
+runtime verify other properties. Finally, conducting experimental evaluations are important to
+understand if the proposed algorithms can provide good performance in practical settings.
+Acknowledgements
+The first author would like to thank Hagit Attiya, Gregory Chockler, Ori Lahav and Serdar Tasiran
+for interesting discussions on this work. Part of this work was done while the first author was on
+sabbatical leave visiting the Department of Computer Science of the University of Surrey. This
+work was partially supported by the research project DGAPA-PAPIIT UNAM IN108723.
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+A
+LINEARIZABILITY FOR SOME OBJECTS IS NOT STRONG VERIFIABLE
+Theorem A.1 (Impossibility of Distributed Runtime Strong Verification). Linearizability
+for queues, stacks, sets, priority queues, counters and the consensus problem (modeled as a sequential
+object) is not distributed runtime strongly verifiable, even if a verifier uses base objects with consensus
+number infinity such as Compare&Swap.
+Proof. The proof is nearly the same as the proof of Theorem 5.1, where executions 𝐸 and 𝐹 of a
+hypothetical wait-free verifier Vqueue (in this case weak) are obtained from the non-linearizable
+queue implementation A defined in the proof. The difference between the proofs is in the last
+step, where it is now observed that the execution 𝐹 can be equally obtained from any wait-free
+linearizable queue implementation B (several such implementations appear in [59]). The proof
+now concludes by observing that processes cannot report ERROR in 𝐹 (which is allowed by strong
+soundness) because there is no witness for B as it is linearizable. Thus, by indistinguishability, no
+process reports ERROR in 𝐸, and hence Vqueue does not satisfy completeness. Therefore, Vqueue
+cannot exist.
+□
+32
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf,len=1901
+page_content='Asynchronous Wait-Free Runtime Verification and  Enforcement of Linearizability  ARMANDO CASTAÑEDA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Instituto de Matemáticas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Universdad Nacional Autónoma de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' México  GILDE VALERIA RODRÍGUEZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Posgrado en Ciencia e Ingeniería de la Computación,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Universidad  Nacional Autónoma de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' México  This paper studies the problem of asynchronous wait-free runtime verification of linearizability for concurrent  shared memory implementations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' where one seeks for an asynchronous wait-free concurrent shared memory  algorithm for verifying at runtime that the current execution of a given concurrent implementation is  linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It proposes an interactive model for distributed runtime verification of correctness conditions, and  shows that it is impossible to runtime verify linearizability for some common sequential objects such as queues,  stacks, sets, priority queues, counters and the consensus problem, regardless of the consensus power of base  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, the paper argues that actually a stronger version of the problem can be solved, if linearizability  is indirectly verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Namely, it shows that (1) linearizability of a class of concurrent implementations can be  distributed runtime strongly verified using only read/write base objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' without the need of consensus),  and (2) any implementation can be transformed to its counterpart in the class using only read/write objects too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  As far as we know, this is the first distributed runtime verification algorithm for any correctness condition that  is fully asynchronous and fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As a by-product, a simple and generic methodology for the design  of self-enforced linearizable implementations is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This type implementations produce outputs that  are guaranteed linearizable, and are able to produce a certificate of it, which allows the design of concurrent  systems in a modular manner with accountable and forensic guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We are not aware of prior concurrent  implementations in the literature with such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' These results hold not only for linearizability but for a  correctness condition that includes generalizations of it such as set-linearizability and interval-linearizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Additional Key Words and Phrases: Concurrent algorithms, Distributed runtime verification, Enforcement,  Fault-tolerance, Linearizability, Lock-freedom, Monitoring, Shared memory, Verification, Wait-freedom  1  INTRODUCTION  Linearizability and its challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Linearizability [60] is the de facto correctness condition for  asynchronous shared memory concurrent implementations of objects defined through sequential  specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Intuitively, an implementation is linearizable if each operation happens atomically at  a single moment of time between its invocation and response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Linearizability is so popular in part  because of its properties: it never forces the use of locks (it is non-blocking) and allows the design of  systems in a modular manner (it is composable) [60, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Designing linearizable implementations is a  simple task due to Herlihy’s Universal Construction [57], although the resulting solutions typically  do not scale well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In contrast, designing linearizable and scalable implementations is  a challenging task, as it requires the use of fine grained synchronization mechanisms in order  to exploit the parallelism in concurrent systems, which typically derives in an large number of  scenarios and subtle corner cases that need to be considered in correctness proofs [59, 72, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is  not uncommon to discover bugs in implementations that were thought to be linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The importance of linearizability, and more broadly of correct concurrent software, naturally  calls for formal verification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Over the past years, a variety of techniques for verifying  linearizability have been developed, using different approaches and providing different levels of  guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' See for example survey [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Despite of all efforts, verifying correctness of linearizable  implementations remains difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Model checking is feasible only for small cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' for a bounded  number of processes, and/or invocations to operations), and finding linearization points, simulations,  invariants, etcetera, is hard, sometimes requiring a great amount of expertise of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The  problem has been also studied from a theoretical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is known that deciding whether  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='02638v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='DC]  6 Jan 2023  Castañeda and Rodríguez  an implementation is linearizable might be EXPSPACE-complete or even undecidable [15], while  deciding if a given finite execution is linearizable might be NP-complete [49, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Distributed runtime verification and its challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Runtime verification is a dynamic, lightweight,  yet rigorous, formal method that complements static verification techniques with a more practical  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It only seeks to verify that the current execution of a system is correct, and maybe  prevent an incorrect action or enforce a correct behaviour otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The system under inspection  can be of any type, from hardware to software, centralized or distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We refer the reader  to [6, 35, 54, 68] for a detailed exposition of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Broadly speaking, in runtime verification, two, non necessarily disjoint tasks need to be accom-  plished [14]: (1) the design of a communication interface that captures the current execution of  the underlaying system under inspection, and (2) the design of a monitoring system that verifies  whether the captured execution is correct with respect to some correctness criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A big source of  difficulty is that the underlaying system and the communication interface are typically decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Namely, the underlaying system is designed, implemented and deployed without considering that  in the future a runtime verification mechanism might be integrated to it, hence it might not export  enough data of the current execution from which a communication interface and a monitoring  system can be built later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The situation gets more difficult when the underlaying system is  distributed as no process of the system “knows” what the current execution is, each process has only  a partial view of what the execution could be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The problem is even worse if we seek for a distributed,  asynchronous and fault-tolerant communication interface, as the processes might not even have the  ability to agree on the partial view of a process of the underlaying system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [36, 69]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In such  scenario, we have several computational entities that exchange information (typically by means of  a shared memory or a network), subject to delays and failures, yet they have to make consistent  decisions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' namely, we have a distributed computing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Designing distributed communication  interfaces that are asynchronous and fault-tolerant is a challenging problem in runtime verification  (see [14, 27, 32, 43, 70, 82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In fact, there are only proposal for runtime verification (of a number  of properties) with distributed communication interfaces that are failure-free and synchronous  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [4, 83]), fault-tolerant with timing assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [8, 10, 11, 13, 28, 33, 41, 42, 63, 80]), or asyn-  chronous failure-free (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [20, 37, 39, 85, 88]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' That is, the known distributed runtime verification  algorithms are not fully asynchronous and fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As far as we know, runtime verification of  linearizability has only been studied in [29, 30], with centralized communication interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Asynchronous wait-free runtime verification of linearizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We are interested in runtime verifi-  cation of linearizability where the underlaying system A is an asynchronous concurrent implemen-  tation of an object, and the communication interface C is an asynchronous wait-free shared memory  concurrent algorithm, where wait-free means that all processes but one can crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover, we  focus on the case where the underlaying system A is a black-box, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', we do not have access to its  specification/pseudocode, and hence the only way to obtain information of the current execution is  by analyzing the sequence of invocations and responses each process obtains from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Why is this setting interesting?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' While arguably A and C being asynchronous and A being a  black-box, model the challenging scenario described above, where A and C are decoupled, C being  asynchronous and wait-free guarantees that the “quality of service” of A is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For example,  if A is non-blocking but C is blocking (maybe because it lock-based), the system that results of  integrating A and C (and the corresponding monitoring system) will be blocking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' non fault-  tolerant, hence loosing A’s progress property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Similarly, A might be designed as an asynchronous  implementation for sake efficiency, hence if C is synchronous or semi-synchronous, A’s timing  property is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Indeed, a target in runtime verification is to design communication interfaces and  2  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  monitoring systems that are “as less intrusive as possible”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ideally they should not interfere with  the behavior of the underlaying system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [6, 12, 25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  p1  p2  Push(1) : true  Pop() : 1  p1  p2  Push(1) : true  Pop() : 1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Two executions of a stack where the two processes have the same partial views, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' sequence of  invocations and responses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' while the execution at the top is linearizable, the execution at the bottom is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Real-time, unaccessible to the processes, is what ultimately defines the executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Under this circumstances, is it possible to capture the actual execution of A in order to verify  correctness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' linearizability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The answer is clearly no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' After the seminal work of Lamport [64], we  know that it is impossible to determine the order of non-causally related events in fully asynchronous  distributed systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' and invocations to and responses from A are local events, which are non-  causally related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, the partial view of a process of A’s actual execution is just the sequence  with its invocation and responses, and for the processes is just impossible to access the real-time  order of these local events, which ultimately defines A’s execution (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This reasoning  suggests that very little should be possible when considering linearizability, as in the end this  correctness condition totally depends on the real-time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' However, there are ways to runtime  verify linearizability, as we show here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We propose an interactive model to study the problem of distributed runtime verification  of linearizability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the model actually serves for studying any correctness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In the model,  an asynchronous concurrent implementation A that is presumably linearizable interacts with a  client C (modelling a communication interface), which, as already said, is an asynchronous wait-free  shared memory concurrent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The client C is required to invoke operations of A, receive  the corresponding responses, and somehow compute relevant information of the current execution  of A in order to decide whether it is linearizable or not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the interaction between A and C is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The client C runtime verifies a correctness property P, if it is sound and complete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', it reports  ERROR if and only if A’s current execution does not satisfy P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Once the interactive model and the distributed runtime verification problem are clearly stated, a  simple indistinguishability argument shows that it is impossible to runtime verify linearizability  for some common objects such as queues, stacks, sets, priority queues, counters and even the  consensus problem, regardless of the consensus number [57] of the base objects used in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus,  the problem of distributed runtime verification of linearizability is beyond consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Then, somewhat surprisingly, we show that a stronger version of the problem can be solved, and  without the need of consensus, if linearizability is indirectly verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Namely, we identify a class of  concurrent implementations, which we call DRV, such that linearizability can be runtime verified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  furthermore, a stronger version of the problem can be solved where any client C is required to  be complete and strongly sound, which means that it might report ERROR when A’s execution is  linearizable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' a false negative), as long as it “discovers” an execution of A that is not linearizable  (hence concluding that A is not linearizable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Intuitively, any implementation in the class DRV  3  Castañeda and Rodríguez  produces an “sketch” of its current execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It turns out that such sketches are good enough to  runtime strongly verify linearizability, for the class DRV, and using only read/write base objects,  which have consensus number one [57] (hence uncapable of solving consensus among two or  more processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Linearizability can then be indirectly strongly verified because any concurrent  implementation A can be transformed into an implementation A∗ in DRV, using only read/write  base objects, such that A and A∗ have the same progress properties, and A is linearizable if and  only if A∗ is linearizable (both w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the same sequential object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover, these results holds  for a correctness condition GenLin that we define here, which includes linearizability, as well as  other correctness conditions such as set-linearizability [75] and interval-linearizability [18], both  generalizations of linearizability for concurrent objects with no sequential specifications [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Finally, we show the usefulness of the proposed interactive model by obtaining concrete dis-  tributed runtime verification algorithms from an interactive client C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For linearizability, we obtain  a simple methodology to derive self-enforced linearizable implementations, intuitively whose re-  sponses are guaranteed linearizable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' verified), and moreover the implementations are able to  produce a certificate of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, self-enforced linearizable implementations allow the design of  systems in a modular manner with accountable and forensic guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We are not aware of prior  concurrent implementations in the literature with such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The methodology takes any  implementation A and produces an implementation B using A and read/write objects such that A  and B have the same progress conditions and A is linearizable if and only if B is self-enforced lin-  earizable (both w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the same sequential object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Furthermore, a concurrent asynchronous wait-free  runtime verification algorithm for linearizability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' a communication interface and a monitoring  system) can be easily obtained from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As far as we know, this is the first distributed runtime  verification algorithm for any correctness condition that is at the same time fully asynchronous  and fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Remarkably, all proposed algorithms are generic and simple to implement, hence  prone to being program synthesized, a property of interests in runtime verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Once Section 2 states the model of computation, the interactive model for  distributed runtime verification and the problem of distributed runtime verification are introduced  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, Section 4 recalls the definition of linearizability and Section 5 shows the  impossibility for runtime verification of linearizability for some objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A high-level perspective of  how the impossibility result is evaded and the definition of the strong version of the problem appear  in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Sections 7 and 8 implement the high-level strategy described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Section 9  explains some extension of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Finally, Section 10 discusses related work, explaining  differences with our results, and Section 11 concludes the paper with a final discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  2  MODEL OF COMPUTATION  We consider a standard concurrent shared memory system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [59, 60]) with 𝑛 ≥ 2 asynchronous  processes, 𝑝1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' , 𝑝𝑛, which may crash at any time during an execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' All but one process can crash  in any execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The index of process 𝑝𝑖 is 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Processes communicate with each other by invoking  atomic operations of shared base objects that reside in the shared memory: either simple Read/Write  operations, or more complex and powerful Read-Modify-Write operations, such as Fetch&Inc or  Compare&Swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Base objects are atomic, hence we consider a sequentially consistent [65] shared  memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Shared base objects are denoted with uppercase letters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' local variables used by a process  for performing its local computations are denoted with lowercase letter subscripted with the index  of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  For ease of exposition, it is assumed that base objects are of unbounded size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Section 11 however  explains how this unrealistic assumption can be removed from the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We consider  the possibility that processes have perfectly synchronized local clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Each process can read its  4  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  local clock in a local computation step, information that then can be written in the shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We stress that the time that elapses between reading a local clock and writing it in the shared  memory is unpredictable, as the system is asynchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This assumption is not relevant in our  algorithms, however it makes our impossibility result stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  An implementation of a concurrent object O (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' a queue or a stack), specified in some way (more  details in Section 4), is a distributed algorithm A consisting of 𝑛 local state machines 𝐴1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ,𝐴𝑛,  some of them possibly non-deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Local machine 𝐴𝑖 specifies which operations on base  objects and local computations 𝑝𝑖 executes in order to return a response when it invokes a high-level  operation of O (sometimes simply called operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Each of these base objects operations and local  computations is a step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Invocations and responses are local computations as well but we do not  refer to them as steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  An execution of A is a possibly infinite sequence of steps, plus invocations and responses to  high-level operations of the concurrent object O, with the following well-formedness properties:  (1) Each process is sequential, namely, it first invokes a high-level operation, and only when it  has a corresponding response, it can invoke another high-level operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (2) A process takes steps only between an invocation and a response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (3) For any invocation of an operation op𝑖 by a process 𝑝𝑖, denoted 𝑖𝑛𝑣𝑖 (op𝑖), the steps of 𝑝𝑖  between that invocation and its corresponding response (if there is one), denoted 𝑟𝑒𝑠𝑖 (op𝑖),  are steps specified by 𝐴𝑖 when 𝑝𝑖 invokes op𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  It is assumed that after a process completes an operation, it non-deterministically picks the operation  it executes next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  For simplicity, and without loos of generality, we assume that every concurrent object provides  a single high-level operation, called Apply, that receives as input op, a description of the actual  operation that is invoked, which includes the inputs to the actual operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We also assume, again  without loss of generality, that Apply is invoked with a given input op only once (a fictitious input  value to the actual operation can be added such that all inputs are different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Typically, invocation and responses of an implementation A also include A as part of their  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We however drop that information from invocations and responses because it facilitates  our discussion, although some ambiguity is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  An implementation A can use other implementation B in order to produce responses, namely,  the processes can invoke high-level operation of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Hence, when a process invokes an operation of  B, it continues executing the steps specified by A only after it receives the corresponding response  from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any execution 𝐸 of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We let 𝐸|B denote the sequence of steps, invocations and  responses of B in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, 𝐸|B is a well-formed execution of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In what follows, unless stated  otherwise, when we talk about operations in 𝐸, we mean only the operations of A, excluding the  nested calls to operations of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  A high-level operation in an execution is complete if both its invocation and response appear in  the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' An operation is pending if only its invocation appears in the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A process  is correct in an infinite execution of an implementation if it takes infinitely many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' When  considering infinite executions, we focus on those that are fair: for every correct process and every  𝐾 ≥ 1, there is a finite prefix with 𝐾 steps of that process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' An implementation A is wait-free  if in every infinite execution, every correct process completes infinitely many operations [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  An implementation A is lock-free if in every infinite execution, infinitely many operations are  complete [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, a wait-free implementation is lock-free but not necessarily vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We  consider only implementations that are at least lock-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The notions of wait-freedom and lock-  freedom naturally extend to specific operations or fragments of pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  5  Castañeda and Rodríguez  The step complexity of an implementation is the maximum number of base operations a process  needs to take to produce a response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Sometimes it will be convenient to think an implementation A as a black-box whose specification  cannot be accessed, and hence the only information that can be obtained from it are the executions  it produces without steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We call such executions without steps histories, namely, sequences of  invocations and responses satisfying the first two well-formedness properties stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As  in [60], we define an abstract implementation as a set of well-formed histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By abuse of notation,  for any execution 𝐸 of an implementation A, we let 𝐸 itself denote the history obtained from 𝐸  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the sequence obtained by removing from 𝐸 all its steps), and let A denote itself the abstract  implementation with all histories of A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', the histories obtained from all its executions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This  abuse of notation will facilitate the discussion, at the cost of introducing some ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The consensus number of a shared object O is the maximum number of processes that can solve  the consensus problem, using any number of instances of O in addition to any number of Read/Write  base objects [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consensus numbers induce the consensus hierarchy where objects are classified  according their consensus numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The simple Read/Write operations stand at the bottom of  the hierarchy, with consensus number one and the lowest coordination power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' At the top of the  hierarchy we find operations with infinite consensus number, like Compare&Swap, that provide  the maximum possible coordination power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  3  AN INTERACTIVE MODEL FOR DISTRIBUTED RUNTIME VERIFICATION  Let us fix a concurrent object O, specified in some way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let A be a lock-free implementation of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Intuitively, a correctness condition is a mechanism to separate the correct implementations of O  from the incorrect ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Basically, it is a predicate PO that all finite executions of A need to satisfy  for A being declared correct, with respect to PO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is known that if PO is linealizability, deciding  whether A is linearizable might be EXPSPACE-complete or even undecidable [15], depending  on the object O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Deciding if a given finite history is linearizable is decidable, but it might be NP-  complete [49, 76], although for some objects this question can be decided in polynomial time [16, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  From now on, we will assume that each process can locally test if a given finite history satisfies PO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Shared Variables:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Shared memory 𝑀  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Operation Verify(A) is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(01) 𝑠𝑒𝑡𝑖 ← ∅  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(02) while true do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(03)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='op𝑖 ← non-deterministically chosen high-level operation that is not in 𝑠𝑒𝑡𝑖  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(04)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='𝑠𝑒𝑡𝑖 ← 𝑠𝑒𝑡𝑖 ∪ {op𝑖 }  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(05)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Encode in 𝑀 the invocation to Apply(op𝑖) of A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(06)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Invoke operation Apply(op𝑖) of A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(07)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='𝑟𝑒𝑠𝑝𝑖 ← response from operation Apply(op𝑖) of A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(08)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Encode the response 𝑟𝑒𝑠𝑝𝑖 in 𝑀  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(09)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='𝑒𝑥𝑒𝑐𝑖 ← description of the current execution of A in 𝑀  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(10)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='if ¬ P𝑂 (𝑒𝑥𝑒𝑐𝑖) then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='(11)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='report (ERROR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='𝑒𝑥𝑒𝑐𝑖)  (12)  end if  (13) end while  end Verify  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Generic structure of a verifier VO for correctness condition PO (code of process 𝑝𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Let us suppose that A is presumably correct with respect to PO, but maybe this fact has not been  formally proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let us also suppose the existence of a client C (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' a concurrent algorithm) that  solves some distributed problem using A, namely, the processes in C invoke high-level operations  6  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We would like to design an intermediate layer VO between C and A that, from time to  time, verifies that the current execution of A is correct, namely, it satisfies the predicate PO, and  reports ERROR otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' If A is indeed correct, then we would like the client C not to be able to  distinguish whether it is interacting with A or VO, and hence we require that VO is asynchronous  and wait-free so that the properties of A are preserved, for example its progress properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We model the intermediate layer just described as a distributed algorithm VO, called verifier, that  interacts with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The generic structure of the interaction appears in Figure 2, where A is a black-  box, and hence the only way VO can obtain information of A is by invoking high-level operations  of it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' namely, the processes in VO only invoke operations of A and only receive responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus,  the verifier VO interacts with an abstract object A, from which receives one of its histories in  every execution of VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  During the interaction, each process invokes a series of non-deterministically chosen high-level  operations of A (modeling the operations of A that any client C might invoke).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For simplicity,  every process tests if the history so far satisfies PO after each of its high-level operations of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  If the predicate is not satisfied, the process reports ERROR together with a witness for A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' a  history of A that does not satisfy PO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' in any case, the interaction continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' (In a practical setting  the interaction would stop and ERROR and the witness would be returned to the client C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' for  simplicity, in our model the interaction continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=') Naturally, the processes in VO need to exchange  information in order to obtain a description of the current history of A (which ultimately creates  causal relations between A’s invocations and responses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, each process might store some  information in the shared memory before and after each of its invocations to and responses from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Since A is a black-box in the generic verifier in Figure 2, VO cannot be designed for a specific  presumably correct implementation for O, it must work for any possibly abstract concurrent  implementation (even if it is correct with respect to an object O′ ≠ O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This requirement is modeled  as conceptually VO taking A as its input to the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We will restrict our attention to verifiers where the segments of the code in Lines 03–05 and  Lines 08–12 are wait-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' we say that such a verifier is wait-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The step complexity of a verifier is  the maximum number of base operations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' excluding the invocation and response in Lines 06  and 07) a process needs to take in order to complete one iteration of the while loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In the definition of the distributed runtime verification problem below, for simplicity, it is assumed  that no process crashes, and hence all executions of a verifier are infinite, since we are assuming  that A is at least lock-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The assumption makes the problem easier to state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The possible crashes  that can occur in the system are captured by asynchrony and the fact that correctness is tested  at finite prefixes of a given infinite execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The soundness and completeness properties that  specify the problem have been considered in the past (see for example [27, 73, 74]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the definition  basically adapts them to fit in our interactive setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (Distributed Runtime Verification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let O be a concurrent object specified  in some way, and consider a correctness condition PO for O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We say that a wait-free verifier VO  distributed runtime verifies PO if the following two requirements are fulfilled in every infinite execution  𝐸 of VO with an arbitrary input (abstract) implementation A:  (1) Soundness: If for every finite prefix 𝐸′ of 𝐸, 𝐸′|A satisfies PO, then no process reports ERROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (2) Completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' If 𝐸 has a finite prefix 𝐸′ such that 𝐸′|A does not satisfy PO, then at least one  process reports ERROR together with a witness for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We say that PO is distributed runtime verifiable if there is a wait-free verifier that distributed  runtime verifies PO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The previous definition is flexible, it is not difficult to modify it to cover the cases where A is  blocking [59] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' it internally uses locks) or obstruction-free [59] (namely, progress is guaranteed  7  Castañeda and Rodríguez  only when a process runs solo), or correctness conditions for one-shot distributed problems such  as tasks [58], where each process invokes one high-level operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The main difference is that in  these cases, the interaction might be only finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Below, for sake of compactness, sometime we will simply say verify/verifiable instead of dis-  tributed runtime verify/verifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  To conclude the section, we observe that the intermediate layer in the situation described above  can be easily obtained from a verifier VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let V𝑂,A denote the implementation whose single  high-level operation Apply(op𝑖) executes Lines 05 to 09 with a fixed implementation A, and returns  𝑟𝑒𝑠𝑝𝑖 if P𝑂 (𝑒𝑥𝑒𝑐𝑖) holds, and returns (ERROR,𝑒𝑥𝑒𝑐𝑖) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Basically, V𝑂,A “wraps” A, and  thus the client C would use VA,O instead of A without noticing the difference, if A is indeed  correct with respect to PO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This claim will be formalized in Section 8 for linearizability and a  correctness condition generalizing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  4  LINEARIZABILITY  Linearizability [60] is the de facto standard correctness condition for concurrent implementations  of objects defined with sequential specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It extends the concept of atomicity introduced by  Lamport [66, 67] to any sequential object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Intuitively, a history of an implementation is linearizable  if its operations can be ordered sequentially, without reordering non-overlapping operations, so  that their responses satisfy the sequential specification of the implemented object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Figure 3 depicts  examples of linearizable and non-linearizable histories of a stack implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (Seqential Specifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A sequential specification of a concurrent object O  is a state machine specified through a (possibly partial) transition function 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Given a state 𝑞 and an  invocation 𝑖𝑛𝑣𝑖 (op𝑖) of process 𝑝𝑖, 𝛿(𝑞,𝑖𝑛𝑣𝑖 (op𝑖)) returns the tuple (𝑞′,𝑟𝑒𝑠𝑖 (op𝑖)) (or a set of tuples if  the machine is non-deterministic) indicating that the machine moves to state 𝑞′ and the response to  op𝑖 is 𝑟𝑒𝑠𝑖 (op𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The sequences of invocation-response tuples, ⟨𝑖𝑛𝑣𝑖 (op𝑖) : 𝑟𝑒𝑠𝑖 (op𝑖)⟩, produced by the  state machine are its sequential histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  p1  p2  Push(1) : true  p3  Push(2) : true  Pop() : 2  Pop() : 1  p1  p2  Push(1) : OK  p3  Push(2) : true  Pop() : empty  Pop() : 1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Two 3-process histories of a stack implementations are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Time goes from left to right, and an  operation is denoted with a double-ended arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For clarity, each operation Apply(op) is simply denoted op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The execution at the top is linearizable with respect to the usual sequential specification of a stack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' a  linearization of it is: ⟨Push(2) : true⟩⟨Push(1) : true⟩⟨Pop() : 1⟩⟨Pop() : 2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The execution at the bottom is  not linearizable because the stack cannot be empty when the operation ⟨Pop() : empty⟩ starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  For sake of clarity, a tuple ⟨𝑖𝑛𝑣𝑖 (op𝑖) : 𝑟𝑒𝑠𝑖 (op𝑖)⟩ is simply denoted op𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Also, subscripts of  invocations and responses are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  8  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  An history 𝐸′ is an extension of a finite history 𝐸, if 𝐸′ can be obtained from 𝐸 by appending zero  or more responses for some of 𝐸’s pending operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  For any history 𝐸 and any process 𝑝𝑖, 𝐸|𝑝𝑖 denotes the sequence of invocations and responses of  𝑝𝑖 in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Two histories 𝐸 and 𝐹 are equivalent if 𝐸|𝑝𝑖 = 𝐹 |𝑝𝑖, for every process 𝑝𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  For any finite history 𝐸 of an implementation A, 𝑐𝑜𝑚𝑝(𝐸) denotes the history obtained by  removing from 𝐸 all invocations of pending operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' note that 𝑐𝑜𝑚𝑝(𝐸) is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' To  formalize linearizability, we define a partial order <𝐸 on the complete operations of any history 𝐸:  op <𝐸 op′ if and only if 𝑟𝑒𝑠(op) precedes 𝑖𝑛𝑣(op′) in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Two complete operations are concurrent if  they are incomparable by <𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The history is sequential if <𝐸 is a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We consider the definition of linearizability in [87], which is a slight variant of the original  definition [60] that fixes some corner cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 (Linearizability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let O be any concurrent object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A finite history 𝐸 is linearizable  with respect to O if there is an extension 𝐸′ of 𝐸 and a sequential history 𝑆 of O such that  (1) 𝑐𝑜𝑚𝑝(𝐸′) and 𝑆 are equivalent and  (2) <𝑐𝑜𝑚𝑝 (𝐸′) ⊆ <𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The sequential history 𝑆 is said to be a linearization of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We say that an implementation A is  linearizable with respect to O, if each of its finite histories is linearizable with respect to O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  5  LINEARIZABILITY FOR SOME OBJECTS IS NOT RUNTIME VERIFIABLE  This section shows that linearizability for some common objects such queues, stacks, priority  queues, counters, and even the fundamental consensus problem, is not distributed runtime verifiable,  regardless of the coordination power of the base objects the processes use to communicate with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The following simple impossibility proof captures informal arguments that have been used in the  past (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [27, 47]) to argue that distributed runtime verification of some correctness conditions in  asynchronous distributed systems is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (Impossibility of Distributed Runtime Verification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Linearizability for queues,  stacks, sets, priority queues, counters and the consensus problem (defined as a sequential object) is not  distributed runtime verifiable, even if a verifier uses base objects with consensus number infinity such  as Compare&Swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We focus on the case of the queues as all other cases are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By contradiction,  suppose that there is a wait-free verifier Vqueue that verifies linearizability for queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider the  following non-linearizable implementation A: every Enqueue operation returns true, and every  Dequeue operation returns empty, for every process 𝑝𝑖 ≠ 𝑝1, and for 𝑝1, it returns 1 in its first  operation, and returns empty in every subsequent operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We will exhibit two executions 𝐸 and 𝐹 of Vqueue with input A and argue that Vqueue cannot  simultaneously satisfy both the soundness and completeness requirements of the distributed  runtime verification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We use the generic structure in Figure 2 to describe the executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Execution 𝐸 is the next:  (1) In its first iteration of the while loop, process 𝑝1 picks 𝑜𝑝1 = Dequeue() in Line 03, executes  the local step in Line 04 and the base operations corresponding to Line 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (2) In its first iteration of the while loop, process 𝑝2 picks 𝑜𝑝2 = Enqueue(1) n Line 03, executes  the local step in Line 04 and the base operations corresponding to Line 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (3) Process 𝑝1 executes Lines 06 and 07 of its first iteration of the while loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus it obtains  response 𝑟𝑒𝑠𝑝1 = 1 for its high-level operation 𝑜𝑝1 = Dequeue() .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (4) Process 𝑝2 executes Lines 06 and 07 of its first iterations of the while loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus it obtains  response 𝑟𝑒𝑠𝑝2 = true for its high-level operation 𝑜𝑝2 = Enqueue(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  9  Castañeda and Rodríguez  (5) Process 𝑝1 executes Lines 08 to 12 of its first iteration of the while loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (6) Process 𝑝2 executes Lines 08 to 12 of its first iteration of the while loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (7) For each 𝑘 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' , ∞ (in this order), 𝑝(𝑘 mod 𝑛)+1 executes a whole iteration of the while  loop where it picks 𝑜𝑝(𝑘 mod 𝑛)+1 = Dequeue() in Line 03 (and hence the response in Line 07  is 𝑟𝑒𝑠𝑝(𝑘 mod 𝑛)+1 = empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Execution 𝐹 is similarly constructed, with the exception that the steps 3 and 4 of the previous  construction appear in the opposite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In other words, in 𝐸, the first high-level operations of 𝑝1  is executed first and then the first high-level 𝑜𝑝2 is executed, whereas in 𝐹, the high-level operations  are executed in the opposite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Observe that the history of A obtained from every finite prefix  of 𝐹 is linearizable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' in contrast, the history of A obtained from every finite prefix of 𝐸 containing  at least the first operation of 𝑝1, is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  By a simple induction, it can be shown that the local state of any process 𝑝𝑖 after its 𝑘-th step  is the same in both executions 𝐸 and 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This means that the executions are indistinguishable to  all processes, and hence in both executions they make the same sequence of decisions in Lines 08  to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' If no process reports ERROR in 𝐸 and 𝐹, then Vqueue does not fulfills completeness due to 𝐸,  and if at least one process reports ERROR in 𝐸 and 𝐹, then Vqueue does not fulfills soundness due  to 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, Vqueue cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  For the other objects, the argument is nearly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For example, for the case of the stack, Pop  operations replace Dequeue and Push operations replace Enqueue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For the case of the consensus,  we define an object with a single Decide operation that can be invoked several times, and the first  operations sets its value as the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  The previous proof can easily be extended to several variations of linearizability whose aim  is modeling relaxed versions of sequential objects or objects with no sequential specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Examples of such variations are quasi-linearizabilty [2], 𝑘-stuttering [56], set-linearizability [75],  interval-linearizability [18] and intermediate value linearizability [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The reason is that all these  relaxations include the sequential executions with the “exact” sequential behavior of the object that  is relaxed, which suffices for the previous proof to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Furthermore, nearly the same proof shows  the impossibility of runtime verification for sequential consistency [65], the only difference in the  argument is that now there is finite prefix of the execution 𝐸|A that is not sequentially consistent,  concretely, the one that only has the first operation of 𝑝1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the rest of the proof is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  6  EVADING THE IMPOSSIBILITY RESULT: THE STRATEGY AT A HIGH-LEVEL  The rest of the paper is devoted to show that it is possible to circumvent the impossibility in  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Roughly speaking, it will argue that linearizability of any implementation A with  respect to any sequential object O, can be indirectly verified through a class of implementations,  called DRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Somewhat surprisingly the class DRV can be verified under a stronger definition of  the distributed runtime verification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This section gives a high-level perspective of the  strategy followed in the next sections, and introduces the stronger version of the distributed runtime  verification problem that will be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The impossibility in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 basically comes from the inability of the processes to capture  the actual history of an arbitrary implementation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since the processes are asynchronous, a delay  of arbitrary length can happen between the steps in Lines 05 and 06, and between the steps in  Lines 07 and 08, in the generic verifier in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, basically the processes are only able to  capture a history of A where the operations might be “stretched”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Figure 4 exemplifies this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Let 𝐸 denote the actual history of A and 𝐸′ denote the history of A captured by the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Hence history 𝐸′ only “sketches” the actual history 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Actually, 𝐸′ might be less restrictive to be  linearized (with respect to some sequential object) as some operations that do not overlap in 𝐸  10  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  p1  p2  enq(1) : true  deq() : 1  Detected history E′:  Actual history E of A:  p1  p2  enq(1) : true  deq() : 1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Due to asynchrony, processes might detect histories where the operations of A are “stretched”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This  phenomenon is exemplified with two histories of a queue implementation where an operation Apply(op)  is simply denoted op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In the history at the top, both histories, the actual one and the detected one, are  linearizable, while in the history at the bottom, the actual history is not linearizable, however the detected  history is linearizable due to a long delay between the event that announce the operation of A that 𝑝1 is  going to call next and the actual moment when the operation is called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  might actually overlap in 𝐸′ (see the history at the bottom in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus we have:  𝐸 is linearizable =⇒ 𝐸′ is linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Arguably, this is the best the processes can do for capturing the actual execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For the time being,  let us suppose that the processes somehow can compute 𝐸′ and each process is able to read it from  the shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' With the help of 𝐸′, the processes can easily satisfy soundness, each process  simply needs to locally test whether 𝐸′ is linearizable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' however completeness cannot be satisfied: if  𝐸 is not linearizable, 𝐸′ might or might not be linearizable (see again the history at the bottom in  Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This discussion suggests a weaker version of the distributed runtime verification problem,  requiring soundness and a weaker version of completeness where processes are allowed to output  false positives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', they do not report ERROR although the actual history of A is not linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In fact, this weaker version of the problem can be formally defined and shown to be solvable [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  This result is not completely satisfactory because, after all, the main motivation of any verification  technique is to detect incorrect solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' But we can do better, as explained next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Detected history E′:  Actual history E of A:  p1  p2  enq(1) : true  deq() : 1  Actual history E∗ of A∗:  p1  p2  enq(1) : true  deq() : 1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The operations of any history of the implementation A∗ might “shrink” in the detected history and in  the actual history of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Two histories of A∗ where A is a queue implementation are shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' an operation  Apply(op) is simply denoted op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In the history at the top, the actual history of A∗ is linearizable but the  detected history is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In contrast, in the history at the bottom, the actual history of A∗ is not linearizable,  which implies that the detected history is not linearizable either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  11  Castañeda and Rodríguez  We can look at the limitation above from a positive perspective, since in some sense the detected  history 𝐸′ might “fix” the incorrect history 𝐸 of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We can take the mechanism that computes 𝐸′  and use it to produce a new implementation, denoted A∗, where processes output the responses  obtained from A, somehow together with the captured history 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In a sense, the idea in A∗ is  to make asynchrony an “ally” instead of an “enemy”: in the case 𝐸, the actual execution of A, is  not linearizable, if the delays are short, then 𝐸′ will not be linearizable, hence the mistake can be  detected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' and if delays are long, then 𝐸′ will be linearizable, hence the mistake cannot be detected  but 𝐸′ “fixes” 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, A can be indirectly verified through A∗ because, as it will be shown, A is  linearizable if and only if A∗ is linearizable (both w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the same sequential object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In every execution 𝐸∗ of A∗, we now have that 𝐸′ “sketches” both the actual history 𝐸∗ of A∗ and  the actual history 𝐸 of A, with the difference that the operations in 𝐸′ and 𝐸 now might “shrink”,  compared to 𝐸∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This situation is schematized in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We thus have the following:  𝐸 is linearizable =⇒ 𝐸′ is linearizable =⇒ 𝐸∗ is linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  From the sequence of implications we obtain that if 𝐸∗ is not linearizable then its sketch 𝐸′ is not  linearizable either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The processes now can easily fulfill completeness for the derived implementation  A∗ by locally analyzing 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, the processes are able to “catch” any possibly non-linearizable  history of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For soundness, however it can be the case that 𝐸′ is not linearizable although 𝐸∗ is  indeed linearizable (see the history at the top in Figure 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' thus the processes might output false  negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It turns out that 𝐸′ is a history of A∗ and then it is a witness for A∗ in those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Hence if processes output a false negative, it is for a reason: the current execution of A∗ might be  linearizable but the processes have “discovered” that A∗ is not linearizable, and they have a history  that witnesses that fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover, no process can distinguish between 𝐸∗ and 𝐸′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the histories  are equivalent), hence it is actually possible that the actual execution of A∗ is 𝐸′ instead of 𝐸∗,  which motivates the processes to take the “preventive” action of reporting ERROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, the  processes can fulfill a stronger version of soundness where false negatives are allowed, as long as  they come with witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The previous discussion motivates the next stronger version of the runtime verification problem,  which will be shown to be solvable for implementations such as A∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' DRV will denote the class  with the implementations like A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (Distributed Runtime Strong Verification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let O be a concurrent object  specified in some way, and consider a correctness condition PO for O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We say that a wait-free verifier VO  distributed runtime strongly verifies PO if the following two requirements are fulfilled in every infinite  execution 𝐸 of VO with an arbitrary input (abstract) implementation A∗:  (1) Strong Soundness: If for every finite prefix 𝐸′ of 𝐸, 𝐸′|A∗ satisfies PO, then either no process  reports ERROR, or at least one process reports ERROR together with a witness for A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (2) Completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' If 𝐸 has a finite prefix 𝐸′ such that 𝐸′|A∗ does not satisfy PO, then at least one  process reports ERROR together with a witness for A∗ in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We say that PO is distributed runtime strongly verifiable if there is a wait-free verifier that  distributed runtime strongly verifies PO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We conclude this section mentioning that a slight varian of the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 shows that  the stronger version of the problem in the definition above is impossible for the general class of  implementations (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  12  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  7  THE CLASS OF DISTRIBUTED RUNTIME STRONGLY VERIFIABLE  IMPLEMENTATIONS (DRV)  This section introduces the class DRV and shows its main properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Roughly speaking, it will  show that (1) every implementation can be easily transformed into its counterpart in DRV and (2)  every implementation in DRV provides an “sketch” of the current execution, additionally to the  outputs of the object it implements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As we will see, the sketches are the key property that makes  any of these implementations strongly verifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Furthermore, the same result holds not only for  linearizability, but for a class of objects, with its companion correctness condition, that generalize  linearizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' After defining the generalized class of objects, the section defines the class DRV and  shows the main properties of the implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1  Generalizing linearizability  In the rest of the paper, similarly to [60], an abstract object, called just object for simplicity, is defined  as a set of well-formed finite histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The associated correctness condition is the membership  predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Namely, a finite history of an implementation is correct with respect to the object if the  history belongs to the set specifying the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, an implementation is correct with respect to  the object if each of its finite histories belong the set specifying the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In order to generalize linearizability, we define a partial order on the set of operations, complete  and pending, of any history 𝐸 of an implementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the relation is denoted ≺𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For any two  operations op and op′ in 𝐸, we have that op ≺𝐸 op′ if and only if 𝑟𝑒𝑠(op) precedes 𝑖𝑛𝑣(op′) in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Thus, the only difference with <𝐸 is that ≺𝐸 also relates pending operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (Similarity between histories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A finite history 𝐸 is similar to a finite history 𝐹  if there is a history 𝐸′ such that:  (1) 𝐸′ can be obtained from 𝐸 by appending responses to some pending operations and removing  invocations of some pending operations,  (2) 𝐸′ and 𝐹 are equivalent, and  (3) ≺𝐸′ ⊆ ≺𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 (The class GenLin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The class GenLin contains every abstract object that is closed  by prefixes and similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Namely, if the set specifying the object contains 𝐹, then it also contains:  (1) every prefix 𝐸 of 𝐹, and  (2) every history 𝐸 that is similar to 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We now show that linearizability belongs to the class GenLin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (Linearizability is in GenLin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐹 be any finite history that is linearizable with  respect to some sequential object O (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' a state machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then,  (1) every prefix 𝐸 of 𝐹 is linearizable with respect to O, and  (2) every history 𝐸 that is similar to 𝐹 is linearizable with respect to O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Therefore, GenLin contains the object (set) with all finite histories that are linearizable with respect  to O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The prefix closure proof in [51][Theorem 4] assumes a definition of linearizability that  is slightly different than the one we use here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' that proof however also holds in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For  completeness, we present the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐹 = 𝐸𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any linearization 𝑆 of 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, there is  an extension 𝐹 ′ of 𝐹 such that 𝑐𝑜𝑚𝑝(𝐹 ′) and 𝑆 are equivalent and <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝑆 = 𝑆𝐸𝑆𝐸′,  where 𝑆𝐸 is the shortest prefix of 𝑆 with all complete operations in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We argue that 𝑆𝐸 is a  linearization of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let us observe first that 𝑆𝐸 does not have an operation whose invocation appears  13  Castañeda and Rodríguez  in 𝐸′: (1) by definition of 𝑆𝐸, the last operation op of 𝑆𝐸 is complete in 𝐸, and (2) if 𝑆𝐸 has such an  operation op′, then op′ <𝑆𝐸 op, which contradicts that <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆 bacause we certainly have  op <𝑐𝑜𝑚𝑝 (𝐹 ′) op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Now, let 𝐸′′ be the extension of 𝐸 obtained by appending all response in 𝑆𝐸 to  pending operations in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is not hard to see that 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑆𝐸 are equivalent: if there is a  𝑝𝑖 such that 𝑐𝑜𝑚𝑝(𝐸′′)|𝑝𝑖 ≠ 𝑆𝐸|𝑝𝑖, then simply 𝑐𝑜𝑚𝑝(𝐹 ′) and 𝑆 are not equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We now argue  that <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑆𝐸, from which we conclude that indeed 𝑆𝐸 is a linearization of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any  op <𝑐𝑜𝑚𝑝 (𝐸′′) op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑆𝐸 are equivalent, both op and op′ appear in 𝑆𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover,  𝑆𝐸 is a sequential history, and hence <𝑆𝐸 must relate op and op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' If both 𝑜𝑝 and 𝑜𝑝′ are complete  in 𝐸, then op <𝐸 op′, hence op <𝐹 op′, and consequently op <𝑐𝑜𝑚𝑝 (𝐹 ′) op′, and as <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆,  op <𝑆 op′, which implies op <𝑆𝐸 op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider now the case where at least one of op and op′ are  pending in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Note that if op is pending in 𝐸, then a response to it is appended in 𝐸′′, and hence it  cannot be the case that op <𝑐𝑜𝑚𝑝 (𝐸′′) op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, op is complete in 𝐸, and op′ is pending in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In 𝐹 ′,  either a response to op′ is appended, or no responses to it is appended because op′ is complete in 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In any case, we have op <𝑐𝑜𝑚𝑝 (𝐹 ′) op′, and hence op <𝑆 op′, as <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆, which ultimately  implies that op <𝑆𝐸 op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, we conclude that <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑆𝐸, and hence 𝑆𝐸 is a linearization  of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  For the second claim, consider any history 𝐸′ obtained from 𝐸 by appending responses to some  pending operations and removing invocations of some pending operations, with 𝐸′ and 𝐹 being  equivalent and ≺𝐸′ ⊆ ≺𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐼 and 𝑅𝐸 denote the sets with the invocations and responses removed  and appended, respectively, to obtain 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any linearization 𝑆 of 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, there is an  extension 𝐹 ′ of 𝐹 such that 𝑐𝑜𝑚𝑝(𝐹 ′) and 𝑆 are equivalent and <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝑅𝐹 denote the  set with the responses appended to obtain 𝐹 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐸′′ be any extension of 𝐸 obtained by appending  to it the responses in 𝑅𝐸 ∪ 𝑅𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We prove that 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑆 are equivalent and <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑆,  from which follows that 𝑆 is a linearization of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As 𝐸′ and 𝐹 are equivalent, it is easy to see  that 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑐𝑜𝑚𝑝(𝐹 ′) are equivalent (just note that the invocations in 𝐼 do not appear in  𝑐𝑜𝑚𝑝(𝐹 ′)), and hence 𝑐𝑜𝑚𝑝(𝐸′′) and 𝑆 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Now, consider any op <𝑐𝑜𝑚𝑝 (𝐸′′) op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Observe  that it cannot be the case that the response of op is in 𝑅𝐸 ∪ 𝑅𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, op is completed in 𝐸;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' however  op′ might be pending or complete in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Note that op <𝑐𝑜𝑚𝑝 (𝐸′′) op′ implies op ≺𝐸′ op′, and hence  op ≺𝐹 op′, as ≺𝐸′ ⊆ ≺𝐹, from which follows op <𝑐𝑜𝑚𝑝 (𝐹 ′) op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We thus have <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑐𝑜𝑚𝑝 (𝐹 ′),  and then <𝑐𝑜𝑚𝑝 (𝐸′′) ⊆ <𝑆 because <𝑐𝑜𝑚𝑝 (𝐹 ′) ⊆ <𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, 𝑆 is a linearization of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  It can similarly be shown that variants of linearizability such as set-linearizability [75] and  interval-linearizability [18], among others, are in the class GenLin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The reason is that these variants  differ from linearizability only in the properties of the linearization 𝑆 of a given execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In some  cases, 𝑆 can “deviate” from sequential executions of state machine O, or it might be the case that 𝑆  is not necessarily a sequential execution, where several operations can occur simultaneously at the  same time, or even an operation can overlap several operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2  The class DRV  Let A be any implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We already have seen that it is impossible to verify whether A  is linearizable with respect to some sequential objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As anticipated, we will see however that  A can be indirectly verified through an implementation A∗ that can be easily obtained from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The implementation A∗ appears in Figure 6, which assumes in Line 05 that the content of all  entries of a shared array can be atomically read with a Snapshot operation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is known that the  atomic Snapshot operation can be wait-free linearizable implemented using only Read/Write base  operations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [1, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, the step in Line 05 can be assumed to be wait-free and atomic, due to  the modular properties of linearizability [60, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  14  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  Shared Variable:  𝑁 [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ,𝑛] = shared array of Read/Write base objects, each initialized to ∅  Local Persistent Variable:  𝑠𝑒𝑡𝑖 = a set initialized to ∅  Operation Apply(op𝑖) is  (01) 𝑠𝑒𝑡𝑖 ← 𝑠𝑒𝑡𝑖 ∪ {(𝑝𝑖, op𝑖) }  (02) 𝑁 [𝑖].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Write(𝑠𝑒𝑡𝑖)  (03) Invoke operation Apply(op𝑖) of A  (04) 𝑦𝑖 ← response from operation Apply(op𝑖) of A  (05) 𝑠𝑖 ← 𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Snapshot()  (06) 𝜆𝑖 ← �  𝑘∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=',𝑛} 𝑠𝑖 [𝑘]  (07) return (𝑦𝑖, 𝜆𝑖)  end Verify  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' From A to A∗ ∈ DRV (code of process 𝑝𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In A∗, every process 𝑝𝑖 simply announces in a shared memory the next high-level operation op𝑖  it wants to execute, then obtains a response for op𝑖 using A, atomically reads all operations that  have been announced so far in the shared memory, storing them all together in a set 𝜆𝑖, and finally  returns the set together with the response obtained from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The set 𝜆𝑖 is called the view of (𝑝𝑖, op𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  As we will see, this simple mechanism, the views, succinctly encode an “sketch” of A’s and A∗’s  current histories, and this sketch is what makes A∗ strongly verifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3 (The class DRV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' DRV denotes the the class of concurrent implementations  obtained through the construction in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3  Analyzing A∗  A∗ preserves A’s properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We first show that A∗ preserves progress and correctness properties  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For the discussion in the rest of the section, we disregard the views in the responses of A∗  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the sets 𝜆𝑖), unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Recall that the execution itself of an algorithm denotes the  history obtained from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 (Correctness of A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any implementation A and any object O in the class  GenLin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, A is correct with respect to O if and only if A∗ is correct with respect to O (disregarding  the views in the responses of A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Furthermore, A∗ possesses the same progress condition as A, and  its step complexity is the step complexity of A plus 𝑂(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' First, it is easy to see the segment of code in Lines 01 to 02 is wait-free, as well as the  segment in Lines 05 to 07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, if A is lock-free (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' wait-free) then A∗ lock-free (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' wait-free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  As for step complexity, it executes one Write before invoking A, and one Snapshot after, which can  be implemented in 𝑂(𝑛) step using the algorithm in [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For correctness, we prove each direction  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  ⇒ Let 𝐸 be any finite execution of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We show that 𝐸 is correct, namely, the history obtained  from it (denoted 𝐸 as well) belongs to O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since 𝐸|A ∈ O and O is closed by similarity, both  by assumption, it suffices to prove that 𝐸 is similar to history 𝐸|A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider the history  𝐸′ obtained from 𝐸 by: (1) appending to 𝐸 the response in 𝐸|A to every operations that is  pending in 𝐸 but complete in 𝐸|A, and (2) removing every invocation of a pending operation  in 𝐸 that does not appear in 𝐸|A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' From the definition of 𝐸′ and the pseudocode in Figure 6, it  can be easily verified that 𝐸′ and 𝐸|A are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Additionally, it holds that ≺𝐸′ ⊆ ≺𝐸 |A: if  op ≺𝐸′ op′, then we must have that 𝑟𝑒𝑠(op) precedes 𝑖𝑛𝑣(op′) in 𝐸|A because in A∗ operation  calls to A are nested in operation calls of A∗, and hence op ≺𝐸 |A op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  15  Castañeda and Rodríguez  ⇐ Let 𝐸 be any finite history of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We argue that 𝐸 ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The asynchrony in the model  guarantees the existence of the following execution 𝐸′ of A∗:  (1) for every process 𝑝𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' for each of its operations Apply(op𝑖),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the corresponding invocation and  the steps from Lines 01 to 02 appear all together right before the invocation to Apply(op𝑖)  of A in Line 02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' and  (2) for every process 𝑝𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' for each of its operations Apply(op𝑖),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the corresponding response and  the steps from Lines 05 to 07 appear all together right after the response from Apply(op𝑖)  of A in Line 04 (if there is one),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' and  (3) 𝐸 and 𝐸′ are the same history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Basically, 𝐸′ is obtained from 𝐸 by adding steps of A∗ right before and after the invocations  and responses in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By assumption, A∗ is correct with respect to O, and hence 𝐸′ ∈ O, from  which follows that 𝐸 ∈ O, as 𝐸 and 𝐸′ are the same history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  p1  p2  enq(1) : true  deq() : 1  Current execution A∗:  Current execution A:  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Due to asynchrony, A∗ is able to “fix” some histories of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The figure depicts a history of A∗ where  the history of A is not linearizable with respect to the queue, however, the history of A∗ is linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Each  operation Apply(op) is simply denoted op and the views of the operations of A∗ are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The implementation A∗ can be alternatively understood as a mechanism that “fixes” some  incorrect histories of A, as the example in Figure 7 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Nevertheless, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 implies that  A∗ cannot fix all incorrect histories of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For those histories that it is not able to fix, the views  provide a mechanism to detect they are incorrect, which will be crucial to fulfill the completeness  requirement of the distributed runtime strong verification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Tight executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let A∗ be any implementation in the class DRV, and consider any finite  execution 𝐸 of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We say that 𝐸 is tight if: (a) each of its pending operations has its Write step  in Line 02 but no Snapshot step in Line 05, (b) the invocation and local step in Line 01 of every  operation, complete or pending, appear in a sequence right before its Write step in Line 02, and (c)  the local steps in Lines 06 and 07 and response of every complete operation appear in a sequence  right after its Snapshot step in Line 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Basically, in a tight execution, the beginning and end of an  operation are identified with the Write and Snapshot steps in Lines 02 and 05, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Any finite execution 𝐸 of A∗ can be “transformed” into a tight execution𝑇 (𝐸) of A∗ as described  next: (a) remove the invocation and local step in Line 01 of every pending operation with no Write  step in Line 02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' (b) for each of the remaining operations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' complete or pending,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' move forward the  invocation and local step in Line 01 in a sequence right before its Write step in Line 02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' (c) for each  of the remaining operations with its Snapshot step in Line 05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' first move backwards the local steps  in Lines 06 and 07 and response (if there are any) in a sequence right after its Snapshot step in  Line 05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' and then insert (if necessary) the missing local steps in Lines 06 and 07 and response to  complete the sequence that makes the operation complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We say that 𝑇 (𝐸) is the tight execution  associated to 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Observe that indeed 𝑇 (𝐸) is an execution of A∗ because all invocations, responses  and steps that are moved, forward of backward, or inserted are local to processes, and hence it  is immaterial when they occur, or if they occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover, in 𝐸 and 𝑇 (𝐸), the operations obtain  16  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  the same responses from A and compute same views as the order of invocations to and responses  from A are not modified to obtain 𝑇 (𝐸), neither the order of Write and Snapshot steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Intuitively,  the difference between 𝐸 and 𝑇 (𝐸) is that operations in 𝑇 (𝐸) span a possibly “shorter” interval of  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3 (Tight executions and actual executions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐸 be any finite execution of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Then, 𝑇 (𝐸) is an execution of A∗, and for any object O in GenLin,  𝐸|A ∈ O =⇒ 𝑇 (𝐸) ∈ O =⇒ 𝐸 ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As already argued, 𝑇 (𝐸) is an execution of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We will show that (history) 𝑇 (𝐸) is  similar to (history) 𝐸|A and (history) 𝐸 is similar to (history) 𝑇 (𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' These two facts will prove the  implications because O is closed by similarity, as it belongs to GenLin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We argue first that 𝐸 is similar to𝑇 (𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐸′ be the history obtained from 𝐸 by: (1) removing the  invocation of every pending operation that is removed from 𝐸 to obtain 𝑇 (𝐸), and (2) appending  the response in 𝑇 (𝐸) of every pending operation in 𝐸 whose response is inserted to obtain 𝑇 (𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Basically, 𝐸′ is obtained following steps (a) and (c) in the construction from 𝐸 to 𝑇 (𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is nor  hard to see that 𝐸 and 𝑇 (𝐸) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider now any op ≺𝐸′ op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Hence 𝑟𝑒𝑠(op) precedes  𝑖𝑛𝑣(op′) in 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Observe that it cannot be that 𝑟𝑒𝑠(op) is appended to 𝐸 to obtain 𝐸′, and 𝑇 (𝐸), and  hence op is complete in 𝐸;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' op′ is complete or pending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We must have that op ≺𝑇 (𝐸) op′ becase to  obtain 𝑇 (𝐸), responses of complete operations might only moved backward and invocations of  complete or pending operations might only moved forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore we have ≺𝐸 ⊆ ≺𝑇 (𝐸), from  which follows that 𝐸 is similar to 𝑇 (𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We show now that 𝑇 (𝐸) is similar to 𝐸|A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider the history 𝐸′ obtained from 𝑇 (𝐸) by: (1)  removing every invocation of a pending operation in 𝑇 (𝐸) that does not appear in 𝐸|A (any such  operation executes its Write step in Line 02 but does not executes the invocation in Line 03), and (2)  appending the response in 𝐸|A to every operations that is pending in 𝑇 (𝐸) but complete in 𝐸|A  (any such operation operation executes its response in Line 04, but does not execute its Snapshot  step in Line 05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is not difficult to verify that 𝐸′ and 𝐸|A are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is also true that  ≺𝐸′ ⊆ ≺𝐸 |A: if op ≺𝐸′ op′, then we must have that 𝑟𝑒𝑠(op) precedes 𝑖𝑛𝑣(op′) in 𝐸|A because in A∗  operation calls to A are nested between the Write and Snapshot steps in Lines 02 and 05, from  which follows that op ≺𝐸 |A op′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, 𝑇 (𝐸) is similar to 𝐸|A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3 suggests a way to strongly verify the implementations of DRV through their tight  executions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the idea is actually very simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Suppose that somehow processes are able to compute  𝑇 (𝐸) of any execution 𝐸 of A∗ ∈ DRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' First, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3 shows that 𝑇 (𝐸) is a history of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' If 𝐸 is  not correct, then the lemma implies that 𝑇 (𝐸) is not correct either, and hence it is a witness for A∗  that can be reported to satisfy completeness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' and if 𝐸 is correct, then 𝑇 (𝐸) might be correct or not,  but in either case strong soundness can be satisfied because if 𝑇 (𝐸) is not correct, it is a witness  for A∗ that can be reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In the rest of the section, we argue that the views encode the tight execution associated to the  actual execution of A∗, which opens the possibility to implement the simple idea just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  From views to tight executions and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐸 be any finite execution of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In the associated  tight execution𝑇 (𝐸), let us replace each invocation of 𝑝𝑖 to operation Apply(op𝑖), with the invocation  pair (𝑝𝑖, op𝑖), and replace each response from operation Apply(op𝑖) to 𝑝𝑖 (if there is one) with the  set with all invocation pairs (𝑝𝑗, op𝑗) that precedes the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Figure 8 depicts an example of  the replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since invocations and responses in 𝑇 (𝐸) are associated to the Write and Snapshot  steps in Lines 05 and 02, respectively, the view returned by any operation in 𝑇 (𝐸) is exactly the set  just defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We show that a “sketch” of the history 𝑇 (𝐸) can be directly obtained from the views  17  Castañeda and Rodríguez  of operations in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' First, from the properties of the Snapshot [1] and the pseudocode of A∗, we  obtain the following:  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider the views 𝜆𝑖 and 𝜆𝑗 in the responses of any pair of completed operations  Apply(op𝑖) and Apply(op𝑗) by 𝑝𝑖 and 𝑝𝑗 (possibly with 𝑝𝑖 = 𝑝𝑗) in any execution of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The next  properties are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (1) Self-inclusion: (𝑝𝑖, op𝑖) ∈ 𝜆𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (2) Containment comparability: 𝜆𝑖 ⊆ 𝜆𝑗 ∨ 𝜆𝑗 ⊆ 𝜆𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (3) Process sequentiality: if 𝑝𝑖 = 𝑝𝑗 ∧ op𝑖 ≠ op𝑗, then (𝑝𝑖, op𝑖) ∉ 𝜆𝑗 ∨ (𝑝𝑗, op𝑗) ∉ 𝜆𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  For any tight execution 𝐸 of A∗, let 𝜆𝐸 will denote the set with all 4-tuples (𝑝𝑖, op𝑖,𝑦𝑖, 𝜆𝑖) such  that (𝑦𝑖, 𝜆𝑖) is the response of operation Apply(op𝑖) of 𝑝𝑖 in 𝐸 (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We now explain that a  well-formed history 𝑋 (𝜆𝐸) can be obtained from 𝜆𝐸, and explain in what sense 𝑋 (𝜆𝐸) is a sketch  of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The construction that follows is from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  By Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (2), all distinct views that appear in 𝜆𝐸 can be ordered in strictly containment  ascending order: 𝜎1 ⊂ 𝜎2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ⊂ 𝜎𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝜎0 denote ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For each 𝑘 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ,𝑚 (in ascending order),  𝑋 (𝜆𝐸) is iteratively obtained following the next two steps in order (see Figure 8 for an example):  (1) For each invocation pair (𝑝𝑖, op𝑖) ∈ 𝜎𝑘 \\ 𝜎𝑘−1, the invocation to operation Apply(op𝑖) by 𝑝𝑖 is  appended to 𝑋 (𝜆𝐸);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the invocations are appended in any arbitrary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (2) For each (𝑝𝑖, op𝑖,𝑦𝑖, 𝜆𝑖) ∈ 𝜆𝐸 with 𝜆𝑖 = 𝜎𝑘, the response from operation Apply(op𝑖) with  output (𝑦𝑖, 𝜆𝑖) by 𝑝𝑖 is appended to 𝑋 (𝜆𝐸);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the responses are appended in any arbitrary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  p1  p2  Apply(op1) : a  p3  Apply(op′  1) : b  Apply(op2) : c  Apply(op3) : d  (p1, op1)  (p1, op′  1)  (p1, op1)  (p2, op2)  (p1, op1)  (p1, op′  1)  (p2, op2)  (p3, op3)  (p1, op1)  (p1, op′  1)  (p2, op2)  (p3, op3)  λE = {(p1, op1, a, view), (p1, op′  1, b, view′), (p3, op3, d, view′′)}  view = {(p1, op1)}  view′ = {(p1, op1), (p1, op′  1), (p2, op2)}  view′′ = {(p1, op1), (p1, op′  1), (p2, op2), (p3, op3)}  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The figure shows the history of a tight execution 𝐸 of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The invocation pair and the view of each  operation is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The set 𝜆𝐸 is depicted too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is easy to check that the construction from 𝜆𝐸 produces a  history 𝑋 (𝜆𝐸) that is equivalent to 𝐸 with ≺𝐸 = ≺𝑋 (𝜆𝐸), and hence 𝐸 and 𝑋 (𝜆𝐸) are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In each of the steps of the construction above, either a set of invocations or responses are placed  in some arbitrary sequential order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For any of these orders, the resulting history has the same  relation ≺ over pending and complete operations, by construction, and hence all possible histories  obtained in this way are similar to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, in fact, 𝑋 (𝜆𝐸) denotes an equivalence class of  histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 By similarity-closure of GenLin, we have:  1In the parlance of [18], 𝑋 (𝜆𝐸) is an interval-sequential history, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', an alternating sequence of non-empty sets with only  either invocations or responses, starting with a set of invocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Interval-sequential histories are used in [18] to define  interval-linearizability, a generalization of linearizability where, roughly speaking, an operation is allowed to overlap other  operations in an interval-linearization of an execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  18  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For every object O in GenLin and every tight execution 𝐸 of A∗, either all histories  of 𝑋 (𝜆𝐸) are in O or none of them is in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  By abuse of notation, we let 𝑋 (𝜆𝐸) denote any history of the equivalence class, unless stated  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The duality between histories and sets of views has been investigated in [18], where it is shown  that the construction above is a bijection between (equivalence classes of) well-formed finite  histories and sets of 4-tuples (𝑝𝑖, op𝑖,𝑦𝑖, 𝜆𝑖) whose views satisfy the properties in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The  views can be understood as a static mechanism that captures the dynamic real-time order of  operations in a history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 in [18] directly implies that 𝑋 (𝜆𝐸) is indeed an accurate sketch  of 𝐸 in the following sense:  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='4 (Views are sketches of tight executions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For any tight execution 𝐸 of A∗,  𝐸 and 𝑋 (𝜆𝐸) are equivalent with ≺𝐸 = ≺𝑋 (𝜆𝐸), hence the histories are similar to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  As every O ∈ GenLin is closed by similarity, we have:  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For any tight execution 𝐸 of A∗ and every object O in GenLin,  𝐸 ∈ O ⇐⇒ 𝑋 (𝜆𝐸) ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' If 𝐻 is a history of an implementation B (not necessarily in the class DRV), then any  history 𝐻 ′ that is equivalent to 𝐻 with ≺𝐻 = ≺𝐻′ is a history of B as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since 𝐻 is a history of B, there is an execution 𝐸 of B such that 𝐻 is the history obtained  from 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' History 𝐻 has the form 𝐼1𝑅1𝐼2𝑅2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', where each 𝐼𝑥 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 𝑅𝑥) is a non-empty sequence  of invocations (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' responses);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' note that the invocations in 𝐼𝑥 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' responses in 𝑅𝑥) does not  necessarily appear in a continuos sequence in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The main observation to prove the claim is that for  every history 𝐻 ′ such that is equivalent to 𝐻 with ≺𝐻 = ≺𝐻′, we must have that 𝐻 ′ = 𝐼 ′  1𝑅′  1𝐼 ′  2𝑅′  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  with 𝐼 ′  𝑥 (resp 𝑅′  𝑥) being a permutation of 𝐼𝑥 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 𝑅𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then, consider the execution 𝐸′ obtained  from 𝐸 as follows: (1) for each 𝐼𝑥, first move all invocations in 𝐼𝑥 to the position of the first invocation  in 𝐼𝑥, and then permute them according to 𝐼 ′  𝑥, and similarly (2) for each 𝑅𝑥, first move all responses  in 𝑅𝑥 to the position of the last response in 𝑅𝑥, and then permute them according to 𝑅′  𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Observe  that 𝐸′ is an execution of B because only invocation and responses of 𝐸, which are local steps, are  modified to obtain 𝐸′, and it is immaterial when these local steps actually occur, as long as the  specification of B is satisfied, as it happens in 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By construction, 𝐻 ′ is the history obtained from  𝐸′, and therefore, 𝐻 ′ is a history of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  Finally, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='4 and Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 imply:  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For any tight execution 𝐸 of A∗, 𝑋 (𝜆𝐸) is a history of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='4  A note on the class DRV and refined task solvability  The views mechanism was introduced in [18] (implicitly defined in [44] too) in order to extend the  task specification formalism [58] such that it is able to capture linearizable sequential long-lived  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The resulting formalism is called multi-shot refined tasks, where processes are required to  produce outputs and implicitly produce views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It turns out that multi-shot refined tasks are strictly  more expressive than linearizability, and are equally expressive as interval-linearizability, also  introduced in [18]: for every interval-sequential object, there is an equivalent multi-shot refined  task, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' From this perspective, the implementations in DRV solve the corresponding  equivalent multi-shot refined tasks, producing explicit views, which is what makes them runtime  verifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  19  Castañeda and Rodríguez  8  THE CLASS DRV IS DISTRIBUTED RUNTIME STRONGLY VERIFIABLE  A wait-free strong verifier for DRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='4 in the previous section are the basis of the  wait-free strong verifier VO in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The idea of the verifier is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For any finite execution 𝐸  of the verifier, the views in 𝜆𝑇 (𝐸 |A∗) sketch the tight execution 𝑇 (𝐸|A∗) associated to the current  execution 𝐸|A∗ of A∗ (Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='4), and tight executions suffice to fulfill strong soundness and  completeness (Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, in VO the processes simply exchange their views using a Write  and a Snapshot (Lines 07 to 09) and then each process locally tests if the execution it reads from  the shared memory is correct (Lines 10 to 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Shared Variables:  𝑀 [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ,𝑛] = shared array of Read/Write base objects, each initialized to ∅  Operation Verify(A∗ ∈ DRV) is  (01) 𝑟𝑒𝑠𝑖 ← ∅  (02) while true do  (03)  𝑜𝑝𝑖 ← non-deterministically chosen operation that does not appear in 𝑟𝑒𝑠𝑖  (04)  Invoke operation Apply(𝑜𝑝𝑖) of A∗  (05)  (𝑦𝑖, 𝜆𝑖) ← response from operation Apply(𝑜𝑝𝑖) of A∗  (06)  𝑟𝑒𝑠𝑖 ← 𝑟𝑒𝑠𝑖 ∪ {(𝑝𝑖,𝑜𝑝𝑖, 𝑦𝑖, 𝜆𝑖) }  (07)  𝑀 [𝑖].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Write(𝑟𝑒𝑠𝑖)  (08)  𝑠𝑖 ← 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='Snapshot()  (09)  𝜏𝑖 ← �  𝑘∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=',𝑛} 𝑠𝑖 [𝑘]  (10)  if 𝑋 (𝜏𝑖) ∉ O then  (11)  report (ERROR,𝑋 (𝜏𝑖))  (12)  end if  (13) end while  end Verify  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A wait-free strong verifier VO for any object O in GenLin, and any implementation of the class DRV  (code of process 𝑝𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Although VO relies on a simple idea, proving it correct is not simple at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The main reason is  that in its Snapshot step in Line 08, a process might obtain only a proper subset of the set 𝜆𝑇 (𝐸 |A∗),  and hence the history 𝑋 (𝜏𝑖) used in the test in Line 10 might not be exactly𝑇 (𝐸|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This situation  can happen due to asynchrony because some processes might have already obtained a response  from A∗ in 𝐸|A∗ but no written yet their responses in the 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' To deal with this issue, Lemmas 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1  and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 below show useful properties of the histories computed in Line 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  As mentioned above, Snapshot can be wait-free implemented using only Read/Write base objects,  and moreover, there are implementations with 𝑂(𝑛) step complexity [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, clearly VO is a  wait-free, and uses only Read/Write base objects, and every iteration of the while loop takes 𝑂(𝑛)  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus we have:  Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The verifier VO in Figure 9 is wait-free, uses only Read/Write base objects with step  complexity 𝑂(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  To prove the correctness of VO, we analyze only its infinite executions in which the sequence of  local computation steps in Lines 09 to 12 of a process 𝑝𝑖 appear all together right after the previous  Snapshot step of 𝑝𝑖 in Line 08 (which is part of the same loop iteration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Restricting our attention  to these executions facilitates the discussion and proves the verifier to be correct in all cases, as  it is immaterial when these steps actually occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This restriction can also be seen from a slightly  different perspective: given any infinite execution of the verifier, the local steps in Lines 09 to 12  of a process 𝑝𝑖 can be moved backwards to be right next to the previous Snapshot step of 𝑝𝑖 in  Line 08, and the processes still make the same decisions in the modified execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  20  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐸 be any infinite execution of the wait-free verifier VO in Figure 9 with an arbitrary  input A∗ ∈ DRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any finite prefix 𝐸′ of 𝐸 whose last sequence of steps correspond to the  steps in Lines 08 to 12 of a process 𝑝𝑖, and let 𝜏 ′ denote the content of 𝜏𝑖 of 𝑝𝑖 at the end of 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then:  (1) 𝑋 (𝜏 ′) is a history of A∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (2) for every object O in GenLin, 𝑇 (𝐸′|A∗) ∈ O =⇒ 𝑋 (𝜏 ′) ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3,𝑇 (𝐸′|A∗) is an execution of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By analyzing𝑇 (𝐸′|A∗), we will conclude  that 𝑋 (𝜏 ′) is a history of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By the definition of 𝐸′, it follows that 𝜏 ′ contains all 4-tuples that  appear in 𝑀 at the end of 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Due to asynchrony, it is possible that not all 4-tuples in 𝜆𝑇 (𝐸′|A∗)  appear in 𝑀 at the end of 𝐸′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the reason is that it can be the case that in 𝐸′ a process executes its  Snapshot in Line 05 of A∗, and hence the 4-tuple of the corresponding operation is in 𝑇 (𝐸′|A∗),  by definition of tight executions, but does not execute its Write step in Line 07 of VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Hence, at the  end of 𝐸′, the response of the last operation in 𝑇 (𝐸′|A∗) of a process in might be “missing” in 𝜏 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We thus have that 𝜏 ′ ⊆ 𝜆𝑇 (𝐸′|A∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' moreover, 𝜆𝑇 (𝐸′|A∗) \\ 𝜏 ′ has at most one 4-tuple for each process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We will modify 𝑇 (𝐸′|A∗) to obtain an execution 𝐹 of A∗ whose history is precisely 𝑋 (𝜏 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As  already said, each 4-tuple that appears in 𝜆𝑇 (𝐸′|A∗) \\𝜏 ′ corresponds to an operation that is complete  in 𝑇 (𝐸′|A∗) and that operation is the last one of the corresponding process in the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐹  be the execution obtained from 𝑇 (𝐸′|A∗) as follows: for each 4-tuple in 𝜆𝑇 (𝐸′|A∗) \\ 𝜏 ′, remove the  steps in Lines 05 to 07 of A∗ (see Figure 6) of the operation corresponding to the 4-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Clearly, 𝐹  is an execution of A∗ as only the last steps and response of the last operation of some processes  are removed from 𝑇 (𝐸′|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover, it is a tight execution as each of its pending operations  does not execute the Snapshot step in Line 05 of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By construction, we have that 𝜏 ′ = 𝜆𝐹, from  which follows that 𝑋 (𝜏 ′) and 𝑋 (𝜆𝐹) are equivalent with ≺𝑋 (𝜏′) = ≺𝑋 (𝜆𝐹 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Finally, 𝐹 and 𝑋 (𝜆𝐹) are  equivalent with ≺𝐹 = ≺𝑋 (𝜆𝐹 ), by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='4, and hence 𝑋 (𝜏 ′) is a history of A∗, by Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 and  as 𝐹 is an execution of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  From the previous discussion, it is easy to see that 𝐹 is similar to 𝑇 (𝐸′|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐹 ′ be the history  obtained from 𝐹 by appending the responses the responses in the 4-tuples of 𝜆𝑇 (𝐸′|A∗) \\ 𝜏 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It  is not difficult to see that 𝐹 ′ and 𝑇 (𝐸′|A∗) are equivalent with ≺𝐹 ′ = ≺𝑇 (𝐸′|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since any object  O ∈ GenLin is closed by similarity, we conclude that𝑇 (𝐸′|A∗) ∈ O =⇒ 𝐹 ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' From the discussion  above, we know that 𝐹 and 𝑋 (𝜏 ′) are similar to one another, and hence 𝐹 ∈ O ⇐⇒ 𝑋 (𝜏 ′), and  therefore 𝑇 (𝐸′|A∗) ∈ O =⇒ 𝑋 (𝜏 ′) ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐸 be any infinite execution of the wait-free verifier VO in Figure 9 with an arbitrary  input A∗ ∈ DRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any finite prefix 𝐸′ of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' There is a finite prefix 𝐹 of 𝐸 such that for every  finite prefix 𝐸′′ of 𝐸 that has 𝐹 as one its prefixes and whose last sequence of steps correspond to the  steps in Lines 08 to 12 of a process 𝑝𝑖, it holds that 𝑇 (𝐸′|A∗) is similar to a prefix of 𝑋 (𝜏 ′′), where 𝜏 ′′  denotes the content of 𝜏𝑖 of 𝑝𝑖 at the end of 𝐸′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='4 implies that we can concentrate on 𝑋 (𝜆𝑇 (𝐸′|A∗)) to reason about 𝑇 (𝐸′|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Observe that due to asynchrony, it is possible that not all 4-tuples in 𝜆𝑇 (𝐸′|A∗) appear in 𝑀 at the  end of 𝐸′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the reason is that it is possible that in 𝐸′ a process executes its Snapshot in Line 05 of A∗  (see Figure 6), and hence the 4-tuple of the corresponding operation is in 𝜆𝑇 (𝐸′|A∗), by definition  of tight executions, but does not execute its Write step in Line 07 of VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, at the end of 𝐸′,  the response of the last operation in 𝜆𝑇 (𝐸′|A∗) of a process might be “missing” in 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since 𝐸 is  infinite and by assumption fair (see Section 2), there is a finite prefix 𝐹 of it in which all 4-tuples in  𝜆𝑇 (𝐸′|A∗) appear in 𝑀 at the end of 𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' note that at the of 𝐹, 𝑀 might contain 4-tuples of operations  that do not appear in 𝜆𝑇 (𝐸′|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We claim that 𝐹 is the prefix of 𝐸 with the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Let 𝐸′′ be any finite prefix of 𝐸 that has 𝐹 as one its prefixes and whose last sequence of steps  correspond to the steps in Lines 08 to 12 of a process 𝑝𝑖, and let 𝜏 ′′ denotes the content of 𝜏𝑖 of  21  Castañeda and Rodríguez  𝑝𝑖 at the end of 𝐸′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is not difficult to verify that 𝐸′ is a prefix of 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Using similar arguments as  above, it can be argued that 𝜏 ′′ ⊆ 𝜆𝑇 (𝐸′′|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover, the election of 𝐹 and the definition of tight  execution give that 𝜆𝑇 (𝐸′|A∗) ⊆ 𝜏 ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It directly follows from the definition of tight execution that if  an operation is pending in 𝑇 (𝐸′|A∗), then in 𝐸′ it does not execute its Snapshot step in Line 05  of A∗ (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We thus have that the view of any 4-tuple in 𝜏 ′′ \\𝜆𝑇 (𝐸′|A∗) contains the largest  view among the views in the 4-tuples in 𝜆𝑇 (𝐸′|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We will now reason how 𝑋 (𝜆𝑇 (𝐸′|A∗)) and 𝑋 (𝜏 ′′) are constructed from 𝜆𝑇 (𝐸′|A∗) and 𝜆𝜏′′, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝜎′  1 ⊂ 𝜎′  2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ⊂ 𝜎′  𝑘′ and 𝜎′′  1 ⊂ 𝜎′′  2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ⊂ 𝜎′′  𝑘′′ be respectively the distinct views in  𝜆𝑇 (𝐸′|A∗) and 𝜏 ′′, ordered in ascending order by containement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For the reasons above exposed, we  have that (1) 𝑘′ ≤ 𝑘′′, (2) 𝜎′  ℓ = 𝜎′′  ℓ , for 1 ≤ ℓ ≤ 𝑘′, and (3) 𝜎′  𝑘′ ⊂ 𝜎′′  ℓ , for 𝑘′ < ℓ ≤ 𝑘′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore,  the construction of 𝑋 (𝜆𝑇 (𝐸′|A∗)) and 𝑋 (𝜏 ′′) from 𝜆𝑇 (𝐸′|A∗) and 𝜏 ′′ use the same first 𝑘′ views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let  𝑆 be the subset of 𝜏 ′′ with all 4-tuples whose views are subset of 𝜎′  𝑘′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Note that 𝜆𝑇 (𝐸′|A∗) ⊆ 𝑆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  intuitively, the difference between 𝑆 and 𝜆𝑇 (𝐸′|A∗) is that 𝜆𝑇 (𝐸′|A∗) might be missing the response  of the last operation of some processes and the views of these operation are contained by 𝜎′  𝑘′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Hence, 𝑆 \\ 𝜆𝑇 (𝐸′|A∗) has at most one 4-tuple for each process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝐻 be the shortest prefix of  𝑋 (𝜆𝑇 (𝐸′|A∗)) containing all operations whose views are subset of 𝜎′  𝑘′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' From the definition of 𝑆 and  the construction of 𝑋 (𝑆), it directly follows that 𝐻 and 𝑋 (𝑆) are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' To conclude  the proof, we argue that 𝑋 (𝜆𝑇 (𝐸′|A∗)) is similar to 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let 𝑋 ′ be the history obtained by appending  to 𝑋 (𝜆𝑇 (𝐸′|A∗)) the responses in the 4-tuples of 𝑆 \\ 𝜆𝑇 (𝐸′|A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' From the discussion above, it can be  concluded that 𝑋 ′ and 𝐻 are equivalent and ≺𝑋 ′ = ≺𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, 𝑋 (𝜆𝑇 (𝐸′|A∗)) is similar to 𝐻, and  hence 𝑇 (𝐸′|A∗) is similar to 𝐻 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  We now are finally ready to prove that GenLin is strongly verifiable for the class DRV of  implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (Distributed runtime strong verifiability of GenLin for the class DRV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Let O be any object in the class GenLin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The verifier VO in Figure 9 is a wait-free strong verifier for  the correctness of O for the class DRV of implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Furthermore, VO satisfies the following  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (1) Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It uses only Read/Write base objects with step complexity 𝑂(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (2) Soundness for correct executions of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In each infinite execution 𝐸 of it with input A∗, if  for every finite prefix 𝐸′ of 𝐸, it holds that 𝐸′|A ∈ O (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the history of A, the underlaying  implementation of A∗, is correct), then no process reports ERROR in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  (3) Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For every infinite execution 𝐸 of it in which at least one process reports ERROR, there  is a finite prefix of it such that it is reported ERROR in every new loop iteration starting after  the prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As it is shown in Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1, the verifier VO is wait-free, uses only Read/Write base  objects with step complexity 𝑂(𝑛), and hence it satisfies the efficiency property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We now argue that VO is a strong verifier for the correctness of O, namely, it satisfies strong  soundness and completeness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' we also argue that it satisfies soundness for correct executions of A  and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any infinite execution 𝐸 of VO with an arbitrary input implementation  A∗ ∈ DRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Strong soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Suppose that there is a finite prefix 𝐸′ of 𝐸, whose last sequence of steps  correspond to the steps in Lines 08 to 12 of a process 𝑝𝑖, and 𝑝𝑖 reports ERROR at the end  of 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Clearly, 𝑝𝑖 reports ERROR because 𝑋 (𝜏 ′) ∉ O, where 𝜏 ′ denotes the content of 𝜏𝑖 of 𝑝𝑖  at the end of 𝐸′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' hence 𝑝𝑖 reports the history 𝑋 (𝜏 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (1), 𝑋 (𝜏 ′) is a history of  A∗, and hence is a witness for A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, strong soundness is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  22  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  Soundness for correct executions of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any finite prefix 𝐸′′ of 𝐸, whose last sequence  of steps correspond to the steps in Lines 08 to 12 of a process 𝑝𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (2),  𝐸′′|A ∈ O =⇒ 𝑇 (𝐸′′|A∗) ∈ O =⇒ 𝑋 (𝜏 ′′) ∈ O, where 𝜏 ′′ denotes the content of 𝜏𝑖 of 𝑝𝑖 at  the end of 𝐸′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, 𝑝𝑖 does not report ERROR at the end of 𝐸′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Completeness and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider any finite prefix 𝐸′ of 𝐸 such that 𝐸′|A∗ ∉ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3, 𝐸′|A∗ ∉ O =⇒ 𝑇 (𝐸′|A∗) ∉ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 guarantees the existence of a finite  prefix 𝐹 of 𝐸 such that for every finite prefix 𝐸′′ of 𝐸 that has 𝐹 as one its prefixes and  whose last sequence of steps correspond to the steps in Lines 08 to 12 of a process 𝑝𝑖, it  holds that 𝑇 (𝐸′|A∗) is similar to a prefix 𝐻 of 𝑋 (𝜏 ′′), where 𝜏 ′′ denotes the content of 𝜏𝑖  of 𝑝𝑖 at the end of 𝐸′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since O is closed by similarity, we have 𝑇 (𝐸′|A∗) ∉ O =⇒ 𝐻 ∉ O,  and since it is closed by prefixes, 𝐻 ∉ O =⇒ 𝑋 (𝜏 ′′) ∉ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus we conclude that 𝑝𝑖 reports  (ERROR,𝑋 (𝜏 ′′)) at the end of 𝐸′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (1), 𝑋 (𝜏 ′′) is a history of A∗, and hence is  a witness for A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, completeness is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Observe that stability is satisfied too  as the analysis holds for every such 𝐸′′ having 𝐹 as a prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  Runtime self-enforced correct implementations for GenLin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A by-product of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 is a  simple generic methodology for developing runtime self-enforced correct implementations for any  object in GenLin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Roughly speaking, a self-enforced implementation produces verfied histories up  to a moment, which might never happen, where only ERROR is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Consider the strong verifier VO in Figure 9 and let A be any implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2  below, we consider the implementation V𝑂,A∗ obtained from VO and A∗ as defined at the end of  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Recall that V𝑂,A∗ has a single high-level operation Apply(op𝑖) that executes Lines 04  to 09 of VO with A∗, and returns 𝑟𝑒𝑠𝑝𝑖 if 𝑋 (𝜏𝑖) ∈ O, and returns (ERROR,𝑋 (𝜏𝑖)) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' By  Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (2), any witness 𝑋 (𝜏𝑖) for A∗ implies that the current execution of A is not  correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, although 𝑋 (𝜏𝑖) is not necessarily a witness for A, it might be helpful for debugging A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Furthermore, at any time V𝑂,A∗ is able to produce a certificate of the current computation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', a  history that might not be the current one but it is indistinguishable to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, our runtime self-  enforced implementations allow the design of concurrent systems with accountable and forensic  guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 (Runtime self-enforced correct implementations for GenLin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Let O be any  object in GenLin and A be any implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Consider the implementation VA∗,O obtained through  the strong verifier VO in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Then,  (1) VA∗,O and A have the same progress condition, and  (2) if A is correct with respect to O, then VA∗,O is correct with respect to O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' otherwise every finite  execution of VA∗,O is correct with respect to O up to a prefix (which does not necessarily exist)  where every new operation returns ERROR together with a witness for A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' First, since VO is wait-free, by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1, and A and A∗ have the same progress  condition, by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2, we have that VA∗,O and A have the same progress condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Observe that, by the definition of VA∗,O, every infinite execution 𝐸 of V𝑂 (A∗) is naturally  mapped to a unique infinite execution 𝐸′ of VA∗,O, where the histories of A and A∗ are exactly  the same in both executions, and every process passes through essentially the same sequence of  local states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Basically, 𝐸′ is obtained from 𝐸 by respectively replacing the beginning and end of a  loop iteration with the invocation and response of the operation invoked in the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Consider the case where A is correct with respect to O, and let 𝐸 be any infinite execution  of VA∗,O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The soundness for correct executions of A property of VO (see Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1) implies  that no process reports ERROR in 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, every operation returns the same response in 𝐸|A and 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  23  Castañeda and Rodríguez  Consider any finite prefix 𝐸′ of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Using a similar reasoning as in previous proofs, it can be shown  that 𝐸′ is similar to 𝐸′|A, and hence 𝐸′|A ∈ O =⇒ 𝐸′ ∈ O, as O is closed by similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore,  VA∗,O is correct with respect to O, as A is correct with respect to O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Suppose now that A is not correct with respect to O, and let 𝐸 be any infinite execution of VA∗,O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Consider any finite prefix 𝐸′ of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The argument in the previous paragraph shows that if 𝐸′|A ∈ O,  then no process reports ERROR in 𝐸′, and hence 𝐸′ ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus consider the case 𝐸′|A ∉ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  The completeness and stability properties of VO (see Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1) implies that eventually all  new operations in 𝐸 report ERROR together with a witness 𝑋 for A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' To conclude the proof,  for the sake of contradiction, suppose that no process reports ERROR in 𝐸′ but 𝐸′ ∉ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Using a  similar reasoning as in previous proofs, it can be shown that 𝐸′ is similar to 𝑇 (𝐸′|A∗), and hence  𝐸′ ∉ O =⇒ 𝑇 (𝐸′|A∗) ∉ O, as O is closed by similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Since no process reports ERROR in 𝐸′, all  invocation an responses that appear in it have been written in the shared memory 𝑀 at the end  of 𝐸′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus the last process that takes its Snapshot in Line 08 of VO reads the views of all these  operations, and hence the history it computes in Line 08 is 𝑇 (𝐸′|A∗), from which follows that it  reports ERROR in 𝐸′, as we already saw that 𝑇 (𝐸′|A∗) ∉ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We have reached a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  At first glance, one might think that the implementation VA∗,O in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='2 is somehow  verifying A, hence contradicting the impossibility in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' there is no contradiction however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  As A∗ is able to “fix” some executions of A (see Section 7), it is possible that in an execution of  VA∗,O the history of A is not correct, but no process reports ERROR, which happens because A∗  is able to fix that execution of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  9  EXTENSIONS  Base objects of bounded size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The implementation A∗ (Figure 6) and the verifier VO (Figure 9)  use shared object of unbounded size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This unrealistic assumption can be removed by representing  sets as linked lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For A∗, each entry in the shared memory 𝑁 contains the first node of a single  linked list with the items in the set of the process associated to that entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Every time a process adds  an item to its set, it creates a new node, links it to the first node of the list (which then becomes  the second node) and then writes the new node in its entry of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A snapshot operation  in A∗ now returns a vector of positions in the linked lists, hence the elements in the sets in the  snapshot are the nodes that are accesible from the positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The sets in VO can be represented in  the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Decoupling clients and verifiers, and lightweight verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Constructing 𝑋 (𝜏𝑖) in Line 10 of VO  (Figure 9) can be locally computed in polynomial time in the number of operations in 𝜏𝑖 (see the  construction in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus the local test in that line is computed in the time it takes to test if  𝑋 (𝜏𝑖) is in O, plus a polynomial overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This polynomial overhead is desirable for linearizability  since it is known that, for some sequential objects, linearizability of an execution can be decided in  polynomial time [16, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  In the verifier VO (Figure 9), the clients (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the processes that use A∗ (Figure 6) to solve some  distributed problem) are in charge of verifying the current execution of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This might slow down  the client’s computation, which in general is undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We can deal with this issue by designating  a group of processes, the verifiers, to exclusively solve the verification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' That is, while the  clients use A∗ and extract the sketch of the current execution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the communication interface),  the verifiers test whether the sketch is correct or not (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' the monitoring system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Furthermore,  using the snapshot implementations in [7, 90], each client only incurs in a constant or logarithmic  step overhead in every invocation to a high-level operation of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' More concretely, by using any  of those wait-free snapshots implementations, each client in A∗ returns a reference of a snapshot  of 𝑁 instead of its view (obtained through a snpashot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Such snapshot references can be obtained  24  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  in constant [90] or logarithmic [7] number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Once a process (a verifier in our case) has  a snapshot reference, it can obtain the actual snapshot (and hence the corresponding view) in  𝑂(𝑛 log𝑛) steps [7], or in a number of steps proportional to the number of updates in 𝑀 after the  reference was taken [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, in VO, each client writes its snapshot reference in 𝑀 and continues  with the next loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', it does not verify the sketch in 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The verifiers periodically read the sketch  so far in 𝑀 of VO, and test if it is correct, namely, they execute in an infinite loop lines 08 to 12  of VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The resulting runtime verified system satisfies strong soundness and completeness, as long  as not all verifiers crash in an execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Verifying obstruction-free and blocking implementations, and task solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The proposed inter-  active model can be modified to include obstruction-free [59] or blocking [59] implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The  difference is that the interaction between the implementation and a verifier might be finite because  the implementation might block;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' thus the required properties, strong soundness and completeness,  need to be satisfied even if the interaction is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Similarly, task solvability can be strongly runtime verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A task is a one-shot distributed  problem [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is known that any task can be equivalently modelled and a one-shot interval-  sequential interval-linearizable object [18], which belongs to GenLin, since GenLin includes interval-  linearizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Again, the difference is that the interaction with this object is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover,  since every process invokes exactly one high-level operation, now a process can declare that the  computation is correct if it reads from the shared memory the views of all processes in the system  and the corresponding history is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Extension to other models of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The proposed implementations assume a fixed number  of processes that can participate in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The implementations can however be adapted  to shared memory models with an unbounded number of processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=', where there is no prior  knowledge on the number of processes that participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We only have to use the wait-free snapshot  implementation in [45] for this kind of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Due to the shared memory simulation algorithm in [5], the same strong verifiability results hold  for fully asynchronous message-passing systems where less than half processes can crash in any  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The reason is simply that A∗ (Figure 6) and the verifier VO (Figure 9) can be simulated  in such message-passing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  10  RELATED WORK  Runtime verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Runtime verification is an active field of research with important advance-  ments in the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It has been employed in academia and industry for testing, verification  and debugging before system deployment, and to ensure reliability, safety, robustness and security  after deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Runtime verification has been used mostly to analyze software, however it  has also been applied to other types of systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' hardware, hybrid and embedded systems,  cyber-physical systems, distributed and concurrent systems, financial transaction systems, and  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For a detailed exposition of the field, we refer the reader to the recent textbook [6] and  surveys [35, 54, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Distributed runtime verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Distributed runtime verification of distributed systems is con-  sidered and emergent and important topic, that poses several challenges that are yet to be solved  (see [14, 27, 32, 43, 70, 82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Designing distributed, asynchronous, fault-tolerant communication  interfaces are regarded as a difficult problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As far as we know, there are no runtime verification al-  gorithms in the literature (for any correctness property) with a distributed communication interface  that is at the same time fully asynchronous and fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' To runtime verify different properties,  25  Castañeda and Rodríguez  there have been proposed distributed runtime verification algorithms that are failure-free synchro-  nous message-passing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [4, 83]), fault-tolerant message-passing where processes have access to  clocks that are synchronized at some level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [8, 10, 11, 28, 33, 63, 80]), and fully asynchronous  message-passing or shared memory with failure-free processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [20, 37, 39, 85, 88]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Distributed fault-tolerant runtime verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Our approach is close to the series of papers [13,  41, 42] initiated by Fraigniaud, Rajsbaum and Travers, who pioneered the study of distributed  fault-tolerant verification [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In them, it is studied a shared memory concurrent system that  solves a series of tasks [58], and the aim is to runtime verify that the outputs for each task are  correct, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' they satisfy the inputs/outputs relation specifying the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' To do so, an asynchronous  wait-free read/write shared memory algorithm runs every time a task in the series is solved, and  it is assumed that the verification algorithm terminates before the processes solve the next task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Thus, the distributed runtime verification algorithm proposed in those papers is distributed and  fault-tolerant but it is not fully asynchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In sharp contrast, we consider a fully asynchrony and  wait-free shared memory system, where some processes might be verifying the current execution  while at the same time others are executing a high-level operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' GenLin includes tasks, hence  task solvability is covered by our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The main concern in [13, 41, 42] is to understand how  many (input,output) pairs of the other processes (opinions in the parlance of those papers) a process  must know in order to make a verdict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Our approach here is slightly different, as a process makes  no verdict (or implicitly makes a “so far so good” verdict) as long as the computation looks correct  from its perspective, and in the worts case when a process “sees” the pairs of all other processes,  it can decide whether the computation is correct or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Another crucial difference is that the  wait-free interactive verifier proposed here can detect validity violations in a finer way, as real-time  relations of the actual execution are taken into account, whereas in [13, 41, 42] only (input,output)  pairs are used in the verification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For example, by observing only (input,output) pairs,  for consensus it is impossible to detect when a process ran solo and decided a value distinct from  its input, which violates validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' That scenario, in contrast, can be detected by our verifier through  the views mechanism of the class DRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Runtime verification of concurrent algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For concurrent shared memory algorithms, runtime  verification has been mostly used to detect data races, serializability violations (also called atomicity  violations) and deadlocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' less studied properties are order instruction violations, missed signals,  starvation, and high-level correctness conditions such as sequential consistency and linearizability  (see [27, 70, 77]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Typically, in these works, asynchronous failure-free processes are assumed, and  achieving a distributed communication interface is not a primary target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For detecting data races and  serializability violations, several algorithms have been proposed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' some algorithms use techniques  based on the assumption that the concurrent algorithm under inspection uses locks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [84]),  and others use some form of vector clocks to capture the happens-before relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' relations  of causally-related events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [39]), or a mixture of both (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It seems to us that none of  these techniques can be adapted to our setting because: (1) we focus on lock-free implementations  and (2) linearizability totally depends on the real-time order of non-causally related events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In  general, a main difficulty is capturing the actual execution of a concurrent algorithm, as explained  in [27, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A simple solution is to serialize events using a lock (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [71]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' we find this type  of solutions undesirable because, first, it might change the progress condition of the algorithm  under inspection (as explained in the Introduction), and second, the lock creates a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Other algorithms rely on bytecode-level added instructions in order to capture the execution  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [55]), and moreover there are dynamic analysis frameworks working at a bycodelevel that  provide information of the current execution for performing dynamic analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A problem  with these techniques is that the moment when an event happens and the moment when the event  26  Asynchronous Wait-Free Runtime Verification and Enforcement of Linearizability  is registered are not the same (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' they do not occur atomically, as in our interactive model), and  hence the actual execution might not be captured (which is the main argument in the proof of the  impossibility in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Runtime verification of linearizabilty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As far as we know, runtime verification of linearizability has  only been studied in [29, 30], with centralized communication interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Those papers study I/O  refinement, which generalizes linearizability for objects without sequential specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' GenLin  includes objects without sequential specifications too as it includes set-linearizability and interval-  linearizability [18, 19, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In [29, 30], specific code is added to a white-box implementation in order  to record the execution in a log (a sequence of events) that later is verified by a single process  (hence the runtime verification algorithm is neither distributed nor fault-tolerant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Events must be  atomically recorded in the log, which necessarily requires consensus or the use of locks [59, 72, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  High-level operations are divided in mutators and observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' For mutators, the user has to add  code that records in the log when the operation takes effect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' its linearization point), and for  observers, invocations and responses are recorded separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' It is known that there are linearizable  implementations whose linearization points are not fixed (see [59, 72, 78]), hence the approach  in [29, 30] is not general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Finally, it is not explained in [29, 30] what the relation is between the  actual execution of the algorithm under inspection and the execution recorded in the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Runtime enforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Runtime enforcement is an extension of runtime verification whose aim  is to evaluate the current execution of a system under inspection, and halt the system whenever  it deviates from a desired property (see survey [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Runtime enforcement initiated with the  security automata formalism of Schneider [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Our interactive model for distributed runtime  verification can be understood as as distributed version of Schneider’s security automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As far as  we know, so far there have been proposed only a few distributed runtime enforcement algorithms  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' [9, 46, 48, 53, 81]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Accountability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In general, accountability requires correct processes to irrevocably detect safety  violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Note that false positives are not allowed: once a violation is detected, the detection cannot  be revoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The concept of accountability in the context of distributed computing was introduced  in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Motivated by blockchain technologies, there have been recently proposed accountable  algorithms for consensus [17, 21, 22, 89] and general tasks [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' All these works consider semi-  synchronous message-passing systems with malicious Byzantine failures, and the main target is to  irrevocably detect Byzantine processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Here we consider concurrent systems with benign crash  failures, hence processes never deviate from its specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In this scenario the safety violation  one can detect are invalid outputs, as our self-linearizable implementations do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  11  FINAL DISCUSSION  This paper studied the problem of distributed runtime verification of linearizability in asynchronous  wait-free shared memory systems, through a novel interactive model for runtime verification of  correctness conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Distributed runtime verification of linearizability is not an agreement  problem: regardless of the consensus number of the base objects used for verification, the problem  is impossible for common sequential objects such as queues, stacks, sets, priority queues, counters  and the consensus problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' However, a stronger version of the problem can be solved for a class DRV  of concurrent implementations, and without the need of consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Moreover, the possibility result  holds for a correctness condition GenLin that includes linearizability and generalizations of it such as  set-linearizability [75] and interval-linearizability [18, 19], the latter known to be expressive enough  to model tasks [58] and any concurrent object satisfying some reasonable assumptions [18, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  GenLin contains any object that is closed by prefixes and similarity, the latter being a property  27  Castañeda and Rodríguez  identified here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Any concurrent implementation can be transformed into its counterpart in DRV,  and there is a wait-free verifier that satisfies strong soundness and completeness, for the class DRV  and any object in GenLin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A crucial building block in the transformation to implementations in  DRV is that of the views mechanism for capturing the real-time in executions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Read/write  objects suffice to solve strong runtime verification, hence consensus is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  A simple and generic methodology for designing self-enforced correct GenLin implementations  was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Given any concurrent implementation for some GenLin object, one can produce a  self-enforced correct concurrent implementation with the same progress properties, and for the  same object such that all outputs are guaranteed correct (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' verified), or the implementation blocks,  reporting error to every new invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' These implementation are able to produce a certificate of  the current computation at any time, hence allowing the design of systems in a modular manner with  accountable and forensic guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' We are not aware of previous concurrent implementations in  the literature with such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Furthermore, a concurrent asynchronous wait-free runtime  verification algorithm for linearizability can be easily obtained from any of our self-linearizable  implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' As far as we know, this is the first distributed runtime verification algorithm for  any correctness condition that is at the same time fully asynchronous and fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  We believe several directions are worth to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' A natural direction is to study other  correctness conditions in our interactive model, such as sequential consistency [65] (which is  impossible to verify, as argued at the end of Section 5) or causal consistency [3], or data races in  white-box concurrent implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Runtime verification of hyperproperties [24] is interesting  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The main algorithmic technique used in our solutions is that of the views mechanism for  sketching the current execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In general it is interesting to explore if the mechanism helps to  runtime verify other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Finally, conducting experimental evaluations are important to  understand if the proposed algorithms can provide good performance in practical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Acknowledgements  The first author would like to thank Hagit Attiya, Gregory Chockler, Ori Lahav and Serdar Tasiran  for interesting discussions on this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Part of this work was done while the first author was on  sabbatical leave visiting the Department of Computer Science of the University of Surrey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' This  work was partially supported by the research project DGAPA-PAPIIT UNAM IN108723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
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+page_content=' Kim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Vigna, and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Shi, editors, CCS ’21: 2021 ACM SIGSAC Conference on Computer and Communications Security, Virtual Event, Republic  of Korea, November 15 - 19, 2021, pages 1722–1743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Blelloch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
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+page_content=' Constant-time snapshots with applications to  concurrent data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' In J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Lee and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Petrank, editors, PPoPP ’21: 26th ACM SIGPLAN Symposium on Principles  and Practice of Parallel Programming, Virtual Event, Republic of Korea, February 27- March 3, 2021, pages 31–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' ACM,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  A  LINEARIZABILITY FOR SOME OBJECTS IS NOT STRONG VERIFIABLE  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1 (Impossibility of Distributed Runtime Strong Verification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Linearizability  for queues, stacks, sets, priority queues, counters and the consensus problem (modeled as a sequential  object) is not distributed runtime strongly verifiable, even if a verifier uses base objects with consensus  number infinity such as Compare&Swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The proof is nearly the same as the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='1, where executions 𝐸 and 𝐹 of a  hypothetical wait-free verifier Vqueue (in this case weak) are obtained from the non-linearizable  queue implementation A defined in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The difference between the proofs is in the last  step, where it is now observed that the execution 𝐹 can be equally obtained from any wait-free  linearizable queue implementation B (several such implementations appear in [59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' The proof  now concludes by observing that processes cannot report ERROR in 𝐹 (which is allowed by strong  soundness) because there is no witness for B as it is linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Thus, by indistinguishability, no  process reports ERROR in 𝐸, and hence Vqueue does not satisfy completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content=' Therefore, Vqueue  cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
+page_content='  □  32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf'}
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+Traffic Steering for 5G Multi-RAT Deployments
+using Deep Reinforcement Learning
+Md Arafat Habib1, Hao Zhou1, Pedro Enrique Iturria-Rivera1, Medhat Elsayed2, Majid Bavand2,
+Raimundas Gaigalas2, Steve Furr2 and Melike Erol-Kantarci1, Senior Member, IEEE
+1School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada
+2Ericsson Inc., Ottawa, Canada
+Emails:{mhabi050, hzhou098, pitur008, melike.erolkantarci}@uottawa.ca,
+{medhat.elsayed, majid.bavand, raimundas.gaigalas, steve.furr}@ericsson.com
+Abstract—In 5G non-standalone mode, traffic steering is a
+critical technique to take full advantage of 5G new radio while
+optimizing dual connectivity of 5G and LTE networks in multiple
+radio access technology (RAT). An intelligent traffic steering
+mechanism can play an important role to maintain seamless
+user experience by choosing appropriate RAT (5G or LTE)
+dynamically for a specific user traffic flow with certain QoS
+requirements. In this paper, we propose a novel traffic steering
+mechanism based on Deep Q-learning that can automate traffic
+steering decisions in a dynamic environment having multiple
+RATs, and maintain diverse QoS requirements for different
+traffic classes. The proposed method is compared with two
+baseline algorithms: a heuristic-based algorithm and Q-learning-
+based traffic steering. Compared to the Q-learning and heuristic
+baselines, our results show that the proposed algorithm achieves
+better performance in terms of 6% and 10% higher average
+system throughput, and 23% and 33% lower network delay,
+respectively.
+Index Terms—Multi-RAT, traffic steering, reinforcement learn-
+ing
+I. INTRODUCTION
+The dual connectivity between long term evolution (LTE)
+and fifth generation new radio (5G NR) results in multiple
+radio access technologies (multi-RAT) [1], [2]. On the other
+hand, each type of RAT is supposed to have distinctive
+capabilities to serve user equipment (UE) with diverse quality-
+of-service (QoS) requirements. This raises the need of steering
+a specific class of traffic to a certain RAT to fulfill the QoS
+demands. For instance, high throughput video traffic can be
+better served by 5G NR. On the contrary, steering voice
+traffic to LTE base station (BS) with wider coverage can be a
+better decision since such traffic is not throughput hungry but
+requires more coverage to avoid frequent handovers. However,
+steering a specific class of traffic continuously to a certain
+RAT may cause several problems. The system may suffer from
+higher delay due to excessive load and reduced throughput
+because of the packet drops. These issues are quite challenging
+to address, especially when 5G NR facilitates dense network
+deployments and an increased number of users.
+To address the above-mentioned challenges, an AI-enabled
+traffic steering scheme emerges as a promising approach to
+manage densely deployed networks with dynamic require-
+ments. In recent years, AI and machine learning have been
+applied to various other problems in 5G [3]. Even though
+the emergence of the 5G non-stand-alone (NSA) mode has
+drawn the attention of researchers recently, most existing
+works linked with traffic steering lack a comprehensive tool
+to overcome the complexity.
+For instance, in [4], the authors propose a traffic steering
+scheme based on some threshold calculated using parameters
+like load at each type of RAT, channel condition, and service
+type but the method lacks the intelligence to handle dynamic
+wireless environments. Compared with conventional model-
+based optimization methods, machine learning, especially re-
+inforcement learning (RL) algorithms, can significantly reduce
+the complexity of defining a dedicated optimization model
+[5]. Advanced machine learning techniques like deep rein-
+forcement learning (DRL) [6] can not only automate traffic
+steering in a dynamic 5G wireless environment, but also it
+can handle larger state-action space compared to traditional
+reinforcement learning. Therefore, unlike previous works, we
+propose a DRL-based traffic steering scheme that tends to per-
+form RAT specific traffic steering in a multi-RAT environment
+to maintain QoS requirements of different traffic classes in a
+dynamic 5G NSA mode to maintain seamless network activity
+and smooth user experience.
+In this paper, we seek to balance the QoS demands of all the
+traffic classes simultaneously by proposing a Deep-Q-network
+(DQN)-based traffic steering scheme. The reward and state
+functions of the proposed DQN-based traffic steering scheme
+is carefully designed to have satisfactory performance based
+on two crucial key performance indicators (KPIs); i.e. network
+delay and average system throughput. Performance of the
+proposed method is compared with two baseline algorithms:
+Q-learning-based method [7] and a heuristic-based algorithm
+adopted from [4]. It gains 6% and 10% increase in average
+system throughput compared to the Q-learning and heuristic-
+based baseline respectively. Furthermore, it achieves 23% and
+33% decrease in network delay compared to the mentioned
+baselines.
+The rest of the paper is organized as follows: Section II
+presents the related works. We discuss the system model and
+the problem formulation in Section III. Section IV covers the
+proposed DQN-based traffic steering scheme along with the
+baselines. The performance evaluation of the proposed DQN-
+based traffic steering method is presented in Section V. Finally,
+the paper is concluded in Section VI.
+arXiv:2301.05316v1  [cs.NI]  12 Jan 2023
+
+II. RELATED WORKS
+In this section, we summarize the state-of-the-art literature
+on traffic steering. Prasad et al. propose a dynamic traffic
+steering scheme for energy efficient radio access network
+moderation in ultra-dense 5G networks [8]. A unified traffic
+steering scheme by Dryjanski et al. is proposed for LTE-
+advanced pro, aiming at optimal radio resource allocation in
+multi-RAT networks [9]. Most recently, Khaled et al. have
+proposed a cell zooming technique to steer traffic in a software
+defined radio-enabled LTE network that uses renewable energy
+sources to lessen on-grid power consumption [10]. Gijon et
+al. propose a data driven approach to perform traffic steering
+in multi-carrier LTE networks in which traffic steering is
+conducted based on reference signal received quality-based
+handover margins [11].
+Nevertheless, 5G deployments have made it more challeng-
+ing to develop an elegant traffic steering scheme because of
+the increased number of users and dual connectivity. Passas
+et al. propose a pricing oriented network selection process
+for distributed heterogeneous networks based on imposed load
+pressure at a particular RAT [12]. A heuristic-based approach
+proposed in [4] performs traffic steering based on a threshold
+level calculated using parameters like channel condition, load
+level at each RAT, and service type. Priscoli et al. address the
+problem of traffic steering using a Q-learning-based solution
+that aims at maintaining QoS, and performs load balancing in
+a 5G heterogeneous network [13]. Different from the previous
+works, this paper provides automation in the system via DRL-
+based traffic steering scheme that can perform RAT specific
+traffic steering in a multi-RAT environment. Furthermore, the
+proposed method can maintain QoS requirements of different
+traffic classes in a dynamic 5G NSA mode to maintain
+seamless network activity and smooth user experience.
+III. SYSTEM MODEL AND PROBLEM FORMULATION
+A. System Model
+In this work, a multi-RAT network is considered having
+Q classes of RATs where each class of RAT, q represents a
+particular access technology (LTE, 5G, etc.). Multiple users are
+associated with different types of RATs via dual connectivity.
+A UE can maintain K types of traffic classes. Fig. 1 presents
+the network model considered in this study. We represent
+three different classes of traffics: voice, gaming, and video
+as TC1, TC2, and TC3 respectively in the figure. We have
+designed our network environment in a way where small
+cells are within the range of a macro-cell. UEs have dual
+connectivity with LTE or 5G RAT and traffic can be steered
+to either one of these RATs based on our proposed method.
+The total downlink bandwidth, B in MHz is divided into
+NRB resource blocks. A resource block contains a set of
+12 contiguous subcarriers. Consecutive resource blocks are
+grouped to constitute resource block group (RBG) as defined
+in [3]. Each RBG, h is allocated a certain transmission power
+ph,b, by a BS, b. Based on our system model, each BS holds a
+number of transmission buffers corresponding to the number of
+Fig. 1.
+Illustration of network environment with one LTE macro cell and
+several 5G small cells.
+users connected to it. Every transmission time interval (TTI),
+the downlink scheduler assigns resources to the users having
+pending data transmissions.
+The link capacity between the UE, u and BS, b can be
+formulated as follows:
+Cu,b =
+H
+�
+h=1
+ωh log2
+�
+1 +
+ph,bxh,u,bgh,u,b
+ωhN0 + �
+m∈B ph,mxh,u,mgh,u,m
+�
+,
+(1)
+where ωh is the bandwidth of the h, ph,b is the transmit power
+of the BS, b on h, gh,u,b is the channel co-efficient and xh,u,b
+is the RBG’s allocation indicator of the link (h, u, b). N0 is
+the additive white Gaussian noise single-sided power spectral
+density. ph,m is the transmit power of the interfering BS, m,
+gh,u,m is the channel co-efficient, and xh,u,m is the allocation
+indicator of link (h, u, m).
+Each link has a capacity limit. Traffic flows passing through
+a link should not exceed the capacity of the link in the system.
+�
+f∈F
+dfxf
+u,b ⩽ Cu,b
+∀(u, b) ∈ L,
+(2)
+where F is the set of all the flows in the network, df is the
+capacity demand of the flow f ∈ F from UE, u to BS b. xf
+u,b
+represents a binary (0, 1) component that is ‘1’ if the link
+(u, b) has been used from UE,u to BS b. It is ‘0’ otherwise.
+L is the set of links and Cu,b is the capacity of link (u, b). as
+presented in eq. (1)
+In our system model, the delay is considered as the summa-
+tion of transmission and queuing delay which is as follows:
+Dk,b = DT rx
+k,b + Dq
+k,b,
+(3)
+where DT rx
+k,b
+is the transmission delay experienced for a
+particular traffic type k and BS b, and Dq
+k,b is the queuing
+delay experienced for a particular traffic type k at BS b for a
+user u. The transmission delay can be calculated as follows:
+DT rx
+k,b = Lu,b
+Cu,b
+,
+(4)
+where Lu,b is the packet length and Cu,b is the link capacity
+as stated in eq. (1).
+
+MBS
+ SBS“
+UE -.- TC1
+TC2
+TC3B. QoS Requirements and Problem Formulation
+To be able to perform traffic steering for different traffic
+classes with QoS requirements for delay and throughput, first
+two parameters are defined based on delay and throughput. The
+delay parameter associated with our traffic steering problem
+is considered as the ratio of the defined QoS requirement for
+delay and the actual delay experienced in the system for a
+particular traffic class being carried by a certain BS. It can be
+stated as follows:
+rD
+k,b = DQoS
+Dk,b
+,
+(5)
+where DQoS is delay requirement defined in the simulation for
+a particular traffic type and Dk,b is the actual delay achieved.
+Similarly, the throughput parameter is defined as the ratio
+of actual throughput achieved and the required throughput as
+stated in eq. (6):
+rT
+k,b = Tk,b
+TQoS
+,
+(6)
+where TQoS is the throughput requirement defined in the
+simulation for a particular traffic class and Tk,b is the actual
+throughput achieved.
+Since our aim is to improve the system performance in
+terms of the delay and throughput, a new variable is formed
+to represent and meet such targets. It combines the delay and
+throughput parameters in eq. (5) and (6) along with some
+weight factors. The declared variable combined with delay,
+throughput, and weight factors (w1 and w2) is as follows:
+M = w1(rD
+k,b) + w2(rT
+k,b).
+(7)
+The traffic steering problem proposed in this paper is formu-
+lated as the maximization of the variable M (presented in eq.
+(7)) which is as follows:
+max
+�
+u∈U
+�
+k∈K
+�
+b∈B
+Mu,f,b,
+s.t.
+�
+(u,b)∈L
+βfk ⩾ βf
+∀f ∈ F,
+�
+(u,b)∈L
+D(u, b)xf
+u,b ⩽ Df
+∀f ∈ F,
+(8)
+where βfk is the required bitrate for a particular type of traffic
+k, and βf is the available bitrate. Also, Df represents the
+latency demand of flow f ∈ F and D(u, b) is the latency of
+link (u, b).
+IV. PROPOSED DQN-BASED TRAFFIC STEERING SCHEME
+A. DQN-based Traffic Steering Scheme
+For a relatively simplistic RL environment, Q-learning is
+a good solution for optimization. However, as the state-
+space increases, the time needed to traverse all these states
+and iteratively update all the Q-values will increase which
+is computationally inefficient and resource consuming. To
+address this issue, DQN can be used to estimate the Q-values
+for each state-action pair in a given environment using a deep
+neural network (DNN) [6].
+During the training stage of DQN, agent’s experiences at
+each time step is stored in a data set called the replay memory.
+At time τ, the agent’s experience eτ is defined as the following
+tuple:
+eτ = (Sτ, Aτ, Rτ+1, Sτ+1).
+(9)
+The tuple contains the state of the environment, the action
+taken from the state, the reward given to the agent as a
+result of previous state-action pair and the next state of the
+environment. In short, the tuple gives us the summary of the
+agent’s experience at time τ. All the agent’s experiences at
+each time step over all the episodes played by the agent are
+stored in the replay memory. In practice, the replay memory
+is set to some finite size unit (N). Therefore, it will only
+store the last N experiences. The replay memory data set is
+the place from where random samples are chosen to train the
+network.
+The DNN in DQN takes states as inputs from the envi-
+ronment and outputs the Q-values for each action that can
+be taken from that state. Before the training starts, first, the
+replay memory data set, D is initialized to capacity, N. Next,
+DNN is initialized with random weights. For each episode,
+the starting state is initialized. For each time step within the
+episode, the agent either explores the environment and selects
+a random action or the agent exploits the environment and
+selects the greedy action for the given state that provides the
+highest Q-value. This epsilon greedy policy is used to balance
+the exploration and exploitation.
+Aτ =
+�
+random
+action,
+if rand ⩽ ϵ
+argmax(qτ(Sτ, Aτ)),
+otherwise
+(10)
+where ϵ is the exploration probability within 0 ⩽ ϵ ⩽ 1 and
+rand represents a random number between 0 to 1.
+After an action is taken, we observe the reward for the action
+along with the next state of the environment. Therefore, the
+state an agent initialized from, action taken, reward observed
+are all put together in a tuple as described in eq. (9).
+For a single sample, the first pass to the network occurs
+for the state from the experience tuple that was sampled.
+The network then outputs the Q-values associated with each
+possible action that can be taken from that state and then the
+loss is calculated between the Q-values for the action from
+the experience tuple and the target Q-value for this action. To
+calculate the target Q-value, it is required to have a second pass
+to the target network with the next state. The target network
+is the clone of the policy network (which is also the main
+network). Its weights are frozen with the weights same as
+the policy network and the weights are updated in the target
+network after every certain amount of time steps. The loss for
+DQN is calculated using the following equation:
+L(w) = Er(Rτ + γ max
+A q(Sτ+1, A, w′) − q(Sτ, Aτ, w)),
+(11)
+where w and w′ are the weights of the main and the target
+network, and Er represents the error function. Having two
+NNs (main and target) ensures stability.
+
+Fig. 2 describes the schematic of the proposed DQN-based
+traffic steering where we have a main network and a target
+network and minibatch from the replay memory is getting
+fetched.
+Fig. 2. Overall system architecture with DQN.
+The mathematical formulation of DQN depends on Markov
+Decision Process (MDP) that is defined by agents, states,
+actions, and a reward function. Tuples associated with DQN
+is defined as follows:
+• Agent: We implement a centralized agent to control the
+macro base station (MBS) and the small cell base stations.
+It is deployed in the MBS and controls all the incoming
+traffic to each BS.
+• State:
+The
+state
+consists
+of
+three
+elements,
+{Tf, LQ(SINR), qL}. Here, Tf
+represents the traffic
+type. It is assumed that each traffic type has fixed
+QoS
+requirements
+and
+we
+can
+perform
+traffic
+steering to a particular RAT based on that. Users
+periodically
+report
+signal-to-interference
+and
+noise
+ratio (SINR) measurements to the 5G base station
+(gNB) and LTE base station (eNB). It indicates the
+quality of the link associated with a UE and a BS.
+Therefore,
+the
+second
+element
+of
+state
+space
+is:
+LQ(SINR)={SINReNB, SINRgNB}. To represent load
+level, queue length of both types of RATs is used. So,
+the last element of the state space is queue length,
+qL={qL(gNB), qL(eNB)}.
+• Action: The action space contains the action of flow
+admission to the RATs. It is defined as: {ALT E, A5G}.
+Here, (ALT E) stands for flow admission to the LTE RAT
+, and (A5G) stands for flow admission to the 5G RAT.
+• Reward: The reward function is based on eq. (7). To
+keep it normalized, sigmoid function is used. Therefore,
+the reward function is as follows:
+R = sigm(M),
+(12)
+where sigm(M) represents the sigmoid function.
+The proposed DQN-based traffic steering algorithm is sum-
+marized as Algorithm 1.
+Algorithm 1 DQN-based traffic steering
+Initialize: Network and DQN parameters
+1: for TTI = 1
+to
+T do
+2:
+for every u, b, k do
+3:
+if (rand ≤ ϵ) then
+4:
+choose action randomly
+5:
+else
+6:
+select Aτ using greedy policy
+7:
+end if
+8:
+BSs are selected for all the UEs for all k ∈ K
+9:
+Traffic admission is performed
+10:
+Reward calculation based on eq. (12)
+11:
+Agent updates its own state Sτ
+12:
+Save (Sτ, Aτ, Rτ+1, Sτ+1)
+13:
+end for
+14:
+Random sample a minibatch from the experience pool
+15:
+Generate target Q-values, qτ(Sτ, Aτ)
+16:
+Update w using gradient descent to minimize the loss,
+L(w) = Er(qτ(Sτ, Aτ) − q(Sτ, Aτ, w))
+17:
+Copy w to w′ after several training
+18: end for
+19: Output: Optimal traffic steering decisions from TTI =
+1
+to
+T
+B. Baseline Algorithms
+In this section, two baseline algorithms are introduced that
+have been used for the performance comparison. The first
+baseline algorithm for RAT selection is based on a predefined
+threshold [4]. This is called the heuristic baseline. Here, the
+threshold is calculated for each UE based on the metrics like
+load at eNB (le) and gNB (lg), channel condition of a user
+under LTE (che,u) and 5G BS (chg,u), service type of a user
+(Su). The channel condition is determined to be good or bad
+considering a threshold of received SINR values. Similarly, the
+load at each RAT is determined based on a threshold value.
+Based on the mentioned metrics, a value Tu is calculated that
+is used for selecting the RAT for a UE after comparing it with
+a predetermined threshold (Tth). Following equation is used
+to calculate the value for Tu:
+Tu(le, lg, che,u, Su) = αle + βlg + γchg,u + δSu,
+(13)
+where α, β, γ, and δ are the weights associated with consid-
+ered parameters that can be modulated based on the impact of
+any certain metric on system performance. Tth is set to be the
+mean of all the possible values of Tu. The decision of steering
+traffic to a particular RAT is taken the following way:
+Ru =
+�
+1, Tu > Tth
+(1 represents gNB)
+0, Tu ⩽ Tth
+(0 represents eNB).
+(14)
+The Q-learning algorithm has been used as another baseline
+in this work [7]. The goal is to investigate how DQN performs
+against the Q-learning algorithm.
+
+Main network
+Hidden layers
+Action
+selector
+LTE
+()
+Hidden layers
+lnput
+5G NR
+Observation
+Target network
+Reward: Throughput,
+delay
+Minibatch from
+New state: Load level
+experience poolV. PERFORMANCE EVALUATION
+A. Simulation setup
+We have conducted MATLAB based simulations consider-
+ing 1 eNB and 4 gNBs with 30 users in total. There are a total
+of 1 macro-cell and 4 small cells facilitated by the gNBs and an
+eNB. A macro-cell and a small-cell have carrier frequencies of
+3.5 GHz and 0.8 GHz respectively. Specifications of the traffic
+classes used in this study have been summarized in TABLE I.
+For the experimental results, the load has been varied between
+5-10 Mbps. Proportion of the voice, video, and gaming traffic
+is 20%, 50%, and 30% respectively. Higher proportion of
+the video traffic is deliberately considered to observe how
+the system performs with the higher throughput requirements.
+Also, gaming traffic has the most stringent delay requirement
+and we wanted to see if the system performs well enough
+to meet such precise requirement. Therefore it has a higher
+percentage compared to the voice traffic. QoS requirements
+associated with delay and throughput for the three types of
+traffic classes are specified based on the existing literature [14]
+and 3GPP specifications (see TABLE I.). We are using multi-
+RAT dual connectivity architecture, an NSA mode where LTE
+and 5G NR BSs serve together. An architecture specified in
+[15] has been used where the dual connectivity is ensured
+via evolved packet core [16]. Transmission power of the LTE
+BS and 5G NR BSs are set to 40W and 20W. Furthermore,
+bandwidth for the LTE and 5G RAT are fixed to 10MHz and
+20MHz.
+TABLE I
+TRAFFIC CLASS DESCRIPTION AND SIMULATION SETTINGS
+Traffic class specification
+Values
+Traffic model
+Poisson distribution, video
+and gaming traffic [14]
+Voice traffic
+Packet size
+30 bytes
+TQoS, DQoS
+0.1 Mbps , 100ms
+Proportion of the traffic
+20%
+Video traffic
+Packet size
+250 bytes
+TQoS, DQoS
+10 Mbps, 80ms
+Proportion of the traffic
+50%
+Gaming traffic
+Packet size (gaming traffic)
+120 bytes
+TQoS, DQoS
+5 Mbps, 40ms
+Proportion of the traffic
+30%
+B. Simulation results
+The performance of the proposed algorithm is evaluated in
+terms of two KPIs: Average system throughput and network
+delay. In Fig 3, we present a comparison in terms of system
+throughput under different user loads. The proposed DQN
+outperforms heuristic and Q-Learning baselines by gaining 6%
+and 10% increased throughput, respectively.
+Fig. 4 presents the performance comparison of the proposed
+DQN-based traffic steering method with the other baselines in
+terms of delay. The DQN-based method achieves 23% and
+Fig. 3. System throughput against traffic load.
+Fig. 4. System delay against traffic load.
+33% decrease in network delay compared to the baselines.
+Note that, the proposed method and the Q-learning, both have
+a reward function formulated based on throughput and delay.
+Whenever high delay is experienced for steering traffic to a
+particular RAT, the system learns. That is why, both of them
+have better performance compared to the heuristic baseline.
+In Fig. 4, delay is calculated considering all the traffic classes
+together at each load.
+It should be mentioned that the main reason of the improved
+performance of the proposed method is the use of DQN,
+that outperforms Q-learning in terms of exploration efficiency
+and achieves higher average reward. Q-learning suffers due to
+longer exploration period and gets lower average reward since
+it does not have a DNN as an approximator which compels
+the agent to cover larger state and action space.
+In this work, we also want to steer a particular type of
+traffic to a specific RAT. For example, steering the voice
+traffic constantly to a gNB is a waste of resources since the
+throughput requirement is not that high for such traffic. Fig. 5
+is presented which shows what percentage of a traffic class is
+processed by a particular RAT and when the traffic gets steered
+due to higher load. In Fig. 5(a), it is observed that most of
+the voice traffic is processed by the eNB, however, a small
+portion of the traffic is processed by the gNB too whenever
+the system experiences higher load. For the video and gaming
+traffic, it is observed that most of the traffic is processed by
+the gNB.
+
+300
+-DQN
+--Q-learning
+-Heuristic baseline
+250
+200
+150
+5
+6
+7
+8
+9
+10
+Load per user (Mbps)20
+-Heuristic baseline
+-Q-learning
++DQN
+15
+(ms)
+Delay (
+10
+5
+0
+5
+6
+7
+8
+9
+10
+Load per user (Mbps)Fig. 5. Data processing percentage for different traffic types.
+Fig. 6. Traffic steered to other RAT as load changed.
+Lastly, Fig. 6 demonstrates how traffic steering occurs
+whenever a high load is experienced in a BS with a particular
+RAT. We start with one UE at the 300th time slot and increase
+the number of UEs in a small cell up to six for different traffic
+classes. The variable L, in the respective figure represents load
+in terms of queue length. At the 1800th time slot, it can be
+seen that four among six UEs are steering different types of
+traffic to the 5G NR BS. This results in higher load and we can
+see that the third and fourth UEs are experiencing high load
+(value of L changed from 0 to 1). So, in the next observed
+time slot, these two UEs steer the traffic to the eNB. In the
+2100th time slot, we can see four UEs steering voice, video,
+and gaming traffic to the only eNB in our system. This incurs
+high load at eNB and in the next observed slot we can see
+that the sixth UE has switched its traffic to the gNB.
+VI. CONCLUSIONS
+In this study, we have proposed a novel method that can
+perform RAT specific and QoS aware traffic steering using
+DQN. It gains 6% and 10% increase in average system
+throughput compared to the Q-learning and heuristic-based
+baseline respectively. Moreover, it achieves 23% and 33%
+times decrease in network delay compared to the baselines.
+Apart from the better performance in terms of the KPIs, the
+proposed method can perform RAT specific traffic steering
+ensuring efficient use of network resources. Lastly, the pro-
+posed DQN-based traffic steering can successfully perform
+load balancing in an optimal way as whenever high load is
+induced to a particular RAT, traffic is steered to another RAT
+dynamically.
+ACKNOWLEDGEMENT
+This work has been supported by MITACS and Ericsson
+Canada, and NSERC Collaborative Research and Training
+Experience Program (CREATE) under Grant 497981.
+REFERENCES
+[1] V. Ramaswamy, J. T. Correia, and D. Swain-Walsh, “Modeling and
+Analysis of Multi-RAT Dual Connectivity Operations in 5G Networks,”
+in 2019 IEEE 2nd 5G World Forum (5GWF), pp. 484–489.
+[2] R. Pirmagomedov, D. Moltchanov, A. Samuylov, A. Orsino, J. Torsner,
+S. Andreev, and Y. Koucheryavy, “Characterizing Throughput and
+Convergence Time in Dynamic Multi-Connectivity 5G Deployments,”
+Computer Communications, vol. 187, pp. 45–58, 2022.
+[3] M. Elsayed and M. Erol-Kantarci, “Radio Resource and Beam Man-
+agement in 5G mmWave Using Clustering and Deep Reinforcement
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+Conference, 2020, pp. 1–6.
+[4] M. Khaturia, P. Jha, and A. Karandikar, “5G-Flow: A Unified Multi-RAT
+RAN Architecture for Beyond 5G Networks,” Computer Networks, vol.
+198, p. 108412, 2021.
+[5] H. Zhou and M. Erol-Kantarci, “RAN Resource Slicing in 5G Using
+Multi-Agent Correlated Q-Learning,” in Proc. IEEE PIMRC, Sep. 2021,
+pp. 1–6.
+[6] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G.
+Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski
+et al., “Human-Level Control Through Deep Reinforcement Learning,”
+nature, vol. 518, no. 7540, pp. 529–533, 2015.
+[7] R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction.
+MIT press, 2018.
+[8] A. Prasad, F. S. Moya, M. Ericson, R. Fantini, and O. Bulakci, “Enabling
+RAN Moderation and Dynamic Traffic Steering in 5G,” in 2016 IEEE
+84th Vehicular Technology Conference (VTC-Fall), 2016, pp. 1–6.
+[9] M. Dryjanski and M. Szydelko, “A Unified Traffic Steering Framework
+for LTE Radio Access Network Coordination,” IEEE Communications
+Magazine, vol. 54, no. 7, pp. 84–92, 2016.
+[10] H. Khaled, I. Ahmad, D. Habibi, and Q. V. Phung, “A Green Traffic
+Steering Solution for Next Generation Communication Networks,” IEEE
+Transactions on Cognitive Communications and Networking, vol. 7,
+no. 1, pp. 222–238, 2020.
+[11] C. Gij´on, M. Toril, S. Luna-Ram´ırez, and M. L. Mar´ı-Altozano, “A Data-
+Driven Traffic Steering Algorithm for Optimizing User Experience in
+Multi-Tier LTE Networks,” IEEE Transactions on Vehicular Technology,
+vol. 68, no. 10, pp. 9414–9424, 2019.
+[12] V. Passas, V. Miliotis, N. Makris, and T. Korakis, “Pricing Based
+Distributed Traffic Allocation for 5G Heterogeneous Networks,” IEEE
+Transactions on Vehicular Technology, vol. 69, no. 10, pp. 12 111–
+12 123, 2020.
+[13] F. D. Priscoli, A. Giuseppi, F. Liberati, and A. Pietrabissa, “Traffic Steer-
+ing and Network Selection in 5G Networks Based on Reinforcement
+Learning,” in 2020 European Control Conference (ECC), 2020, pp. 595–
+601.
+[14] J. Navarro-Ortiz, P. Romero-Diaz, S. Sendra, P. Ameigeiras, J. J. Ramos-
+Munoz, and J. M. Lopez-Soler, “A Survey on 5G Usage Scenarios and
+Traffic Models,” IEEE Communications Surveys & Tutorials, vol. 22,
+no. 2, pp. 905–929, 2020.
+[15] P.
+Frenger
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+Tano.
+(2019)
+A
+Technical
+Look
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+5G
+Energy Consumption and Performance. [Online]. Available: https:
+//www.ericsson.com/en/blog/2019/9/energy-consumption-5{G}-nr
+[16] M. Agiwal, H. Kwon, S. Park, and H. Jin, “A Survey on 4G-5G Dual
+Connectivity: Road to 5G Implementation,” IEEE Access, vol. 9, pp.
+16 193–16 210, 2021.
+
+TCi(Voice traffic)
+TC2(Gamingtraffic)
+100
+100
+-eNB
+Higher load atgNB
+80
++gNB
+Mostofthetraffic
+80
++Packet drop rate
+processedbyeNB
+-gNB
+60
+60
++eNB
++Packet drop rate
+40
+40
+SteeredtraffictoeNB
+Smallpartofthetraffic
+Data
+20
+processed bygNB
+20
+0
+0
+6
+7
+8
+9
+10
+6
+7
+8
+9
+10
+Load per user (Mbps)
+Load peruser (Mbps)
+(b)
+(a)
+TC3(Video traffic)
+100
+80
++gNB
++eNB
+Mostofthetraffic
+-Packet drop rate
+60
+processed by gNB
+40
+Smallpartofthetraffic
+Data
+20
+processedbyeNB
+0
+6
+8
+9
+10
+Load per user (Mbps)
+(c)Traffic steered
+to gNB due to
+eNB
+eNB
+L=Load ("o"
+gNB
+6
+high load
+indicates "low"
+L=0
+L=1
+L=0
+1 indicates “high")
+gNB
+gNB
+gNB
+gNB
+5
+Traffic steered
+△ eNB
+to eNB due to
+=O
+L=0
+L=0
+L=0
+Number of UEs
+O gNB
+high load
+gNB
+gNB
+gNB
+eNB
+eNB
+Tcl (Voice)
+4
+Tc2 (Video)
+TC3 (Gaming)
+L=0
+L=0
+L=1
+L=O
+L=0
+gNB
+gNB
+gNb
+gNB
+eNB
+eNB
+3
+L=0
+L=0
+L=0
+L=1
+L=0
+L=0
+gNB
+gNB
+gNB
+gNB
+gNB
+gNB
+gNB
+2
+L=0
+L=0
+L=0
+L=0
+0=1
+L=0
+L=0
+eNB
+eNB
+eNB
+eNB
+eNB
+eNB
+eNB
+gNB
+1
+L=0
+L=0
+L=0
+L=0
+L=0
+L=0
+L=1
+L=0
+0
+0
+300
+600
+900
+1200 1500 1800 2100 2400 ....
+Time slots
\ No newline at end of file
diff --git a/ANE4T4oBgHgl3EQf4w6U/content/tmp_files/load_file.txt b/ANE4T4oBgHgl3EQf4w6U/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..11fab006a71e962c29c2c8165c8a38ceaf622f95
--- /dev/null
+++ b/ANE4T4oBgHgl3EQf4w6U/content/tmp_files/load_file.txt
@@ -0,0 +1,535 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf,len=534
+page_content='Traffic Steering for 5G Multi-RAT Deployments  using Deep Reinforcement Learning  Md Arafat Habib1, Hao Zhou1, Pedro Enrique Iturria-Rivera1, Medhat Elsayed2, Majid Bavand2,  Raimundas Gaigalas2, Steve Furr2 and Melike Erol-Kantarci1, Senior Member, IEEE  1School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada  2Ericsson Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=', Ottawa, Canada  Emails:{mhabi050, hzhou098, pitur008, melike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='erolkantarci}@uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='ca,  {medhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='elsayed, majid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='bavand, raimundas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='gaigalas, steve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='furr}@ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='com  Abstract—In 5G non-standalone mode, traffic steering is a  critical technique to take full advantage of 5G new radio while  optimizing dual connectivity of 5G and LTE networks in multiple  radio access technology (RAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' An intelligent traffic steering  mechanism can play an important role to maintain seamless  user experience by choosing appropriate RAT (5G or LTE)  dynamically for a specific user traffic flow with certain QoS  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' In this paper, we propose a novel traffic steering  mechanism based on Deep Q-learning that can automate traffic  steering decisions in a dynamic environment having multiple  RATs, and maintain diverse QoS requirements for different  traffic classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The proposed method is compared with two  baseline algorithms: a heuristic-based algorithm and Q-learning-  based traffic steering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Compared to the Q-learning and heuristic  baselines, our results show that the proposed algorithm achieves  better performance in terms of 6% and 10% higher average  system throughput, and 23% and 33% lower network delay,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Index Terms—Multi-RAT, traffic steering, reinforcement learn-  ing  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' INTRODUCTION  The dual connectivity between long term evolution (LTE)  and fifth generation new radio (5G NR) results in multiple  radio access technologies (multi-RAT) [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' On the other  hand, each type of RAT is supposed to have distinctive  capabilities to serve user equipment (UE) with diverse quality-  of-service (QoS) requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' This raises the need of steering  a specific class of traffic to a certain RAT to fulfill the QoS  demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' For instance, high throughput video traffic can be  better served by 5G NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' On the contrary, steering voice  traffic to LTE base station (BS) with wider coverage can be a  better decision since such traffic is not throughput hungry but  requires more coverage to avoid frequent handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' However,  steering a specific class of traffic continuously to a certain  RAT may cause several problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The system may suffer from  higher delay due to excessive load and reduced throughput  because of the packet drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' These issues are quite challenging  to address, especially when 5G NR facilitates dense network  deployments and an increased number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  To address the above-mentioned challenges, an AI-enabled  traffic steering scheme emerges as a promising approach to  manage densely deployed networks with dynamic require-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' In recent years, AI and machine learning have been  applied to various other problems in 5G [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Even though  the emergence of the 5G non-stand-alone (NSA) mode has  drawn the attention of researchers recently, most existing  works linked with traffic steering lack a comprehensive tool  to overcome the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  For instance, in [4], the authors propose a traffic steering  scheme based on some threshold calculated using parameters  like load at each type of RAT, channel condition, and service  type but the method lacks the intelligence to handle dynamic  wireless environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Compared with conventional model-  based optimization methods, machine learning, especially re-  inforcement learning (RL) algorithms, can significantly reduce  the complexity of defining a dedicated optimization model  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Advanced machine learning techniques like deep rein-  forcement learning (DRL) [6] can not only automate traffic  steering in a dynamic 5G wireless environment, but also it  can handle larger state-action space compared to traditional  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Therefore, unlike previous works, we  propose a DRL-based traffic steering scheme that tends to per-  form RAT specific traffic steering in a multi-RAT environment  to maintain QoS requirements of different traffic classes in a  dynamic 5G NSA mode to maintain seamless network activity  and smooth user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  In this paper, we seek to balance the QoS demands of all the  traffic classes simultaneously by proposing a Deep-Q-network  (DQN)-based traffic steering scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The reward and state  functions of the proposed DQN-based traffic steering scheme  is carefully designed to have satisfactory performance based  on two crucial key performance indicators (KPIs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' network  delay and average system throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Performance of the  proposed method is compared with two baseline algorithms:  Q-learning-based method [7] and a heuristic-based algorithm  adopted from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' It gains 6% and 10% increase in average  system throughput compared to the Q-learning and heuristic-  based baseline respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Furthermore, it achieves 23% and  33% decrease in network delay compared to the mentioned  baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  The rest of the paper is organized as follows: Section II  presents the related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' We discuss the system model and  the problem formulation in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Section IV covers the  proposed DQN-based traffic steering scheme along with the  baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The performance evaluation of the proposed DQN-  based traffic steering method is presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Finally,  the paper is concluded in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='05316v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='NI]  12 Jan 2023  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' RELATED WORKS  In this section, we summarize the state-of-the-art literature  on traffic steering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' propose a dynamic traffic  steering scheme for energy efficient radio access network  moderation in ultra-dense 5G networks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' A unified traffic  steering scheme by Dryjanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' is proposed for LTE-  advanced pro, aiming at optimal radio resource allocation in  multi-RAT networks [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Most recently, Khaled et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' have  proposed a cell zooming technique to steer traffic in a software  defined radio-enabled LTE network that uses renewable energy  sources to lessen on-grid power consumption [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Gijon et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' propose a data driven approach to perform traffic steering  in multi-carrier LTE networks in which traffic steering is  conducted based on reference signal received quality-based  handover margins [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Nevertheless, 5G deployments have made it more challeng-  ing to develop an elegant traffic steering scheme because of  the increased number of users and dual connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Passas  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' propose a pricing oriented network selection process  for distributed heterogeneous networks based on imposed load  pressure at a particular RAT [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' A heuristic-based approach  proposed in [4] performs traffic steering based on a threshold  level calculated using parameters like channel condition, load  level at each RAT, and service type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Priscoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' address the  problem of traffic steering using a Q-learning-based solution  that aims at maintaining QoS, and performs load balancing in  a 5G heterogeneous network [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Different from the previous  works, this paper provides automation in the system via DRL-  based traffic steering scheme that can perform RAT specific  traffic steering in a multi-RAT environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Furthermore, the  proposed method can maintain QoS requirements of different  traffic classes in a dynamic 5G NSA mode to maintain  seamless network activity and smooth user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' System Model  In this work, a multi-RAT network is considered having  Q classes of RATs where each class of RAT, q represents a  particular access technology (LTE, 5G, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Multiple users are  associated with different types of RATs via dual connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  A UE can maintain K types of traffic classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 1 presents  the network model considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' We represent  three different classes of traffics: voice, gaming, and video  as TC1, TC2, and TC3 respectively in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' We have  designed our network environment in a way where small  cells are within the range of a macro-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' UEs have dual  connectivity with LTE or 5G RAT and traffic can be steered  to either one of these RATs based on our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  The total downlink bandwidth, B in MHz is divided into  NRB resource blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' A resource block contains a set of  12 contiguous subcarriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Consecutive resource blocks are  grouped to constitute resource block group (RBG) as defined  in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Each RBG, h is allocated a certain transmission power  ph,b, by a BS, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Based on our system model, each BS holds a  number of transmission buffers corresponding to the number of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Illustration of network environment with one LTE macro cell and  several 5G small cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  users connected to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Every transmission time interval (TTI),  the downlink scheduler assigns resources to the users having  pending data transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  The link capacity between the UE, u and BS, b can be  formulated as follows:  Cu,b =  H  �  h=1  ωh log2  �  1 +  ph,bxh,u,bgh,u,b  ωhN0 + �  m∈B ph,mxh,u,mgh,u,m  �  ,  (1)  where ωh is the bandwidth of the h, ph,b is the transmit power  of the BS, b on h, gh,u,b is the channel co-efficient and xh,u,b  is the RBG’s allocation indicator of the link (h, u, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' N0 is  the additive white Gaussian noise single-sided power spectral  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' ph,m is the transmit power of the interfering BS, m,  gh,u,m is the channel co-efficient, and xh,u,m is the allocation  indicator of link (h, u, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Each link has a capacity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Traffic flows passing through  a link should not exceed the capacity of the link in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  �  f∈F  dfxf  u,b ⩽ Cu,b  ∀(u, b) ∈ L,  (2)  where F is the set of all the flows in the network, df is the  capacity demand of the flow f ∈ F from UE, u to BS b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' xf  u,b  represents a binary (0, 1) component that is ‘1’ if the link  (u, b) has been used from UE,u to BS b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' It is ‘0’ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  L is the set of links and Cu,b is the capacity of link (u, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' as  presented in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' (1)  In our system model, the delay is considered as the summa-  tion of transmission and queuing delay which is as follows:  Dk,b = DT rx  k,b + Dq  k,b,  (3)  where DT rx  k,b  is the transmission delay experienced for a  particular traffic type k and BS b, and Dq  k,b is the queuing  delay experienced for a particular traffic type k at BS b for a  user u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The transmission delay can be calculated as follows:  DT rx  k,b = Lu,b  Cu,b  ,  (4)  where Lu,b is the packet length and Cu,b is the link capacity  as stated in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  MBS  SBS“  UE -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='- TC1  TC2  TC3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' QoS Requirements and Problem Formulation  To be able to perform traffic steering for different traffic  classes with QoS requirements for delay and throughput, first  two parameters are defined based on delay and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The  delay parameter associated with our traffic steering problem  is considered as the ratio of the defined QoS requirement for  delay and the actual delay experienced in the system for a  particular traffic class being carried by a certain BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' It can be  stated as follows:  rD  k,b = DQoS  Dk,b  ,  (5)  where DQoS is delay requirement defined in the simulation for  a particular traffic type and Dk,b is the actual delay achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Similarly, the throughput parameter is defined as the ratio  of actual throughput achieved and the required throughput as  stated in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' (6):  rT  k,b = Tk,b  TQoS  ,  (6)  where TQoS is the throughput requirement defined in the  simulation for a particular traffic class and Tk,b is the actual  throughput achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Since our aim is to improve the system performance in  terms of the delay and throughput, a new variable is formed  to represent and meet such targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' It combines the delay and  throughput parameters in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' (5) and (6) along with some  weight factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The declared variable combined with delay,  throughput, and weight factors (w1 and w2) is as follows:  M = w1(rD  k,b) + w2(rT  k,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  (7)  The traffic steering problem proposed in this paper is formu-  lated as the maximization of the variable M (presented in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  (7)) which is as follows:  max  �  u∈U  �  k∈K  �  b∈B  Mu,f,b,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  �  (u,b)∈L  βfk ⩾ βf  ∀f ∈ F,  �  (u,b)∈L  D(u, b)xf  u,b ⩽ Df  ∀f ∈ F,  (8)  where βfk is the required bitrate for a particular type of traffic  k, and βf is the available bitrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Also, Df represents the  latency demand of flow f ∈ F and D(u, b) is the latency of  link (u, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' PROPOSED DQN-BASED TRAFFIC STEERING SCHEME  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' DQN-based Traffic Steering Scheme  For a relatively simplistic RL environment, Q-learning is  a good solution for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' However, as the state-  space increases, the time needed to traverse all these states  and iteratively update all the Q-values will increase which  is computationally inefficient and resource consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' To  address this issue, DQN can be used to estimate the Q-values  for each state-action pair in a given environment using a deep  neural network (DNN) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  During the training stage of DQN, agent’s experiences at  each time step is stored in a data set called the replay memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  At time τ, the agent’s experience eτ is defined as the following  tuple:  eτ = (Sτ, Aτ, Rτ+1, Sτ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  (9)  The tuple contains the state of the environment, the action  taken from the state, the reward given to the agent as a  result of previous state-action pair and the next state of the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' In short, the tuple gives us the summary of the  agent’s experience at time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' All the agent’s experiences at  each time step over all the episodes played by the agent are  stored in the replay memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' In practice, the replay memory  is set to some finite size unit (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Therefore, it will only  store the last N experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The replay memory data set is  the place from where random samples are chosen to train the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  The DNN in DQN takes states as inputs from the envi-  ronment and outputs the Q-values for each action that can  be taken from that state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Before the training starts, first, the  replay memory data set, D is initialized to capacity, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Next,  DNN is initialized with random weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' For each episode,  the starting state is initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' For each time step within the  episode, the agent either explores the environment and selects  a random action or the agent exploits the environment and  selects the greedy action for the given state that provides the  highest Q-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' This epsilon greedy policy is used to balance  the exploration and exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Aτ =  �  random  action,  if rand ⩽ ϵ  argmax(qτ(Sτ, Aτ)),  otherwise  (10)  where ϵ is the exploration probability within 0 ⩽ ϵ ⩽ 1 and  rand represents a random number between 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  After an action is taken, we observe the reward for the action  along with the next state of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Therefore, the  state an agent initialized from, action taken, reward observed  are all put together in a tuple as described in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  For a single sample, the first pass to the network occurs  for the state from the experience tuple that was sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  The network then outputs the Q-values associated with each  possible action that can be taken from that state and then the  loss is calculated between the Q-values for the action from  the experience tuple and the target Q-value for this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' To  calculate the target Q-value, it is required to have a second pass  to the target network with the next state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The target network  is the clone of the policy network (which is also the main  network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Its weights are frozen with the weights same as  the policy network and the weights are updated in the target  network after every certain amount of time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The loss for  DQN is calculated using the following equation:  L(w) = Er(Rτ + γ max  A q(Sτ+1, A, w′) − q(Sτ, Aτ, w)),  (11)  where w and w′ are the weights of the main and the target  network, and Er represents the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Having two  NNs (main and target) ensures stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 2 describes the schematic of the proposed DQN-based  traffic steering where we have a main network and a target  network and minibatch from the replay memory is getting  fetched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Overall system architecture with DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  The mathematical formulation of DQN depends on Markov  Decision Process (MDP) that is defined by agents, states,  actions, and a reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Tuples associated with DQN  is defined as follows:  Agent: We implement a centralized agent to control the  macro base station (MBS) and the small cell base stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  It is deployed in the MBS and controls all the incoming  traffic to each BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  State:  The  state  consists  of  three  elements,  {Tf, LQ(SINR), qL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Here, Tf  represents the traffic  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' It is assumed that each traffic type has fixed  QoS  requirements  and  we  can  perform  traffic  steering to a particular RAT based on that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Users  periodically  report  signal-to-interference  and  noise  ratio (SINR) measurements to the 5G base station  (gNB) and LTE base station (eNB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' It indicates the  quality of the link associated with a UE and a BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Therefore,  the  second  element  of  state  space  is:  LQ(SINR)={SINReNB, SINRgNB}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' To represent load  level, queue length of both types of RATs is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' So,  the last element of the state space is queue length,  qL={qL(gNB), qL(eNB)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Action: The action space contains the action of flow  admission to the RATs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' It is defined as: {ALT E, A5G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Here, (ALT E) stands for flow admission to the LTE RAT  , and (A5G) stands for flow admission to the 5G RAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Reward: The reward function is based on eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' To  keep it normalized, sigmoid function is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Therefore,  the reward function is as follows:  R = sigm(M),  (12)  where sigm(M) represents the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  The proposed DQN-based traffic steering algorithm is sum-  marized as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Algorithm 1 DQN-based traffic steering  Initialize: Network and DQN parameters  1: for TTI = 1  to  T do  2:  for every u, b, k do  3:  if (rand ≤ ϵ) then  4:  choose action randomly  5:  else  6:  select Aτ using greedy policy  7:  end if  8:  BSs are selected for all the UEs for all k ∈ K  9:  Traffic admission is performed  10:  Reward calculation based on eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' (12)  11:  Agent updates its own state Sτ  12:  Save (Sτ, Aτ, Rτ+1, Sτ+1)  13:  end for  14:  Random sample a minibatch from the experience pool  15:  Generate target Q-values, qτ(Sτ, Aτ)  16:  Update w using gradient descent to minimize the loss,  L(w) = Er(qτ(Sτ, Aτ) − q(Sτ, Aτ, w))  17:  Copy w to w′ after several training  18: end for  19: Output: Optimal traffic steering decisions from TTI =  1  to  T  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Baseline Algorithms  In this section, two baseline algorithms are introduced that  have been used for the performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The first  baseline algorithm for RAT selection is based on a predefined  threshold [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' This is called the heuristic baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Here, the  threshold is calculated for each UE based on the metrics like  load at eNB (le) and gNB (lg), channel condition of a user  under LTE (che,u) and 5G BS (chg,u), service type of a user  (Su).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The channel condition is determined to be good or bad  considering a threshold of received SINR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Similarly, the  load at each RAT is determined based on a threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Based on the mentioned metrics, a value Tu is calculated that  is used for selecting the RAT for a UE after comparing it with  a predetermined threshold (Tth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Following equation is used  to calculate the value for Tu:  Tu(le, lg, che,u, Su) = αle + βlg + γchg,u + δSu,  (13)  where α, β, γ, and δ are the weights associated with consid-  ered parameters that can be modulated based on the impact of  any certain metric on system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Tth is set to be the  mean of all the possible values of Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The decision of steering  traffic to a particular RAT is taken the following way:  Ru =  �  1, Tu > Tth  (1 represents gNB)  0, Tu ⩽ Tth  (0 represents eNB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  (14)  The Q-learning algorithm has been used as another baseline  in this work [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The goal is to investigate how DQN performs  against the Q-learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Main network  Hidden layers  Action  selector  LTE  ()  Hidden layers  lnput  5G NR  Observation  Target network  Reward: Throughput,  delay  Minibatch from  New state: Load level  experience poolV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' PERFORMANCE EVALUATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Simulation setup  We have conducted MATLAB based simulations consider-  ing 1 eNB and 4 gNBs with 30 users in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' There are a total  of 1 macro-cell and 4 small cells facilitated by the gNBs and an  eNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' A macro-cell and a small-cell have carrier frequencies of  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='5 GHz and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='8 GHz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Specifications of the traffic  classes used in this study have been summarized in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  For the experimental results, the load has been varied between  5-10 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Proportion of the voice, video, and gaming traffic  is 20%, 50%, and 30% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Higher proportion of  the video traffic is deliberately considered to observe how  the system performs with the higher throughput requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Also, gaming traffic has the most stringent delay requirement  and we wanted to see if the system performs well enough  to meet such precise requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Therefore it has a higher  percentage compared to the voice traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' QoS requirements  associated with delay and throughput for the three types of  traffic classes are specified based on the existing literature [14]  and 3GPP specifications (see TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' We are using multi-  RAT dual connectivity architecture, an NSA mode where LTE  and 5G NR BSs serve together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' An architecture specified in  [15] has been used where the dual connectivity is ensured  via evolved packet core [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Transmission power of the LTE  BS and 5G NR BSs are set to 40W and 20W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Furthermore,  bandwidth for the LTE and 5G RAT are fixed to 10MHz and  20MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  TABLE I  TRAFFIC CLASS DESCRIPTION AND SIMULATION SETTINGS  Traffic class specification  Values  Traffic model  Poisson distribution, video  and gaming traffic [14]  Voice traffic  Packet size  30 bytes  TQoS, DQoS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='1 Mbps , 100ms  Proportion of the traffic  20%  Video traffic  Packet size  250 bytes  TQoS, DQoS  10 Mbps, 80ms  Proportion of the traffic  50%  Gaming traffic  Packet size (gaming traffic)  120 bytes  TQoS, DQoS  5 Mbps, 40ms  Proportion of the traffic  30%  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Simulation results  The performance of the proposed algorithm is evaluated in  terms of two KPIs: Average system throughput and network  delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' In Fig 3, we present a comparison in terms of system  throughput under different user loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The proposed DQN  outperforms heuristic and Q-Learning baselines by gaining 6%  and 10% increased throughput, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 4 presents the performance comparison of the proposed  DQN-based traffic steering method with the other baselines in  terms of delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The DQN-based method achieves 23% and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' System throughput against traffic load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' System delay against traffic load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  33% decrease in network delay compared to the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Note that, the proposed method and the Q-learning, both have  a reward function formulated based on throughput and delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Whenever high delay is experienced for steering traffic to a  particular RAT, the system learns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' That is why, both of them  have better performance compared to the heuristic baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 4, delay is calculated considering all the traffic classes  together at each load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  It should be mentioned that the main reason of the improved  performance of the proposed method is the use of DQN,  that outperforms Q-learning in terms of exploration efficiency  and achieves higher average reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Q-learning suffers due to  longer exploration period and gets lower average reward since  it does not have a DNN as an approximator which compels  the agent to cover larger state and action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  In this work, we also want to steer a particular type of  traffic to a specific RAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' For example, steering the voice  traffic constantly to a gNB is a waste of resources since the  throughput requirement is not that high for such traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 5  is presented which shows what percentage of a traffic class is  processed by a particular RAT and when the traffic gets steered  due to higher load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 5(a), it is observed that most of  the voice traffic is processed by the eNB, however, a small  portion of the traffic is processed by the gNB too whenever  the system experiences higher load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' For the video and gaming  traffic, it is observed that most of the traffic is processed by  the gNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  300  DQN  --Q-learning  Heuristic baseline  250  200  150  5  6  7  8  9  10  Load per user (Mbps)20  Heuristic baseline  Q-learning  +DQN  15  (ms)  Delay (  10  5  0  5  6  7  8  9  10  Load per user (Mbps)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Data processing percentage for different traffic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Traffic steered to other RAT as load changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Lastly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 6 demonstrates how traffic steering occurs  whenever a high load is experienced in a BS with a particular  RAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' We start with one UE at the 300th time slot and increase  the number of UEs in a small cell up to six for different traffic  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' The variable L, in the respective figure represents load  in terms of queue length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' At the 1800th time slot, it can be  seen that four among six UEs are steering different types of  traffic to the 5G NR BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' This results in higher load and we can  see that the third and fourth UEs are experiencing high load  (value of L changed from 0 to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' So, in the next observed  time slot, these two UEs steer the traffic to the eNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' In the  2100th time slot, we can see four UEs steering voice, video,  and gaming traffic to the only eNB in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' This incurs  high load at eNB and in the next observed slot we can see  that the sixth UE has switched its traffic to the gNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' CONCLUSIONS  In this study, we have proposed a novel method that can  perform RAT specific and QoS aware traffic steering using  DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' It gains 6% and 10% increase in average system  throughput compared to the Q-learning and heuristic-based  baseline respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Moreover, it achieves 23% and 33%  times decrease in network delay compared to the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  Apart from the better performance in terms of the KPIs, the  proposed method can perform RAT specific traffic steering  ensuring efficient use of network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Lastly, the pro-  posed DQN-based traffic steering can successfully perform  load balancing in an optimal way as whenever high load is  induced to a particular RAT, traffic is steered to another RAT  dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  This work has been supported by MITACS and Ericsson  Canada, and NSERC Collaborative Research and Training  Experience Program (CREATE) under Grant 497981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 9414–9424, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' Makris, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Korakis, “Pricing Based  Distributed Traffic Allocation for 5G Heterogeneous Networks,” IEEE  Transactions on Vehicular Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 12 111–  12 123, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' Giuseppi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Liberati, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' 595–  601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' Romero-Diaz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' Ameigeiras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Ramos-  Munoz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' 22,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 905–929, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content='  Tano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  (2019)  A  Technical  Look  at  5G  Energy Consumption and Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Available: https:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='com/en/blog/2019/9/energy-consumption-5{G}-nr  [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content=' Kwon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Park, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' Jin, “A Survey on 4G-5G Dual  Connectivity: Road to 5G Implementation,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  16 193–16 210, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='TCi(Voice traffic)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='eNB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
+page_content='Higher load atgNB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf'}
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+Discovering Sound Free-choice Workflow Nets
+With Non-block Structures
+Tsung-Hao Huang[0000−0002−3011−9999] and Wil M. P. van der
+Aalst[0000−0002−0955−6940]
+Process and Data Science (PADS), RWTH Aachen University, Aachen, Germany
+{tsunghao.huang, wvdaalst}@pads.rwth-aachen.de
+Abstract. Process discovery aims to discover models that can explain
+the behaviors of event logs extracted from information systems. While
+various approaches have been proposed, only a few guarantee desirable
+properties such as soundness and free-choice. State-of-the-art approaches
+that exploit the representational bias of process trees to provide the guar-
+antees are constrained to be block-structured. Such constructs limit the
+expressive power of the discovered models, i.e., only a subset of sound
+free-choice workflow nets can be discovered. To support a more flexible
+structural representation, we aim to discover process models that provide
+the same guarantees but also allow for non-block structures. Inspired by
+existing works that utilize synthesis rules from the free-choice nets the-
+ory, we propose an automatic approach that incrementally adds activities
+to an existing process model with predefined patterns. Playing by the
+rules ensures that the resulting models are always sound and free-choice.
+Furthermore, the discovered models are not restricted to block struc-
+tures and are thus more flexible. The approach has been implemented
+in Python and tested using various real-life event logs. The experiments
+show that our approach can indeed discover models with competitive
+quality and more flexible structures compared to the existing approach.
+Keywords: Process Discovery · Free-choice Net · Synthesis Rules.
+1
+Introduction
+Process discovery aims to construct process models that reflect the behaviors of
+a given event log extracted from information systems [2]. As it is a non-trivial
+problem, many challenges remain. In most cases, the one and only "best model"
+does not exist as there are trade-offs among the four model quality metrics,
+namely fitness, precision, generalization, and simplicity [2]. In addition to the
+quality metrics, there exist properties that one would like to have for the discov-
+ered models. One of the important properties is being a sound workflow net as
+soundness ensures the absence of deadlocks, proper completion, etc. [1] and it is
+a prerequisite for many crucial automated analyses such as conformance check-
+ing. The other desirable structural property is being free-choice [3]. In free-choice
+nets, choices and synchronizations are separated. This provides an easy conver-
+sion between the discovered models and many process modeling languages such
+arXiv:2301.02185v1  [cs.DB]  3 Jan 2023
+
+2
+T. Huang and W. M. P. van der Aalst
+as Business Process Modeling Notation (BPMN) since the equivalent constructs
+(dedicated split and join connectors) are naturally embedded. Furthermore, free-
+choice nets have been studied extensively and thus supported by an abundance
+of theories [10], which provide efficient analysis techniques.
+While various discovery algorithms have been proposed, only a handful of
+them provides such guarantees. State-of-the-art discovery algorithms like the In-
+ductive Miner (IM) [15] are able to discover sound free-choice workflow nets by
+exploiting its representational bias. However, due to the same reason, the discov-
+ered models are constrained to be block-structured. This limits the expressive
+power of such models, i.e., only a subset of the sound free-choice workflow nets
+can be discovered. As an example, Fig. 1a shows a sound free-choice workflow
+net (with non-block structures) discovered by our approach1. The same language
+can never be expressed by the model discovered by IM, as shown in Fig. 1b.
+b
+c
+d
+f
+g
+a
+e
+h
+(a) A model discovered by our approach. The
+same language cannot be expressed by the
+models discovered using the Inductive Miner,
+which uses process trees internally.
+b
+c
+d
+f
+g
+a
+e
+h
+(b) A model discovered by the IM using the
+log generated by the model in (a). The two
+branches before c need to be synchronized first
+before d can be executed.
+Fig. 1: Examples showing the need for the non-block process models discovery. Note
+that the trace ⟨a, b, c, d, e, f, g, h⟩ that is possible in (a) cannot be replayed by (b).
+In this paper, we aim to discover sound free-choice workflow nets with non-
+block structures. Inspired by the interactive process discovery approach in [11,12],
+we develop an automatic process discovery algorithm that incrementally adds
+activities to an existing net using synthesis rules [10]. Since checking the feasibil-
+ity for the application of the synthesis rules is computationally expensive, we use
+log heuristics to locate the most possible position for the to-be-added activity
+on the existing process model instead of evaluating all possible applications of
+synthesis rules as in [11]. Additionally, we identify the need for an additional
+rule and extend the set of patterns introduced in [12].
+Playing by the rules ensures that the discovered process models by our ap-
+proach are guaranteed to be sound free-choice workflow nets [10,11]. Moreover,
+the discovered models are not constrained to block structures. Last but not least,
+the level of replay fitness is guaranteed via a threshold set by the users. The
+approach has been implemented in Python and evaluated using various public-
+available real-life event logs. The evaluation shows that our approach is able to
+discover non-block structured models with competitive qualities compared to the
+state-of-the-art discovery algorithm.
+1 The proposed approach has dedicated silent transitions for start and end as defined
+later in Def. 5. We dropped them here for ease of comparison.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+3
+The remainder of the paper is organized as follows. We review the related
+work in Sec. 2 and introduce necessary concepts in Sec. 3. Sec. 4 introduces the
+approach. Sec. 5 presents the experiment and Sec. 6 concludes the paper.
+2
+Related Work
+An overview of process discovery is out of the scope of this paper, we refer to
+[7,14] for more details. In this section, we focus on process discovery techniques
+that guarantee soundness (and free-choice) properties. Approaches like [6,8] can
+discover non-block structured models but cannot guarantee both properties.
+While Split Miner discovers models that are deadlock-free, they are not nec-
+essarily sound [8].
+The family of Inductive Miner (IM) algorithms [15] guarantees sound and
+free-choice of the discovered models by exploiting the representational bias of
+the process tree. By design, a process tree represents a sound workflow net. It is
+a rooted tree where the leaf nodes are activities and the non-leaf nodes are the
+operators. The hierarchical representation has a straightforward translation to
+Petri net. However, the resulting models are limited to being block-structured as
+a process tree can only represent process models that can be separated into parts
+that have a single entry and exit [15]. Consequently, process trees can only rep-
+resent a subset of sound workflow nets. The same arguments hold for approaches
+that are based on process trees such as the Evolutionary Tree Miner (ETM) [9]
+and the recently developed incremental process discovery approach [16].
+Applying the synthesis rules [10], the interactive process discovery approaches
+developed in [12,13,11] ensure soundness and free-choice properties. A semi-
+automatic interactive tool, ProDiGy, is proposed in [12] to recommend the best
+possible ways to add an activity to an existing model.
+Our approach differs from [12,13,11] in several ways. First, we adopt an
+automatic setting as the order of adding activities is predetermined and the best
+modification to the existing net is selected based on the model quality. Second,
+we use log heuristics to locate the most suitable position for adding the new
+activity instead of evaluating all the possibilities of synthesis rules applications.
+Moreover, we identify the need for a new rule as the desired models often cannot
+be discovered without going back and forth by a combination of reduction and
+synthesis rules [13]. Lastly, the set of patterns is extended and formally defined.
+3
+Preliminaries
+We denote the set of all sequences over some set A as A∗, the power set of
+A as P(A), and the set of all multisets over A as B(A). For some multiset
+b ∈ B(A), b(a) denotes the number of times a ∈ A appears in b. For a given
+sequence σ = ⟨a1, a2, ..., an⟩ ∈ A∗, |σ| = n is the length of σ and dom(σ) =
+{1, 2, ..., |σ|} is the domain of σ. ⟨⟩ is the empty sequence. σ(i) = ai denotes the
+i-th element of σ. Given sequences σ1 and σ2, σ1 · σ2 denotes the concatenation
+of the two. Let A be a set and X ⊆ A be a subset of A. For σ ∈ A∗ and a ∈ A,
+
+4
+T. Huang and W. M. P. van der Aalst
+we define ↾X∈ A∗→X∗ as a projection function recursively with ⟨⟩↾X = ⟨⟩,
+(⟨a⟩ · σ)↾X = ⟨a⟩ · σ↾X if a ∈ X and (⟨a⟩ · σ)↾X = σ↾X if a /∈ X. For example,
+⟨x, y, x⟩↾{x,z} = ⟨x, x⟩. Projection can also be applied to multisets of sequences,
+e.g., [⟨a, b, a⟩6, ⟨a, b, c⟩6, ⟨b, a, c⟩2]↾{b,c} = [⟨b⟩6, ⟨b, c⟩8].
+Definition 1 (Trace, Log). A trace σ ∈ U∗
+A is a sequence of activities, where
+UA is the universe of activities. A log L ∈ B(U∗
+A) is a multiset of traces.
+Definition 2 (Log Properties). Let L ∈ B(U∗
+A) and a, b ∈ UA.
+– #(a, L) = Σσ∈L|{i ∈ dom(σ)|σ(i) = a}| is the times a occurred in L.
+– #(a, b, L) = Σσ∈L|{i ∈ dom(σ)\{|σ|}|σ(i) = a∧σ(i+1) = b}| is the number
+of direct successions from a to b in L.
+– caus(a, b, L) =
+�
+#(a,b,L)−#(b,a,L)
+#(a,b,L)+#(b,a,L)+1
+if a ̸= b
+#(a,b,L)
+#(a,b,L)+1
+if a = b is the strength of causal rela-
+tion (a, b).
+– Apre
+c
+(a, L) = {apre ∈ UA|caus(apre, a, L) ≥ c} is the set of a’s preceding
+activities, determined by threshold c.
+– Afol
+c
+(a, L) = {afol ∈ UA|caus(a, afol, L) ≥ c} is the set of a’s following
+activities, determined by threshold c.
+– As(L) = {σ(1) | σ ∈ L ∧ σ ̸= ⟨⟩} is the set of start activities in L.
+– Ae(L) = {σ(|σ|) | σ ∈ L ∧ σ ̸= ⟨⟩} is the set of end activities in L.
+Definition 3 (Petri Net, Labeled Petri Net). A Petri net N = (P, T, F) is
+a tuple, where P is the set of places, T is the set of transitions, P ∩ T = ∅, and
+F ⊆ (P × T) ∪ (T × P) is the set of arcs. A labeled Petri net N = (P, T, F, l) is
+a Petri net (P, T, F) with a labeling function l ∈ T ↛ UA that maps a subset of
+transitions to activities. A t ∈ T is called invisible if t is not in the domain of l.
+For any x ∈ P ∪ T,
+N•x = {y|(y, x) ∈ F} denotes the set of input nodes and
+x
+N• = {y|(x, y) ∈ F} denotes the set of output nodes. The superscript N is
+dropped if it is clear from the context. The notation can be generalized to set.
+For any X ⊆ P ∪ T, •X = {y|∃x∈X(y, x) ∈ F} and X• = {y|∃x∈X(x, y) ∈ F}.
+Definition 4 (Free-choice Net). Let N = (P, T, F) be a Petri net. N is a
+free-choice net if for any t1, t2 ∈ T : •t1 = •t2 or •t1 ∩ •t2 = ∅.
+Definition 5 (Workflow Net (WF-net) [1,11]). Let N = (P, T, F, l) be a
+labeled Petri net. W = (P, T, F, l, ps, pe, ⊤, ⊥) is a WF-net iff (1) it has a dedi-
+cated source place ps ∈ P: •ps = ∅ and a dedicated sink place pe ∈ P: pe• = ∅
+(2) ⊤ ∈ T: •⊤ = {ps}∧ps• = {⊤} and ⊥ ∈ T: ⊥• = {pe}∧•pe = {⊥} (3) every
+node x is on some path from ps to pe, i.e., ∀x∈P ∪T (ps, x) ∈ F ∗ ∧ (x, pe) ∈ F ∗,
+where F ∗ is the reflexive transitive closure of F.
+Definition 6 (Short-circuited WF-net [1]). Let W = (P, T, F, l, ps, pe, ⊤, ⊥)
+be a WF-net. The short-circuited WF-net of W, denoted by SC(W), is con-
+structed by SC(W)=(P, T ∪{t′}, F ∪{(⊥, t′), (t′, ⊤)}, l, ps, pe, ⊤, ⊥), where t′ /∈ T.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+5
+Definition 7 (Paths, Elementary Paths). A path of a Petri net N = (P, T, F)
+is a non-empty sequence of nodes ρ = ⟨x1, x2, ..., xn⟩ such that (xi, xi+1) ∈ F for
+1 ≤ i < n. ρ is an elementary path if xi ̸= xj for 1 ≤ i < j ≤ n.
+Definition 8 (Incidence Matrix [10]). Let N = (P, T, F) be a Petri net. The
+incidence matrix N : (P × T) → {−1, 0, 1} of N is defined as
+N(p, t) =
+�
+�
+�
+�
+�
+0
+if ((p, t) /∈ F ∧ (t, p) /∈ F) ∨ ((p, t) ∈ F ∧ (t, p) ∈ F)
+−1
+if (p, t) ∈ F ∧ (t, p) /∈ F
+1
+if (p, t) /∈ F ∧ (t, p) ∈ F
+For a Petri net N = (P, T, F) and its corresponding incidence matrix N, we use
+N(p) to denote the row vector of the corresponding p ∈ P and N(t) to denote
+the column vector of the corresponding t ∈ T.
+Definition 9 (Linearly Dependent Nodes [10]). Let N = (P, T, F) be a
+Petri net. Q is the set of rational numbers. A place p is linearly dependent if
+there exists a row vector ⃗v : P → Q such that ⃗v(p) = 0 and ⃗v · N = N(p). A
+transition t is linearly dependent if there exists a column vector ⃗v : T → Q such
+that ⃗v(t) = 0 and ⃗v · N = N(t).
+Definition 10 (Synthesis Rules [10,11]). Let W and W ′ be two free-choice
+workflow nets, and let SC(W) = (P, T, F, l, ps, pe, ⊤, ⊥) and SC(W ′) = (P ′, T ′,
+F ′, l′, ps, pe, ⊤, ⊥) be the corresponding short-circuited WF-nets:
+– Linear Dependent Place Rule ψP : W ′ is derived from W using ψP , i.e.,
+(W, W ′) ∈ ψP if (1) T ′ = T, P ′ = P ∪ {p} and p /∈ P is linear dependent in
+SC(W ′), F ′ = F ∪ �F where �F ⊆ (({p} × T) ∪ (T × {p})) (2) Every siphon
+in SC(W ′) contains ps.
+– Linear Dependent Transition Rule ψT : W ′ is derived from W using ψT , i.e.,
+(W, W ′) ∈ ψT if P ′ = P, T ′ = T ∪ {t} and t /∈ T is linear dependent in
+SC(W ′) and F ′ = F ∪ �F where �F ⊆ ((P ×{t})∪({t}×P)), and ∀t∈T ∩T ′l(t) =
+l′(t).
+– Abstraction Rule ψA: (W, W ′) ∈ ψA if (1) there exists a set of transitions
+R ⊆ T and a set of places S ⊆ P such that (R × S ⊆ F) ∧ (R × S ̸= ∅). (2)
+SC(W ′) is constructed by adding an additional place p /∈ P and a transition
+t /∈ T such that P ′ = P ∪ {p}, T ′ = T ∪ {t}, F ′ = (F\(R × S)) ∪ ((R × {p}) ∪
+({p} × {t}) ∪ ({t} × S)), and ∀t∈T ∩T ′l(t) = l′(t).
+Applying the three synthesis rules (ψP , ψT , ψA) to derive W ′ from a sound
+free-choice workflow net W ensures that W ′ is also sound [13,11]. Three proper-
+ties need to be hold for a WF-net to be sound (1) safeness: places cannot hold
+multiple tokens at the same time (2) option to complete: it is always possible to
+reach the marking in which only the sink place is marked. (3) no dead transitions.
+Next, we introduce the initial net [11] and show some examples of synthesis rules
+applications.
+
+6
+T. Huang and W. M. P. van der Aalst
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+(a)
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑡1
+𝑝2
+(b)
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+(c)
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+(d)
+Fig. 2: Examples of synthesis rules applications starting from (a) The initial net. (b)
+Using ψA, p2 and t1 are added to the initial net with R = {⊤} and S = {p1}. (c) Using
+ψA, p3 and t2 are added to previous net with R = {⊤} and S = {p2}. (d) p4 is added
+using ψp as p4 is a linear combination of p3 and p2.
+Definition 11 (Initial Net [13]). Let W = (P, T, F, l, ps, pe, ⊤, ⊥) be a free-
+choice WF-net. W is an initial net if P = {ps, p1, pe}, T = {⊤, ⊥}, F =
+{(ps, ⊤), (⊤, p1), (p1, ⊥), (⊥, pe)}.
+The initial net is shown in Fig. 2a. Clearly, it is a sound free-choice workflow
+net. Starting from the initial net, one can incrementally add additional nodes
+according to the synthesis rules. Fig. 2 shows example applications of synthesis
+rules starting from the initial net.
+4
+Approach
+With the necessary concepts introduced, we are now ready to introduce the ap-
+proach. We start by showing the basic idea of the approach with the help of
+Fig. 3 before diving into each step in detail. Internally, the approach incremen-
+tally adds a new activity to an existing net. The figure shows a single iteration.
+In each iteration, we have an existing model from the previous iteration and a
+log projected on the already added activities so far and the to-be-added one.
+We start by locating the most likely position to add the new activity deter-
+mined by log heuristics. The result of this step is a subset of nodes of the existing
+model. The set of nodes will then be used to prune the search space. Then, the
+predefined patterns are applied to the existing net to get a set of candidate nets.
+Lastly, we select the best net (next existing net) out of the candidates in terms of
+fitness and precision. Note that the existing net in the first iteration is initiated
+by the initial net (Def. 11). As a running example, consider the correspond-
+ing log that is used to discover the Petri net in Fig.1a by our approach: Ls =
+[⟨a, b, c, d, f, g, h⟩22, ⟨a, b, c, f, d, g, h⟩14, ⟨a, e, b, c, d, f, g, h⟩13, ⟨a, e, b, c, f, d, g, h⟩13,
+⟨a, e, b, c, f, g, d, h⟩10, ⟨a, b, c, f, g, d, h⟩10, ⟨a, b, e, c, d, f, g, h⟩6, ⟨a, b, e, c, f, g, d, h⟩3,
+⟨a, b, e, c, f, d, g, h⟩3, ⟨a, b, c, d, e, f, g, h⟩2, ⟨a, b, c, e, d, f, g, h⟩2, ⟨a, b, c, e, f, g, d, h⟩1,
+⟨a, b, c, e, f, d, g, h⟩1]. The instances provided in Fig. 3 shows the 3rd iteration for
+the running example Ls. In the following subsections, we introduce the details
+of each step.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+7
+(1) Pruning search 
+space using log 
+heuristics
+(2) Add new activity to 
+the existing net with 
+pre-defined patterns
+[ 𝑑, 𝑔, ℎ 76, 𝑔, 𝑑, ℎ 24]
+Projected Log 𝐿𝑖
+Existing Net
+𝑊𝑖 = (𝑃𝑖, 𝑇𝑖, 𝐹𝑖, 𝑙𝑖, 𝑝𝑠, 𝑝𝑒, ⊤, ⊥)
+(3) Select the best net
+for the next iteration
+Next Existing Net
+𝑊𝑖+1 = (𝑃𝑖+1, 𝑇𝑖+1, 𝐹𝑖+1, 𝑙𝑖+1, 𝑝𝑠, 𝑝𝑒, ⊤, ⊥)
+To-be-added Activity 𝛾(𝑖)
+𝑉𝑖 ⊆ 𝑃𝑖 ∪ 𝑇𝑖
+Set of Candidate Nets 𝐶𝑖
+skip
+loop
+…
+…
+Fig. 3: An example of a single iteration of our approach.
+4.1
+Ordering Strategies for Adding Activities
+Before starting any iteration, we need to come up with an order for adding ac-
+tivities based on a given log L. It is important as the quality of the discovered
+models often depends on the order of adding activities [11]. Moreover, in combi-
+nation with the search space pruning, it can influence the computation time for
+each iteration significantly. In this paper, we introduce two ordering strategies.
+The first one is relatively straightforward. The activities in L are simply ordered
+by their frequency.
+Definition 12 (Activities-Adding Order, Frequency-Based Ordering).
+Let L ∈ B(U∗
+A) and A = �
+σ∈L{a ∈ σ}. γ ∈ A∗ is an activities-adding order for
+L if {a ∈ γ} = A and |γ| = |A|. The frequency-based ordering is orderfreq(L) = γ
+such that γ is an activities-adding order and ∀1≤i<j≤|γ|#(γ(i), L) ≥ #(γ(j), L).
+The second ordering strategy is similar to the Breadth-first Search (BFS) al-
+gorithm. The advantage of this is that it also considers the closeness between
+activities in the log, rather than just frequency. To explain this ordering strategy,
+we first define a sub-function.
+Definition 13 (Directly-Precedes Activities Sorting). Let L ∈ B(U∗
+A) and
+a ∈ UA. A={b ∈ UA|#(b, a, L)>0} is the set of activities directly-precede a in L at
+least once and σ ∈ A∗. Directly-precedes activities sorting is sortPreceded(a, L)=σ
+such that {b ∈ σ} = A and |σ| = |A| and ∀1≤i<j≤|σ| #(σ(i), a, L) ≥ #(σ(j), a, L).
+The function sortPreceded takes an activity a and a log L to return a sequence of
+a’s directly-preceded activities b that are sorted by the frequency of #(b, a, L).
+Finally, we can define the BFS-based ordering strategy.
+Definition 14 (Breadth-First-Search-Based Ordering). Let L ∈ B(U∗
+A)
+and A= �
+σ∈L{a ∈ σ}. BFS-based ordering is defined as orderBFS(L)=γ, where
+γ is an activities adding order for L and γ=γ1 · γ2 · ... · γ|γ|, for each 1 ≤ j ≤ |γ|,
+γj =
+�
+orderfreq(L↾Ae(L))
+if j = 1
+sortPreceded(γ(j − 1), L)↾A\{γ(1),γ(2),...,γ(j−1)}
+otherwise
+
+d
+h
+gdps
+4
+0
+d
+h
+gd
+hh
+g8
+T. Huang and W. M. P. van der Aalst
+The function starts by sorting the end activities Ae(L) according to their fre-
+quency in the log. Then, it enumerates through the sequence γ and sorts the
+preceded activities of γ(j − 1) by the frequency of direct successions. The pro-
+jection function in the second case of Def.14 filters out the activities that are
+already in γ.
+Compared to the frequency-based ordering, the BFS-based ordering considers
+the closeness of the activities. This allows us to add activities that are close
+together. Together with the effect of the search space pruning, it is expected that
+BFS-based ordering would have less computation time. Applying the function
+orderBFS to our running example, Ls, we get the activities adding order as
+γ = orderBFS(Ls) = ⟨h⟩ · ⟨g, d⟩ · ⟨f⟩ · ⟨c, e⟩ · ⟨⟩ · ⟨b⟩ · ⟨a⟩ · ⟨⟩ = ⟨h, g, d, f, c, e, b, a⟩.
+γ is then used to determine the order of adding activities. Given the activities
+adding order γ, we define the artifacts for each iteration i as followed.
+Definition 15 (Projected Log). Let L ∈ B(U∗
+A) and γ be a activities adding
+order for L. The projected log for L in the i-th iteration is Li = L↾{γ(1),γ(2),...γ(i)}.
+For instance, the projected log for the running example Ls for the 3rd iteration
+is then L3
+s = Ls↾{h,g,d} = [⟨d, g, h⟩76, ⟨g, d, h⟩24], as shown in Fig. 3. The to-be-
+added activity is denoted as γ(i), which is γ(3) = d for the 3rd iteration. Also,
+we denote the existing sound free-choice workflow net for iteration i as W i. Note
+that for the running example, W 1, W 2, and W 3 are visualized in Fig. 2a, 2b,
+and 2c, respectively.
+4.2
+Search Space Pruning
+As checking the feasibility of applying linear dependent rules ψT , ψP is compu-
+tationally expensive [11], it is impractical to compute all possible applications
+of the synthesis rules. Also, some of them are not of interest. For example, as
+shown in Fig. 3, it is clear that the to-be-added activity d never happens after
+h in the projected log. Using such information, we can already eliminate the
+constructs (applications of synthesis rules) that allow activity d to be executed
+after h. Therefore, in each iteration i we start by locating the most likely posi-
+tion to add γ(i). This helps us to restrict the application of synthesis rules on
+only a subset of nodes, denoted as V i ⊆ P i ∪ T i, in the existing net W i. To do
+that, we first identify the set of preceding and following activities of γ(i) in the
+projected log Li, which would be Apre
+c
+(γ(i), Li) and Afol
+c
+(γ(i), Li) respectively.
+Recall that c is a threshold for the causal relation and can be given by users
+as an input. We use c = 0.9 as the default value. Then, the corresponding la-
+belled transitions are identified in W i. Finally, V i is the set of all the nodes on
+the elementary paths from the preceding transitions to the following transitions.
+If Apre
+c
+(γ(i), L)/Afol
+c
+(γ(i), L) is an empty set, we use the ⊤/⊥ transitions. For
+instance, in Fig. 3, we identify that Apre
+c
+(d, L3
+s)=∅ and Afol
+c
+(d, L3
+s)={h}. There-
+fore, we find all the nodes on the elementary paths between every node in {⊤}
+and every node in {t1}. As a result, the set V 3 = {⊤, p3, t2, p2, t1} is used to
+prune the search space, i.e., to constrain the application of synthesis rules.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+9
+Constraining synthesis rules For the abstraction rule ψA, this means that the
+set of transitions R and the set of places S used as the preconditions for applying
+ψA need to be a subset of V , i.e., S ⊆ V ∧R ⊆ V . For the linear dependent rules
+ψP /ψT , the new place/transition (p′/t′) cannot have arcs connected to any node
+outside V . This shortens the computation time as certain rules applications can
+be removed and there is no need to check their feasibility.
+4.3
+Patterns
+In this section, we introduce the patterns that are used to add activity γ(i) to
+the existing free-choice workflow net W i. First, we motivate the need for an
+additional rule.
+The need for an additional rule It is proven that any sound free-choice
+workflow net can be constructed by the three synthesis rules ψA, ψP , ψT [10,11].
+However, when applying to discover process models, the desirable model is often
+not possible to derive due to the existing construct. An example is shown in
+Fig. 4. While it is possible to add a transition labeled by a in Fig. 4a, it is not
+possible to derive the same net in Fig. 4b as there is no rule allowing such trans-
+formation. One possible workaround is to go back and forth by a combination
+of reduction and synthesis rules as suggested in [13]. However, once the existing
+net becomes more complex, such a solution becomes infeasible to track.
+b
+c
+L↾〈c,b〉 = [〈c〉3, 〈b〉3]
+b
+c
+a
+L↾〈a,b,c〉 = [〈c,a〉3, 〈b〉3]
+(a)
+L↾〈a,b〉 = [〈a〉3, 〈b〉3]
+L↾〈a,b,c〉 = [〈c,a〉3, 〈b〉3]
+b
+c
+a
+b
+a
+?
+(b)
+Fig. 4: Examples showing the motivation for the dual abstraction rule ψD. Although
+the desirable nets on the right-hand side of (a) and (b) are semantically the same, the
+existing synthesis rules only allow the transformation in (a). There is no rule defined
+for the transformation in (b).
+We observe that in many situations (including the example in Fig. 4b), the
+desired models cannot be constructed because there is no rule allowing one to
+introduce a new transition t and a new place p in between a set of places S and
+a set of transitions R that are fully connected, i.e., S × R ⊆ F. Therefore, we
+define the dual abstraction rule to allow such construct.
+Definition 16 (Dual Abstraction Rule ψD). Let W = (P, T, F, l, ps, pe, ⊤, ⊥)
+and W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be two free-choice workflow nets. (W, W ′) ∈
+ψD if (1) there exists a set of places S ⊆ P and a set of transitions R ⊆ T such
+that S × R ⊆ F ∧ S × R ̸= ∅. (2) W ′ is constructed by adding an additional
+
+10
+T. Huang and W. M. P. van der Aalst
+transition t /∈ T and a place p /∈ P such that P ′ = P ∪ {p}, T ′ = T ∪ {t}, F ′ =
+(F \ (S × R)) ∪ ((S × {t}) ∪ ({t} × {p}) ∪ ({p} × R)), and ∀t∈T ∩T ′l(t) = l′(t).
+As we are only interested in sound free-choice workflow nets, we need to make
+sure that the dual abstraction rule ψD preserves soundness.
+Proposition 1 (ψD preserves soundness). Let W = (P, T, F, l, ps, pe, ⊤, ⊥),
+W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be free-choice workflow nets, and W ′ is derived
+from W using ψD, i.e., (W, W ′) ∈ ψD. Then W ′ is sound if W is sound.
+Proof. Let t′ ∈ T ′\T and p′ ∈ P ′\P be the new transition and place in W ′. Let
+R = p′• and S = •t′. The new net W ′ is free-choice in only two cases. Either
+S =
+W• R or R = S
+W• . In either case, any reachable marking in (W, [ps]) that
+does not need to fire tR ∈ R is still reachable in (W ′, [ps]). Also, the reachable
+markings in (W, [ps]) that need to fire tR ∈ R can be reached in (W ′, [ps]) as one
+can just add t′ somewhere before tR in the corresponding firing sequence. Then,
+it is trivial to see that W ′ fulfils the three conditions of soundness if W is also
+sound.
+□
+Next, we extent the linear dependent place rule ψP . As we aim to add a transition
+labeled by γ(i) to the existing labeled free-choice workflow net W i, only adding
+a place p′ by ψP does not suffice. Hence, in our approach, an application of ψP
+is always coupled with a directly followed application of abstraction rule ψA to
+include a transition. ψA is applied between the added place p′ and its preset •p′.
+This is possible because every transition in •p′ is connected to every place in
+{p′} by definition, which satisfies the precondition of ψA. An example is shown
+in Fig. 5, p5 and t3 are added by ψA directly after the addition of p4 by ψP . To
+be more precise, we define the extended rule, ψ′
+P , that describes the pattern.
+Definition 17 (Extended Linear Dependent Place Rule ψ′
+P ). Let W=(P,
+T, F, l, ps, pe, ⊤, ⊥) and W ′′=(P ′′, T ′′, F ′′, l′′, ps, pe, ⊤, ⊥) be free-choice workflow
+nets. (W, W ′′)∈ψ′
+P if (1) ∃W ′=(P ′,T ′,F ′,l′,ps,pe,⊤,⊥)(W, W ′)∈ψP ∧ (W ′, W ′′)∈ψA
+and (2) ∃!p∗∈P ′′({p∗}=P ′\P) ∧ (((T ′′\T ′) × {p∗}) ⊂ F ′′) ∧ ((T ′ × {p∗}) ̸⊂ F ′′).
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+(a)
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+d
+𝑝5
+𝑡3
+(b)
+Fig. 5: (a) ψP adds a place p4. (b) As every transition in •p4 has an arc to every place
+in {p4}, one can directly apply ψA to add p5 and t3.
+Then, we define the set of nets constructed by every possible single application
+of the rules ψA, ψ′
+P , ψT , ψD.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+11
+Definition 18 (Base Candidates Set). Let W=(P, T, F, l, ps, pe, ⊤, ⊥), W ′ =
+(P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be free-choice workflow nets. Let X=(P ′∪T ′)\(P∪T),
+V ⊆P∪T, V ′=(P∪T)\V , and let a ∈ UA be an activity label. The base candidates
+set is base(W, V, a)={W ′|((W, W ′) ∈ (ψA ∪ψT ∪ψ′
+P ∪ψD))∧(∄x∈X(({x}×V ′)∪
+(V ′ × {x})) ⊆ F ′) ∧ (l′=l ∪ ((T ′ \ T) × {a}))}.
+The base candidates set Ci
+base=base(W i, V i, γ(i)) consists of the nets that are
+constructed by every possible single application of the rules ψA, ψ′
+P , ψT , ψD to
+add a transition labeled by γ(i) to W i considering the the constraints on V i.
+Next, we introduce three patterns that make a transition skippable, in a strict
+loop, or in an optional (tau) loop. A transition in a strict loop means that the
+execution of the transition is required, otherwise it is an optional loop.
+Definition 19 (Pattern-Building Functions). Let W=(P, T, F, l, ps, pe, ⊤, ⊥)
+and W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be two free-choice workflow nets. Let a ∈ UA
+be an activity label and ta ∈ T : l(ta) = a be the corresponding transition in W.
+We define the three pattern-building functions2 as
+– skip(W, a) = W ′ such that
+– (W, W ′) ∈ ψT
+– F ′ = F ∪ ({t′} × ta•) ∪ (•ta × {t′}) (where t′ ∈ T ′\T)
+– l′ = l (t′ is a silent transition)
+– loops(LW, a) is defined by two cases:
+1. if ∄t∗∈((ta•)•)(|•t∗| > 1)∧(•t∗\ta• ̸= ∅), then loops(W, a) = W ′ such that
+– (W, W ′) ∈ ψT
+– F ′ = F ∪ (ta• × {t′}) ∪ ({t′} × •ta) (where t′ ∈ T ′\T)
+– l′ = l (t′ is a silent transition)
+2. otherwise, return loops(W ′, a) such that
+– (W, W ′) ∈ ψA
+– (({ta} × (P ′\P)) ∈ F ′) ∧ (({ta} × P) /∈ F ′)
+– l′ = l
+– loopτ(W, a) is defined by two cases:
+1. if ∄t∗∈((ta•)•)(|•t∗| > 1) ∧ (•t∗\ta• ̸= ∅), then loopτ(W, a)=W ′ such that
+– (W, W ′) ∈ ψT
+– F ′ = F ∪ (ta• × {t′}) ∪ ({t′} × •ta) (where t′ ∈ T ′\T)
+– l′ = (l\{(ta, a)}) ∪ {(t′, a)} (the labels of ta and t′ are swapped)
+2. otherwise, return loopτ(W ′, a) such that
+– (W, W ′) ∈ ψA
+– (({ta} × (P ′\P)) ∈ F ′) ∧ (({ta} × P) /∈ F ′)
+– l′ = l
+The second case of the loop functions is there to keep the free-choice property.
+To illustrate the ideas using the running example, consider the net shown in
+Fig. 6a as the input net W and t3 (labeled by d) is the transition for which we
+are going to apply the functions to derive patterns. Fig. 6b shows that function
+2 The input/output nodes notations (•) used in Def. 19 refer to the input net W. We
+drop the superscript for readability.
+
+12
+T. Huang and W. M. P. van der Aalst
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+d
+𝑝5
+𝑡3
+(a) the net (W ). t3 (labeled by d)
+is the target transition.
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝4
+d
+𝑝5
+𝑡3
+𝑡4
+(b) skip(W, d) adds a silent tran-
+sition t4 that makes t3 skippable.
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝6
+d
+𝑝5
+𝑡3
+𝑝4
+𝑡4
+(c) an intermediate net W ′ (be-
+tween (a) and (d)) constructed to
+keep the free-choice property.
+g
+h
+𝑝𝑠
+𝑝𝑒
+⊤
+⊥
+𝑝1
+𝑝2
+𝑝3
+𝑡2
+𝑡1
+𝑝6
+d
+𝑝5
+𝑡3
+𝑝4
+𝑡4
+𝑡5
+(d)
+the
+resulting
+net
+of
+loops(W, d),
+which
+makes
+t3
+in a loop.
+Fig. 6: Examples showing how the functions are applied to derive patterns.
+skip(W, d) simply adds a silent transition t4 with the same connection as t3 to
+W. Fig. 6c and 6d show an application of loops(W, d) and illustrate the need
+for the two cases for the loop functions. As shown in Fig. 6c, the second case
+of loops is applied since there exists a transition t1 ∈ ((t3•)•) with more than
+one place in its preset (| • t1| > 1) and •t1\t3• ̸= ∅. Therefore, W ′ (Fig. 6c) is
+first constructed by adding p6 and t4. Then, the function returns loops(W ′, d).
+Now, the first case should be applied. In this case, t5 is added with the reverse
+connections of t3. As indicated, the second case in the loop functions helps to
+keep the free-choice property. Imagine a net that is constructed by adding t′ to
+the net in Fig. 6a with connections (p4, t′) and (t′, p5). Such a net makes t3 in a
+loop but it is no longer a free-choice net. The constructs of loops and loopτ are
+almost the same, the difference is that the labels of t3 and the silent transition
+t5 are swapped.
+Finally, to get the set of candidate nets Ci, we apply the three pattern-
+building functions to every net W ∈ Ci
+base. Observe that all the nets in Fig. 6
+are elements of C3.
+4.4
+Selection and Fall-through
+Selection In the last step, we select the next existing net W i+1 from the set
+of candidates Ci evaluated by the projected log Li. The selection is done in a
+stepwise manner. We first try to filter out the candidates that do not reach a
+user-defined replay fitness threshold θ and then select the best net out of the
+rest in terms of F1 score, which is calculated as the harmonic mean of fitness
+and precision. We use alignment-based fitness [4] and precision [5].
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+13
+Fall-through If none of the nets in Ci reach the fitness threshold θ, we adopt
+a fall-through. This is done by going back to Step 2, where γ(i) is added to
+W i = (P i, T i, F i, li, ps, pe, ⊤, ⊥), but without the constraints of V i. This can
+also be seen as setting V i = P i ∪ T i. In this case, a new place p′ with arcs
+{(⊤, p′), (p′, ⊥)} can be always added by ψP as p is linear dependent on ps and
+pe. Then, the patterns building functions can be applied to ensure that the fitness
+threshold θ is guaranteed in every iteration.
+5
+Evaluation
+In this section, we present the experiments conducted to evaluate our approach.
+The presented approach in this paper is implemented in Python using PM4Py3
+and can be accessed here4. As mentioned, the algorithm takes as inputs a log
+and three parameters including two thresholds θ, c, and the types of ordering
+strategy. Using this implementation, we conduct three experiments to address
+the following questions (1) How effective are the pre-defined patterns? (2) What
+are the effects of the ordering strategy on the model quality and the execution
+time? (3) Can the model quality be improved by the non-block structures?
+5.1
+Experiment Setup
+Dataset: We use four public available real-life event logs, which are BPI20175,
+helpdesk6, hospitalBilling7, and traffic8 respectively. BPI2017 is split into two
+sub logs, BPI2017A and BPI2017O, using the event prefixes. To focus on the
+mainstream behaviors, the logs are filtered to include at least 95% of the cases.
+Experiment 1 (Effectiveness of patterns): The first experiment aims to
+evaluate how effective are the pre-defined patterns. As our approach is based on
+[11], this can be evaluated by comparing the quality of the intermediate models
+of our approach to the ones from ProDiGy [12], which adopts a similar setting.
+To conduct the experiment, we follow the top recommendation of ProDiGy in
+every step to get the intermediate models and compare the models’ quality with
+ours. We use the projected log of every iteration to evaluate the model obtained
+after adding additional activity to the model. To have a fair comparison, we
+force our approach to use the same order of adding activities from ProDiGy.
+Experiment 2 (Effects of Ordering Strategy & Search Space Pruning):
+The order of adding activities to the log is crucial to our approach as model
+quality is highly dependent on the order [11]. Moreover, the order can influence
+the execution time due to its influence on the search space pruning. Therefore, we
+would like to investigate the effects of the ordering strategy on the model quality
+and the execution time. To set up the experiment, we apply the approach to the
+3 https://pm4py.fit.fraunhofer.de/
+4 https://github.com/tsunghao-huang/synthesisRulesMiner
+5 https://doi.org/10.4121/uuid:3926db30-f712-4394-aebc-75976070e91f
+6 https://doi.org/10.4121/uuid:0c60edf1-6f83-4e75-9367-4c63b3e9d5bb
+7 https://doi.org/10.4121/uuid:76c46b83-c930-4798-a1c9-4be94dfeb741
+8 https://doi.org/10.4121/uuid:270fd440-1057-4fb9-89a9-b699b47990f5
+
+14
+T. Huang and W. M. P. van der Aalst
+five event logs using the two different ordering strategies while keeping the other
+two parameters at the same values. We evaluate the model quality in terms of
+fitness, precision, and F1 score. In addition, we keep track of the ratio of the
+reduced nodes, which is calculated by
+|V i|
+|P i∪T i|. This gives us an indication of the
+effectiveness of search space pruning.
+Experiment 3 (Effects of non-block structures): In this experiment, we
+compare our approach to the state-of-the-art: Inductive Miner - Infrequent (IMf)
+[15]. As the models discovered by IMf are guaranteed to be sound free-choice
+workflow net as well, comparing our approach with IMf enables us to see if the
+models can benefit from the non-block structures discovered by our approach.
+For each event log, we apply IMf using five different values ([0.1, 0.2, 0.3, 0.4, 0.5])
+for the filter threshold and choose the best model (by F1 score) to compare the
+quality with the ones discovered by our approach in experiment 2.
+For all the experiments, we use the alignment-based approaches to calculate
+fitness [4] and precision [5]. We also calculate the F1 score as the harmonic mean
+of the fitness and precision.
+5.2
+Results
+Effectiveness of Patterns Fig. 7 shows the result of the comparison. The
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+Number of activities added
+0.4
+0.6
+0.8
+1.0
+Fitness
+our approach
+ProDiGy
+(a) Fitness
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+Number of activities added
+0.4
+0.6
+0.8
+1.0
+Precision
+our approach
+ProDiGy
+(b) Precision
+Fig. 7: Results on fitness and precision comparison for the effectiveness of patterns
+fitness and precision are the average values of the five event logs. As one can
+see from the figures, both approaches can capture the behaviors quite well for
+the first three activities added. When adding more activities to the model,
+our approach has consistently higher values for both fitness and precision than
+ProDiGy. One might think that this is expected as we extend the set of patterns
+used in ProDiGy. However, note that ProDiGy evaluates every possible synthesis
+rules applications while we only focus on a subset of the nodes using log heuris-
+tics. There is a trade-off between optimal solution and time in our approach.
+Nevertheless, the results show that the extended patterns enable us to discover
+models with higher quality compared to the existing approach, ProDiGy, while
+limiting the search space.
+
+Discovering Sound Free-choice Workflow Nets With Non-block Structures
+15
+Effects of Ordering Strategy and Search Space Pruning Tab. 1 shows the
+results of experiments 2 and 3. We observe that the BFS-based ordering strat-
+egy performs better than the frequency-based strategy (in terms of F1 score and
+time) for four of the five logs. We further investigate the reason for the shorter
+execution time of BFS-based ordering. As shown in Fig. 8, it turns out that the
+BFS-ordering strategy is more effective (lower
+|V i|
+|P i∪T i|) in reducing the search
+space
+at
+the
+later
+stage
+of
+the
+discovery
+process.
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+Number of activities added
+0.1
+0.3
+0.5
+0.7
+Ratio of nodes reduced 
+|Vi|
+|Pi
+Ti|
+frequency
+BFS
+Fig. 8: Comparison of the to-be-considered
+nodes ratio for each iteration between the
+two ordering strategies.
+As the model grows, checking the pre-
+conditions of an application for the
+linear dependent place or transition
+rule becomes more expensive. Reduc-
+ing the search space more effectively
+at the later stage is more beneficial in
+terms of execution time in most cases.
+BFS-based ordering achieves this by
+considering the closeness of activities
+in the process. In such case, activities
+that are closer together are added first
+and it is more likely for BFS-based
+ordering to focus on a smaller subset
+of nodes on the existing net when pruning the search space compared to the
+frequency-based one.
+Table 1: Results about effects of ordering strategy and comparison to IMf
+Log
+Miner Ordering
+Strategy
+IMf
+filter Fitness Precision
+F1
+Time (s)
+BPI2017A
+ours frequency
+-
+0.970
+0.947
+0.958
+734
+ours
+BFS
+-
+0.989
+0.935
+0.961
+342
+IMf
+-
+0.2
+0.999
+0.936
+0.967
+10
+BPI2017O
+ours frequency
+-
+0.994
+0.962
+0.978
+560
+ours
+BFS
+-
+0.989
+1.000
+0.994
+240
+IMf
+-
+0.2
+0.997
+0.907
+0.950
+7
+helpdesk
+ours frequency
+-
+0.972
+0.984
+0.977
+54
+ours
+BFS
+-
+0.981
+0.976
+0.978
+44
+IMf
+-
+0.2
+0.967
+0.950
+0.958
+1
+hospital
+billing
+ours frequency
+-
+0.961
+0.810
+0.879
+567
+ours
+BFS
+-
+0.989
+0.935
+0.961
+407
+IMf
+-
+0.2
+0.982
+0.906
+0.943
+45
+traffic
+ours frequency
+-
+0.960
+0.930
+0.945
+321
+ours
+BFS
+-
+0.964
+0.720
+0.825
+427
+IMf
+-
+0.4
+0.904
+0.720
+0.801
+28
+Effects of Non-block Structures Table 1 shows that compared to IMf, the
+models discovered by our approach have higher F1 scores for four of the five
+logs. Note that the fitness values of the models discovered by our approach are
+
+16
+T. Huang and W. M. P. van der Aalst
+all higher than the defined threshold 0.95. In general, IMf tends to discover mod-
+els with higher fitness values while our approach discovers models with higher
+precision. In IMf, one can use the filter threshold to balance fitness and precision.
+This is also the case in our approach, the user can set a lower fitness thresh-
+old to include more candidate nets that are less fitting but more precise. Fig. 9
+shows the discovered models from the two approaches for the hospitalBilling
+log. While the overall structure of Fig. 9a is similar to its counterpart in Fig. 9b,
+our approach discovered non-block structures at the later stage of the process.
+Such construct is not possible to model by IMf. The result shows that our ap-
+proach can discover sound free-choice workflow nets with non-block structures
+and produce competitive model quality as the state-of-the-art algorithm.
+(a) The discovered model using our approach. Due to the more flexible structure, one can exe-
+cute EMPTY, BILLED, or REOPEN after CODE NOK while only BILLED or REOPEN are
+executable after CODE OK. The construct is not discoverable by IMf.
+(b) The discovered model using IMf. Note that activity REOPEN is dropped by the filter of IMf.
+Fig. 9: The models discovered by our approach and IMf for the hospitalBilling log.
+6
+Conclusion and Future Work
+In this paper, we present a discovery algorithm that aims to discover sound free-
+choice workflow nets with non-block structures. The algorithm utilizes the syn-
+thesis rules to incrementally add activities with predefined patterns to discover
+models that are guaranteed to be sound and free-choice. Moreover, a certain
+level of replay fitness is guaranteed by a user-defined threshold.
+The approach has been implemented and evaluated using various real-life
+event logs. The results show that the process models discovered by our approach
+have higher model quality (in terms of both replay fitness and precision) than
+the existing approach [12], which also depends on synthesis rules. Moreover, our
+approach produces competitive model quality compared to the state-of-the-art:
+Inductive Miner - infrequent. For future work, we plan to explore more advanced
+ordering strategies and investigate their influences on the model quality and
+computation time. The other direction is to further speed up the approach as
+the long execution time is a clear limitation. This could be done by exploiting
+the log-based heuristics further.
+
+DELETE
+CHANGE
+IAGN
+CODE
+REOPEIDELETE
+CHANGE
+DIAGN
+CODE NOK
+EMPTY
+NEW
+CODE OK
+End
+FIN
+RELEASE
+BLLEDDiscovering Sound Free-choice Workflow Nets With Non-block Structures
+17
+Acknowledgements. We thank the Alexander von Humboldt (AvH) Stiftung
+for supporting our research.
+References
+1. van der Aalst, W.M.P.: The application of Petri nets to workflow management. J.
+Circuits Syst. Comput. 8(1), 21–66 (1998)
+2. van der Aalst, W.M.P.: Process Mining - Data Science in Action, Second Edition.
+Springer (2016)
+3. van der Aalst, W.M.P.: Using free-choice nets for process mining and business
+process management. In: FedCSIS 2021. vol. 25, pp. 9–15 (2021)
+4. van der Aalst, W.M.P., Adriansyah, A., van Dongen, B.F.: Replaying history on
+process models for conformance checking and performance analysis. WIREs Data
+Mining Knowl. Discov. 2(2), 182–192 (2012)
+5. Adriansyah, A., Munoz-Gama, J., Carmona, J., van Dongen, B.F., van der Aalst,
+W.M.P.: Measuring precision of modeled behavior. Inf. Syst. E Bus. Manag. 13(1),
+37–67 (2015)
+6. Augusto, A., Conforti, R., Dumas, M., Rosa, M.L., Bruno, G.: Automated discovery
+of structured process models from event logs:the discover-and-structure approach.
+Data Knowl. Eng. 117, 373–392 (2018)
+7. Augusto, A., Conforti, R., Dumas, M., Rosa, M.L., Maggi, F.M., Marrella, A.,
+Mecella, M., Soo, A.: Automated discovery of process models from event logs:
+Review and benchmark. IEEE Trans. Knowl. Data Eng. 31(4), 686–705 (2019)
+8. Augusto, A., Conforti, R., Dumas, M., Rosa, M.L., Polyvyanyy, A.: Split miner:
+automated discovery of accurate and simple business process models from event
+logs. Knowl. Inf. Syst. 59(2), 251–284 (2019)
+9. Buijs, J.C.A.M., van Dongen, B.F., van der Aalst, W.M.P.: A genetic algorithm
+for discovering process trees. In: CEC 2012. pp. 1–8. IEEE (2012)
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+(1995)
+11. Dixit, P.M.: Interactive Process Mining. Ph.D. thesis, Technische Universiteit Eind-
+hoven (2019)
+12. Dixit, P.M., Buijs, J.C.A.M., van der Aalst, W.M.P.: Prodigy : Human-in-the-loop
+process discovery. In: RCIS 2018. pp. 1–12. IEEE (2018)
+13. Dixit, P.M., Verbeek, H.M.W., Buijs, J.C.A.M., van der Aalst, W.M.P.: Interactive
+data-driven process model construction. In: ER 2018. vol. 11157, pp. 251–265.
+Springer (2018)
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+outlook of Petri net discovery algorithms. Trans. Petri Nets Other Model. Concurr.
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+
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+page_content='Discovering Sound Free-choice Workflow Nets  With Non-block Structures  Tsung-Hao Huang[0000−0002−3011−9999] and Wil M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der  Aalst[0000−0002−0955−6940]  Process and Data Science (PADS), RWTH Aachen University, Aachen, Germany  {tsunghao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='huang, wvdaalst}@pads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='de  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Process discovery aims to discover models that can explain  the behaviors of event logs extracted from information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' While  various approaches have been proposed, only a few guarantee desirable  properties such as soundness and free-choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' State-of-the-art approaches  that exploit the representational bias of process trees to provide the guar-  antees are constrained to be block-structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Such constructs limit the  expressive power of the discovered models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', only a subset of sound  free-choice workflow nets can be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' To support a more flexible  structural representation, we aim to discover process models that provide  the same guarantees but also allow for non-block structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Inspired by  existing works that utilize synthesis rules from the free-choice nets the-  ory, we propose an automatic approach that incrementally adds activities  to an existing process model with predefined patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Playing by the  rules ensures that the resulting models are always sound and free-choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Furthermore, the discovered models are not restricted to block struc-  tures and are thus more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The approach has been implemented  in Python and tested using various real-life event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The experiments  show that our approach can indeed discover models with competitive  quality and more flexible structures compared to the existing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Keywords: Process Discovery · Free-choice Net · Synthesis Rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  1  Introduction  Process discovery aims to construct process models that reflect the behaviors of  a given event log extracted from information systems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As it is a non-trivial  problem, many challenges remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In most cases, the one and only "best model"  does not exist as there are trade-offs among the four model quality metrics,  namely fitness, precision, generalization, and simplicity [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In addition to the  quality metrics, there exist properties that one would like to have for the discov-  ered models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' One of the important properties is being a sound workflow net as  soundness ensures the absence of deadlocks, proper completion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' [1] and it is  a prerequisite for many crucial automated analyses such as conformance check-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The other desirable structural property is being free-choice [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In free-choice  nets, choices and synchronizations are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This provides an easy conver-  sion between the discovered models and many process modeling languages such  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='02185v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='DB]  3 Jan 2023  2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der Aalst  as Business Process Modeling Notation (BPMN) since the equivalent constructs  (dedicated split and join connectors) are naturally embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Furthermore, free-  choice nets have been studied extensively and thus supported by an abundance  of theories [10], which provide efficient analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  While various discovery algorithms have been proposed, only a handful of  them provides such guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' State-of-the-art discovery algorithms like the In-  ductive Miner (IM) [15] are able to discover sound free-choice workflow nets by  exploiting its representational bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' However, due to the same reason, the discov-  ered models are constrained to be block-structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This limits the expressive  power of such models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', only a subset of the sound free-choice workflow nets  can be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 1a shows a sound free-choice workflow  net (with non-block structures) discovered by our approach1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The same language  can never be expressed by the model discovered by IM, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  b  c  d  f  g  a  e  h  (a) A model discovered by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The  same language cannot be expressed by the  models discovered using the Inductive Miner,  which uses process trees internally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  b  c  d  f  g  a  e  h  (b) A model discovered by the IM using the  log generated by the model in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The two  branches before c need to be synchronized first  before d can be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 1: Examples showing the need for the non-block process models discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Note  that the trace ⟨a, b, c, d, e, f, g, h⟩ that is possible in (a) cannot be replayed by (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  In this paper, we aim to discover sound free-choice workflow nets with non-  block structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Inspired by the interactive process discovery approach in [11,12],  we develop an automatic process discovery algorithm that incrementally adds  activities to an existing net using synthesis rules [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Since checking the feasibil-  ity for the application of the synthesis rules is computationally expensive, we use  log heuristics to locate the most possible position for the to-be-added activity  on the existing process model instead of evaluating all possible applications of  synthesis rules as in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Additionally, we identify the need for an additional  rule and extend the set of patterns introduced in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Playing by the rules ensures that the discovered process models by our ap-  proach are guaranteed to be sound free-choice workflow nets [10,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Moreover,  the discovered models are not constrained to block structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Last but not least,  the level of replay fitness is guaranteed via a threshold set by the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The  approach has been implemented in Python and evaluated using various public-  available real-life event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The evaluation shows that our approach is able to  discover non-block structured models with competitive qualities compared to the  state-of-the-art discovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  1 The proposed approach has dedicated silent transitions for start and end as defined  later in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We dropped them here for ease of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Discovering Sound Free-choice Workflow Nets With Non-block Structures  3  The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We review the related  work in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 2 and introduce necessary concepts in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 4 introduces the  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 5 presents the experiment and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  2  Related Work  An overview of process discovery is out of the scope of this paper, we refer to  [7,14] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In this section, we focus on process discovery techniques  that guarantee soundness (and free-choice) properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Approaches like [6,8] can  discover non-block structured models but cannot guarantee both properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  While Split Miner discovers models that are deadlock-free, they are not nec-  essarily sound [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The family of Inductive Miner (IM) algorithms [15] guarantees sound and  free-choice of the discovered models by exploiting the representational bias of  the process tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' By design, a process tree represents a sound workflow net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' It is  a rooted tree where the leaf nodes are activities and the non-leaf nodes are the  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The hierarchical representation has a straightforward translation to  Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' However, the resulting models are limited to being block-structured as  a process tree can only represent process models that can be separated into parts  that have a single entry and exit [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Consequently, process trees can only rep-  resent a subset of sound workflow nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The same arguments hold for approaches  that are based on process trees such as the Evolutionary Tree Miner (ETM) [9]  and the recently developed incremental process discovery approach [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Applying the synthesis rules [10], the interactive process discovery approaches  developed in [12,13,11] ensure soundness and free-choice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A semi-  automatic interactive tool, ProDiGy, is proposed in [12] to recommend the best  possible ways to add an activity to an existing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Our approach differs from [12,13,11] in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' First, we adopt an  automatic setting as the order of adding activities is predetermined and the best  modification to the existing net is selected based on the model quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Second,  we use log heuristics to locate the most suitable position for adding the new  activity instead of evaluating all the possibilities of synthesis rules applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Moreover, we identify the need for a new rule as the desired models often cannot  be discovered without going back and forth by a combination of reduction and  synthesis rules [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Lastly, the set of patterns is extended and formally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  3  Preliminaries  We denote the set of all sequences over some set A as A∗, the power set of  A as P(A), and the set of all multisets over A as B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' For some multiset  b ∈ B(A), b(a) denotes the number of times a ∈ A appears in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' For a given  sequence σ = ⟨a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', an⟩ ∈ A∗, |σ| = n is the length of σ and dom(σ) =  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', |σ|} is the domain of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' ⟨⟩ is the empty sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' σ(i) = ai denotes the  i-th element of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Given sequences σ1 and σ2, σ1 · σ2 denotes the concatenation  of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let A be a set and X ⊆ A be a subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' For σ ∈ A∗ and a ∈ A,  4  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der Aalst  we define ↾X∈ A∗→X∗ as a projection function recursively with ⟨⟩↾X = ⟨⟩,  (⟨a⟩ · σ)↾X = ⟨a⟩ · σ↾X if a ∈ X and (⟨a⟩ · σ)↾X = σ↾X if a /∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' For example,  ⟨x, y, x⟩↾{x,z} = ⟨x, x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Projection can also be applied to multisets of sequences,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', [⟨a, b, a⟩6, ⟨a, b, c⟩6, ⟨b, a, c⟩2]↾{b,c} = [⟨b⟩6, ⟨b, c⟩8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 1 (Trace, Log).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A trace σ ∈ U∗  A is a sequence of activities, where  UA is the universe of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A log L ∈ B(U∗  A) is a multiset of traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 2 (Log Properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let L ∈ B(U∗  A) and a, b ∈ UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – #(a, L) = Σσ∈L|{i ∈ dom(σ)|σ(i) = a}| is the times a occurred in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – #(a, b, L) = Σσ∈L|{i ∈ dom(σ)\\{|σ|}|σ(i) = a∧σ(i+1) = b}| is the number  of direct successions from a to b in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – caus(a, b, L) =  �  #(a,b,L)−#(b,a,L)  #(a,b,L)+#(b,a,L)+1  if a ̸= b  #(a,b,L)  #(a,b,L)+1  if a = b is the strength of causal rela-  tion (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – Apre  c  (a, L) = {apre ∈ UA|caus(apre, a, L) ≥ c} is the set of a’s preceding  activities, determined by threshold c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – Afol  c  (a, L) = {afol ∈ UA|caus(a, afol, L) ≥ c} is the set of a’s following  activities, determined by threshold c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – As(L) = {σ(1) | σ ∈ L ∧ σ ̸= ⟨⟩} is the set of start activities in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – Ae(L) = {σ(|σ|) | σ ∈ L ∧ σ ̸= ⟨⟩} is the set of end activities in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 3 (Petri Net, Labeled Petri Net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A Petri net N = (P, T, F) is  a tuple, where P is the set of places, T is the set of transitions, P ∩ T = ∅, and  F ⊆ (P × T) ∪ (T × P) is the set of arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A labeled Petri net N = (P, T, F, l) is  a Petri net (P, T, F) with a labeling function l ∈ T ↛ UA that maps a subset of  transitions to activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A t ∈ T is called invisible if t is not in the domain of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  For any x ∈ P ∪ T,  N•x = {y|(y, x) ∈ F} denotes the set of input nodes and  x  N• = {y|(x, y) ∈ F} denotes the set of output nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The superscript N is  dropped if it is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The notation can be generalized to set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  For any X ⊆ P ∪ T, •X = {y|∃x∈X(y, x) ∈ F} and X• = {y|∃x∈X(x, y) ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 4 (Free-choice Net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let N = (P, T, F) be a Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' N is a  free-choice net if for any t1, t2 ∈ T : •t1 = •t2 or •t1 ∩ •t2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 5 (Workflow Net (WF-net) [1,11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let N = (P, T, F, l) be a  labeled Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' W = (P, T, F, l, ps, pe, ⊤, ⊥) is a WF-net iff (1) it has a dedi-  cated source place ps ∈ P: •ps = ∅ and a dedicated sink place pe ∈ P: pe• = ∅  (2) ⊤ ∈ T: •⊤ = {ps}∧ps• = {⊤} and ⊥ ∈ T: ⊥• = {pe}∧•pe = {⊥} (3) every  node x is on some path from ps to pe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', ∀x∈P ∪T (ps, x) ∈ F ∗ ∧ (x, pe) ∈ F ∗,  where F ∗ is the reflexive transitive closure of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 6 (Short-circuited WF-net [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let W = (P, T, F, l, ps, pe, ⊤, ⊥)  be a WF-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The short-circuited WF-net of W, denoted by SC(W), is con-  structed by SC(W)=(P, T ∪{t′}, F ∪{(⊥, t′), (t′, ⊤)}, l, ps, pe, ⊤, ⊥), where t′ /∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Discovering Sound Free-choice Workflow Nets With Non-block Structures  5  Definition 7 (Paths, Elementary Paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A path of a Petri net N = (P, T, F)  is a non-empty sequence of nodes ρ = ⟨x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', xn⟩ such that (xi, xi+1) ∈ F for  1 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' ρ is an elementary path if xi ̸= xj for 1 ≤ i < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 8 (Incidence Matrix [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let N = (P, T, F) be a Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The  incidence matrix N : (P × T) → {−1, 0, 1} of N is defined as  N(p, t) =  �  �  �  �  �  0  if ((p, t) /∈ F ∧ (t, p) /∈ F) ∨ ((p, t) ∈ F ∧ (t, p) ∈ F)  −1  if (p, t) ∈ F ∧ (t, p) /∈ F  1  if (p, t) /∈ F ∧ (t, p) ∈ F  For a Petri net N = (P, T, F) and its corresponding incidence matrix N, we use  N(p) to denote the row vector of the corresponding p ∈ P and N(t) to denote  the column vector of the corresponding t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 9 (Linearly Dependent Nodes [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let N = (P, T, F) be a  Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Q is the set of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A place p is linearly dependent if  there exists a row vector ⃗v : P → Q such that ⃗v(p) = 0 and ⃗v · N = N(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A  transition t is linearly dependent if there exists a column vector ⃗v : T → Q such  that ⃗v(t) = 0 and ⃗v · N = N(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 10 (Synthesis Rules [10,11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let W and W ′ be two free-choice  workflow nets, and let SC(W) = (P, T, F, l, ps, pe, ⊤, ⊥) and SC(W ′) = (P ′, T ′,  F ′, l′, ps, pe, ⊤, ⊥) be the corresponding short-circuited WF-nets:  – Linear Dependent Place Rule ψP : W ′ is derived from W using ψP , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=',  (W, W ′) ∈ ψP if (1) T ′ = T, P ′ = P ∪ {p} and p /∈ P is linear dependent in  SC(W ′), F ′ = F ∪ �F where �F ⊆ (({p} × T) ∪ (T × {p})) (2) Every siphon  in SC(W ′) contains ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – Linear Dependent Transition Rule ψT : W ′ is derived from W using ψT , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=',  (W, W ′) ∈ ψT if P ′ = P, T ′ = T ∪ {t} and t /∈ T is linear dependent in  SC(W ′) and F ′ = F ∪ �F where �F ⊆ ((P ×{t})∪({t}×P)), and ∀t∈T ∩T ′l(t) =  l′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  – Abstraction Rule ψA: (W, W ′) ∈ ψA if (1) there exists a set of transitions  R ⊆ T and a set of places S ⊆ P such that (R × S ⊆ F) ∧ (R × S ̸= ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (2)  SC(W ′) is constructed by adding an additional place p /∈ P and a transition  t /∈ T such that P ′ = P ∪ {p}, T ′ = T ∪ {t}, F ′ = (F\\(R × S)) ∪ ((R × {p}) ∪  ({p} × {t}) ∪ ({t} × S)), and ∀t∈T ∩T ′l(t) = l′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Applying the three synthesis rules (ψP , ψT , ψA) to derive W ′ from a sound  free-choice workflow net W ensures that W ′ is also sound [13,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Three proper-  ties need to be hold for a WF-net to be sound (1) safeness: places cannot hold  multiple tokens at the same time (2) option to complete: it is always possible to  reach the marking in which only the sink place is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (3) no dead transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Next, we introduce the initial net [11] and show some examples of synthesis rules  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  6  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der Aalst  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  (a)  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑡1  𝑝2  (b)  g  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑝2  𝑝3  𝑡2  𝑡1  (c)  g  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑝2  𝑝3  𝑡2  𝑡1  𝑝4  (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 2: Examples of synthesis rules applications starting from (a) The initial net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (b)  Using ψA, p2 and t1 are added to the initial net with R = {⊤} and S = {p1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (c) Using  ψA, p3 and t2 are added to previous net with R = {⊤} and S = {p2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (d) p4 is added  using ψp as p4 is a linear combination of p3 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 11 (Initial Net [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let W = (P, T, F, l, ps, pe, ⊤, ⊥) be a free-  choice WF-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' W is an initial net if P = {ps, p1, pe}, T = {⊤, ⊥}, F =  {(ps, ⊤), (⊤, p1), (p1, ⊥), (⊥, pe)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The initial net is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Clearly, it is a sound free-choice workflow  net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Starting from the initial net, one can incrementally add additional nodes  according to the synthesis rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 2 shows example applications of synthesis  rules starting from the initial net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  4  Approach  With the necessary concepts introduced, we are now ready to introduce the ap-  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We start by showing the basic idea of the approach with the help of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 3 before diving into each step in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Internally, the approach incremen-  tally adds a new activity to an existing net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The figure shows a single iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  In each iteration, we have an existing model from the previous iteration and a  log projected on the already added activities so far and the to-be-added one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  We start by locating the most likely position to add the new activity deter-  mined by log heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The result of this step is a subset of nodes of the existing  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The set of nodes will then be used to prune the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Then, the  predefined patterns are applied to the existing net to get a set of candidate nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Lastly, we select the best net (next existing net) out of the candidates in terms of  fitness and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Note that the existing net in the first iteration is initiated  by the initial net (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As a running example, consider the correspond-  ing log that is used to discover the Petri net in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='1a by our approach: Ls =  [⟨a, b, c, d, f, g, h⟩22, ⟨a, b, c, f, d, g, h⟩14, ⟨a, e, b, c, d, f, g, h⟩13, ⟨a, e, b, c, f, d, g, h⟩13,  ⟨a, e, b, c, f, g, d, h⟩10, ⟨a, b, c, f, g, d, h⟩10, ⟨a, b, e, c, d, f, g, h⟩6, ⟨a, b, e, c, f, g, d, h⟩3,  ⟨a, b, e, c, f, d, g, h⟩3, ⟨a, b, c, d, e, f, g, h⟩2, ⟨a, b, c, e, d, f, g, h⟩2, ⟨a, b, c, e, f, g, d, h⟩1,  ⟨a, b, c, e, f, d, g, h⟩1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The instances provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 3 shows the 3rd iteration for  the running example Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In the following subsections, we introduce the details  of each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Discovering Sound Free-choice Workflow Nets With Non-block Structures  7  (1) Pruning search  space using log  heuristics  (2) Add new activity to  the existing net with  pre-defined patterns  [ 𝑑, 𝑔, ℎ 76, 𝑔, 𝑑, ℎ 24]  Projected Log 𝐿𝑖  Existing Net  𝑊𝑖 = (𝑃𝑖, 𝑇𝑖, 𝐹𝑖, 𝑙𝑖, 𝑝𝑠, 𝑝𝑒, ⊤, ⊥)  (3) Select the best net  for the next iteration  Next Existing Net  𝑊𝑖+1 = (𝑃𝑖+1, 𝑇𝑖+1, 𝐹𝑖+1, 𝑙𝑖+1, 𝑝𝑠, 𝑝𝑒, ⊤, ⊥)  To-be-added Activity 𝛾(𝑖)  𝑉𝑖 ⊆ 𝑃𝑖 ∪ 𝑇𝑖  Set of Candidate Nets 𝐶𝑖  skip  loop  …  …  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 3: An example of a single iteration of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='1  Ordering Strategies for Adding Activities  Before starting any iteration, we need to come up with an order for adding ac-  tivities based on a given log L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' It is important as the quality of the discovered  models often depends on the order of adding activities [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Moreover, in combi-  nation with the search space pruning, it can influence the computation time for  each iteration significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In this paper, we introduce two ordering strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The first one is relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The activities in L are simply ordered  by their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 12 (Activities-Adding Order, Frequency-Based Ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Let L ∈ B(U∗  A) and A = �  σ∈L{a ∈ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' γ ∈ A∗ is an activities-adding order for  L if {a ∈ γ} = A and |γ| = |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The frequency-based ordering is orderfreq(L) = γ  such that γ is an activities-adding order and ∀1≤i<j≤|γ|#(γ(i), L) ≥ #(γ(j), L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The second ordering strategy is similar to the Breadth-first Search (BFS) al-  gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The advantage of this is that it also considers the closeness between  activities in the log, rather than just frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' To explain this ordering strategy,  we first define a sub-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 13 (Directly-Precedes Activities Sorting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let L ∈ B(U∗  A) and  a ∈ UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A={b ∈ UA|#(b, a, L)>0} is the set of activities directly-precede a in L at  least once and σ ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Directly-precedes activities sorting is sortPreceded(a, L)=σ  such that {b ∈ σ} = A and |σ| = |A| and ∀1≤i<j≤|σ| #(σ(i), a, L) ≥ #(σ(j), a, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The function sortPreceded takes an activity a and a log L to return a sequence of  a’s directly-preceded activities b that are sorted by the frequency of #(b, a, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Finally, we can define the BFS-based ordering strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 14 (Breadth-First-Search-Based Ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let L ∈ B(U∗  A)  and A= �  σ∈L{a ∈ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' BFS-based ordering is defined as orderBFS(L)=γ, where  γ is an activities adding order for L and γ=γ1 · γ2 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' · γ|γ|, for each 1 ≤ j ≤ |γ|,  γj =  �  orderfreq(L↾Ae(L))  if j = 1  sortPreceded(γ(j − 1), L)↾A\\{γ(1),γ(2),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=',γ(j−1)}  otherwise  d  h  gdps  4  0  d  h  gd  hh  g8  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der Aalst  The function starts by sorting the end activities Ae(L) according to their fre-  quency in the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Then, it enumerates through the sequence γ and sorts the  preceded activities of γ(j − 1) by the frequency of direct successions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The pro-  jection function in the second case of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='14 filters out the activities that are  already in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Compared to the frequency-based ordering, the BFS-based ordering considers  the closeness of the activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This allows us to add activities that are close  together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Together with the effect of the search space pruning, it is expected that  BFS-based ordering would have less computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Applying the function  orderBFS to our running example, Ls, we get the activities adding order as  γ = orderBFS(Ls) = ⟨h⟩ · ⟨g, d⟩ · ⟨f⟩ · ⟨c, e⟩ · ⟨⟩ · ⟨b⟩ · ⟨a⟩ · ⟨⟩ = ⟨h, g, d, f, c, e, b, a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  γ is then used to determine the order of adding activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Given the activities  adding order γ, we define the artifacts for each iteration i as followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 15 (Projected Log).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let L ∈ B(U∗  A) and γ be a activities adding  order for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The projected log for L in the i-th iteration is Li = L↾{γ(1),γ(2),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='γ(i)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  For instance, the projected log for the running example Ls for the 3rd iteration  is then L3  s = Ls↾{h,g,d} = [⟨d, g, h⟩76, ⟨g, d, h⟩24], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The to-be-  added activity is denoted as γ(i), which is γ(3) = d for the 3rd iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Also,  we denote the existing sound free-choice workflow net for iteration i as W i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Note  that for the running example, W 1, W 2, and W 3 are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 2a, 2b,  and 2c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='2  Search Space Pruning  As checking the feasibility of applying linear dependent rules ψT , ψP is compu-  tationally expensive [11], it is impractical to compute all possible applications  of the synthesis rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Also, some of them are not of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' For example, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 3, it is clear that the to-be-added activity d never happens after  h in the projected log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Using such information, we can already eliminate the  constructs (applications of synthesis rules) that allow activity d to be executed  after h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Therefore, in each iteration i we start by locating the most likely posi-  tion to add γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This helps us to restrict the application of synthesis rules on  only a subset of nodes, denoted as V i ⊆ P i ∪ T i, in the existing net W i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' To do  that, we first identify the set of preceding and following activities of γ(i) in the  projected log Li, which would be Apre  c  (γ(i), Li) and Afol  c  (γ(i), Li) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Recall that c is a threshold for the causal relation and can be given by users  as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We use c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='9 as the default value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Then, the corresponding la-  belled transitions are identified in W i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Finally, V i is the set of all the nodes on  the elementary paths from the preceding transitions to the following transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  If Apre  c  (γ(i), L)/Afol  c  (γ(i), L) is an empty set, we use the ⊤/⊥ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' For  instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 3, we identify that Apre  c  (d, L3  s)=∅ and Afol  c  (d, L3  s)={h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' There-  fore, we find all the nodes on the elementary paths between every node in {⊤}  and every node in {t1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As a result, the set V 3 = {⊤, p3, t2, p2, t1} is used to  prune the search space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', to constrain the application of synthesis rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Discovering Sound Free-choice Workflow Nets With Non-block Structures  9  Constraining synthesis rules For the abstraction rule ψA, this means that the  set of transitions R and the set of places S used as the preconditions for applying  ψA need to be a subset of V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', S ⊆ V ∧R ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' For the linear dependent rules  ψP /ψT , the new place/transition (p′/t′) cannot have arcs connected to any node  outside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This shortens the computation time as certain rules applications can  be removed and there is no need to check their feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='3  Patterns  In this section, we introduce the patterns that are used to add activity γ(i) to  the existing free-choice workflow net W i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' First, we motivate the need for an  additional rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The need for an additional rule It is proven that any sound free-choice  workflow net can be constructed by the three synthesis rules ψA, ψP , ψT [10,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  However, when applying to discover process models, the desirable model is often  not possible to derive due to the existing construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' An example is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' While it is possible to add a transition labeled by a in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 4a, it is not  possible to derive the same net in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 4b as there is no rule allowing such trans-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' One possible workaround is to go back and forth by a combination  of reduction and synthesis rules as suggested in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' However, once the existing  net becomes more complex, such a solution becomes infeasible to track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  b  c  L↾〈c,b〉 = [〈c〉3, 〈b〉3]  b  c  a  L↾〈a,b,c〉 = [〈c,a〉3, 〈b〉3]  (a)  L↾〈a,b〉 = [〈a〉3, 〈b〉3]  L↾〈a,b,c〉 = [〈c,a〉3, 〈b〉3]  b  c  a  b  a  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 4: Examples showing the motivation for the dual abstraction rule ψD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Although  the desirable nets on the right-hand side of (a) and (b) are semantically the same, the  existing synthesis rules only allow the transformation in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' There is no rule defined  for the transformation in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  We observe that in many situations (including the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 4b), the  desired models cannot be constructed because there is no rule allowing one to  introduce a new transition t and a new place p in between a set of places S and  a set of transitions R that are fully connected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', S × R ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Therefore, we  define the dual abstraction rule to allow such construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 16 (Dual Abstraction Rule ψD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let W = (P, T, F, l, ps, pe, ⊤, ⊥)  and W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be two free-choice workflow nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (W, W ′) ∈  ψD if (1) there exists a set of places S ⊆ P and a set of transitions R ⊆ T such  that S × R ⊆ F ∧ S × R ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (2) W ′ is constructed by adding an additional  10  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der Aalst  transition t /∈ T and a place p /∈ P such that P ′ = P ∪ {p}, T ′ = T ∪ {t}, F ′ =  (F \\ (S × R)) ∪ ((S × {t}) ∪ ({t} × {p}) ∪ ({p} × R)), and ∀t∈T ∩T ′l(t) = l′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  As we are only interested in sound free-choice workflow nets, we need to make  sure that the dual abstraction rule ψD preserves soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Proposition 1 (ψD preserves soundness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let W = (P, T, F, l, ps, pe, ⊤, ⊥),  W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be free-choice workflow nets, and W ′ is derived  from W using ψD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', (W, W ′) ∈ ψD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Then W ′ is sound if W is sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let t′ ∈ T ′\\T and p′ ∈ P ′\\P be the new transition and place in W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let  R = p′• and S = •t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The new net W ′ is free-choice in only two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Either  S =  W• R or R = S  W• .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In either case, any reachable marking in (W, [ps]) that  does not need to fire tR ∈ R is still reachable in (W ′, [ps]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Also, the reachable  markings in (W, [ps]) that need to fire tR ∈ R can be reached in (W ′, [ps]) as one  can just add t′ somewhere before tR in the corresponding firing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Then,  it is trivial to see that W ′ fulfils the three conditions of soundness if W is also  sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  □  Next, we extent the linear dependent place rule ψP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As we aim to add a transition  labeled by γ(i) to the existing labeled free-choice workflow net W i, only adding  a place p′ by ψP does not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Hence, in our approach, an application of ψP  is always coupled with a directly followed application of abstraction rule ψA to  include a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' ψA is applied between the added place p′ and its preset •p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  This is possible because every transition in •p′ is connected to every place in  {p′} by definition, which satisfies the precondition of ψA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' An example is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 5, p5 and t3 are added by ψA directly after the addition of p4 by ψP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' To  be more precise, we define the extended rule, ψ′  P , that describes the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 17 (Extended Linear Dependent Place Rule ψ′  P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let W=(P,  T, F, l, ps, pe, ⊤, ⊥) and W ′′=(P ′′, T ′′, F ′′, l′′, ps, pe, ⊤, ⊥) be free-choice workflow  nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (W, W ′′)∈ψ′  P if (1) ∃W ′=(P ′,T ′,F ′,l′,ps,pe,⊤,⊥)(W, W ′)∈ψP ∧ (W ′, W ′′)∈ψA  and (2) ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='p∗∈P ′′({p∗}=P ′\\P) ∧ (((T ′′\\T ′) × {p∗}) ⊂ F ′′) ∧ ((T ′ × {p∗}) ̸⊂ F ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  g  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑝2  𝑝3  𝑡2  𝑡1  𝑝4  (a)  g  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑝2  𝑝3  𝑡2  𝑡1  𝑝4  d  𝑝5  𝑡3  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 5: (a) ψP adds a place p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (b) As every transition in •p4 has an arc to every place  in {p4}, one can directly apply ψA to add p5 and t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Then, we define the set of nets constructed by every possible single application  of the rules ψA, ψ′  P , ψT , ψD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Discovering Sound Free-choice Workflow Nets With Non-block Structures  11  Definition 18 (Base Candidates Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let W=(P, T, F, l, ps, pe, ⊤, ⊥), W ′ =  (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be free-choice workflow nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let X=(P ′∪T ′)\\(P∪T),  V ⊆P∪T, V ′=(P∪T)\\V , and let a ∈ UA be an activity label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The base candidates  set is base(W, V, a)={W ′|((W, W ′) ∈ (ψA ∪ψT ∪ψ′  P ∪ψD))∧(∄x∈X(({x}×V ′)∪  (V ′ × {x})) ⊆ F ′) ∧ (l′=l ∪ ((T ′ \\ T) × {a}))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The base candidates set Ci  base=base(W i, V i, γ(i)) consists of the nets that are  constructed by every possible single application of the rules ψA, ψ′  P , ψT , ψD to  add a transition labeled by γ(i) to W i considering the the constraints on V i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Next, we introduce three patterns that make a transition skippable, in a strict  loop, or in an optional (tau) loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' A transition in a strict loop means that the  execution of the transition is required, otherwise it is an optional loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Definition 19 (Pattern-Building Functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let W=(P, T, F, l, ps, pe, ⊤, ⊥)  and W ′ = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be two free-choice workflow nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Let a ∈ UA  be an activity label and ta ∈ T : l(ta) = a be the corresponding transition in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  We define the three pattern-building functions2 as  – skip(W, a) = W ′ such that  – (W, W ′) ∈ ψT  – F ′ = F ∪ ({t′} × ta•) ∪ (•ta × {t′}) (where t′ ∈ T ′\\T)  – l′ = l (t′ is a silent transition)  – loops(LW, a) is defined by two cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' if ∄t∗∈((ta•)•)(|•t∗| > 1)∧(•t∗\\ta• ̸= ∅), then loops(W, a) = W ′ such that  – (W, W ′) ∈ ψT  – F ′ = F ∪ (ta• × {t′}) ∪ ({t′} × •ta) (where t′ ∈ T ′\\T)  – l′ = l (t′ is a silent transition)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' otherwise, return loops(W ′, a) such that  – (W, W ′) ∈ ψA  – (({ta} × (P ′\\P)) ∈ F ′) ∧ (({ta} × P) /∈ F ′)  – l′ = l  – loopτ(W, a) is defined by two cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' if ∄t∗∈((ta•)•)(|•t∗| > 1) ∧ (•t∗\\ta• ̸= ∅), then loopτ(W, a)=W ′ such that  – (W, W ′) ∈ ψT  – F ′ = F ∪ (ta• × {t′}) ∪ ({t′} × •ta) (where t′ ∈ T ′\\T)  – l′ = (l\\{(ta, a)}) ∪ {(t′, a)} (the labels of ta and t′ are swapped)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' otherwise, return loopτ(W ′, a) such that  – (W, W ′) ∈ ψA  – (({ta} × (P ′\\P)) ∈ F ′) ∧ (({ta} × P) /∈ F ′)  – l′ = l  The second case of the loop functions is there to keep the free-choice property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  To illustrate the ideas using the running example, consider the net shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6a as the input net W and t3 (labeled by d) is the transition for which we  are going to apply the functions to derive patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6b shows that function  2 The input/output nodes notations (•) used in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 19 refer to the input net W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We  drop the superscript for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  12  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der Aalst  g  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑝2  𝑝3  𝑡2  𝑡1  𝑝4  d  𝑝5  𝑡3  (a) the net (W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' t3 (labeled by d)  is the target transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  g  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑝2  𝑝3  𝑡2  𝑡1  𝑝4  d  𝑝5  𝑡3  𝑡4  (b) skip(W, d) adds a silent tran-  sition t4 that makes t3 skippable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  g  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑝2  𝑝3  𝑡2  𝑡1  𝑝6  d  𝑝5  𝑡3  𝑝4  𝑡4  (c) an intermediate net W ′ (be-  tween (a) and (d)) constructed to  keep the free-choice property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  g  h  𝑝𝑠  𝑝𝑒  ⊤  ⊥  𝑝1  𝑝2  𝑝3  𝑡2  𝑡1  𝑝6  d  𝑝5  𝑡3  𝑝4  𝑡4  𝑡5  (d)  the  resulting  net  of  loops(W, d),  which  makes  t3  in a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6: Examples showing how the functions are applied to derive patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  skip(W, d) simply adds a silent transition t4 with the same connection as t3 to  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6c and 6d show an application of loops(W, d) and illustrate the need  for the two cases for the loop functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6c, the second case  of loops is applied since there exists a transition t1 ∈ ((t3•)•) with more than  one place in its preset (| • t1| > 1) and •t1\\t3• ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Therefore, W ′ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6c) is  first constructed by adding p6 and t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Then, the function returns loops(W ′, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Now, the first case should be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In this case, t5 is added with the reverse  connections of t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As indicated, the second case in the loop functions helps to  keep the free-choice property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Imagine a net that is constructed by adding t′ to  the net in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6a with connections (p4, t′) and (t′, p5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Such a net makes t3 in a  loop but it is no longer a free-choice net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The constructs of loops and loopτ are  almost the same, the difference is that the labels of t3 and the silent transition  t5 are swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Finally, to get the set of candidate nets Ci, we apply the three pattern-  building functions to every net W ∈ Ci  base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Observe that all the nets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 6  are elements of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='4  Selection and Fall-through  Selection In the last step, we select the next existing net W i+1 from the set  of candidates Ci evaluated by the projected log Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The selection is done in a  stepwise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We first try to filter out the candidates that do not reach a  user-defined replay fitness threshold θ and then select the best net out of the  rest in terms of F1 score, which is calculated as the harmonic mean of fitness  and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We use alignment-based fitness [4] and precision [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Discovering Sound Free-choice Workflow Nets With Non-block Structures  13  Fall-through If none of the nets in Ci reach the fitness threshold θ, we adopt  a fall-through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This is done by going back to Step 2, where γ(i) is added to  W i = (P i, T i, F i, li, ps, pe, ⊤, ⊥), but without the constraints of V i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This can  also be seen as setting V i = P i ∪ T i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In this case, a new place p′ with arcs  {(⊤, p′), (p′, ⊥)} can be always added by ψP as p is linear dependent on ps and  pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Then, the patterns building functions can be applied to ensure that the fitness  threshold θ is guaranteed in every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  5  Evaluation  In this section, we present the experiments conducted to evaluate our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The presented approach in this paper is implemented in Python using PM4Py3  and can be accessed here4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As mentioned, the algorithm takes as inputs a log  and three parameters including two thresholds θ, c, and the types of ordering  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Using this implementation, we conduct three experiments to address  the following questions (1) How effective are the pre-defined patterns?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (2) What  are the effects of the ordering strategy on the model quality and the execution  time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' (3) Can the model quality be improved by the non-block structures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='1  Experiment Setup  Dataset: We use four public available real-life event logs, which are BPI20175,  helpdesk6, hospitalBilling7, and traffic8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' BPI2017 is split into two  sub logs, BPI2017A and BPI2017O, using the event prefixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' To focus on the  mainstream behaviors, the logs are filtered to include at least 95% of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Experiment 1 (Effectiveness of patterns): The first experiment aims to  evaluate how effective are the pre-defined patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As our approach is based on  [11], this can be evaluated by comparing the quality of the intermediate models  of our approach to the ones from ProDiGy [12], which adopts a similar setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  To conduct the experiment, we follow the top recommendation of ProDiGy in  every step to get the intermediate models and compare the models’ quality with  ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We use the projected log of every iteration to evaluate the model obtained  after adding additional activity to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' To have a fair comparison, we  force our approach to use the same order of adding activities from ProDiGy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Experiment 2 (Effects of Ordering Strategy & Search Space Pruning):  The order of adding activities to the log is crucial to our approach as model  quality is highly dependent on the order [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Moreover, the order can influence  the execution time due to its influence on the search space pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Therefore, we  would like to investigate the effects of the ordering strategy on the model quality  and the execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' To set up the experiment, we apply the approach to the  3 https://pm4py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='fraunhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='de/  4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='com/tsunghao-huang/synthesisRulesMiner  5 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='4121/uuid:3926db30-f712-4394-aebc-75976070e91f  6 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='4121/uuid:0c60edf1-6f83-4e75-9367-4c63b3e9d5bb  7 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='4121/uuid:76c46b83-c930-4798-a1c9-4be94dfeb741  8 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='4121/uuid:270fd440-1057-4fb9-89a9-b699b47990f5  14  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der Aalst  five event logs using the two different ordering strategies while keeping the other  two parameters at the same values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We evaluate the model quality in terms of  fitness, precision, and F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In addition, we keep track of the ratio of the  reduced nodes, which is calculated by  |V i|  |P i∪T i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This gives us an indication of the  effectiveness of search space pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Experiment 3 (Effects of non-block structures): In this experiment, we  compare our approach to the state-of-the-art: Inductive Miner - Infrequent (IMf)  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As the models discovered by IMf are guaranteed to be sound free-choice  workflow net as well, comparing our approach with IMf enables us to see if the  models can benefit from the non-block structures discovered by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  For each event log, we apply IMf using five different values ([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='5])  for the filter threshold and choose the best model (by F1 score) to compare the  quality with the ones discovered by our approach in experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  For all the experiments, we use the alignment-based approaches to calculate  fitness [4] and precision [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We also calculate the F1 score as the harmonic mean  of the fitness and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='2  Results  Effectiveness of Patterns Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 7 shows the result of the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The  1  2  3  4  5  6  7  8  9  10  11  Number of activities added  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='0  Fitness  our approach  ProDiGy  (a) Fitness  1  2  3  4  5  6  7  8  9  10  11  Number of activities added  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='0  Precision  our approach  ProDiGy  (b) Precision  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 7: Results on fitness and precision comparison for the effectiveness of patterns  fitness and precision are the average values of the five event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As one can  see from the figures, both approaches can capture the behaviors quite well for  the first three activities added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' When adding more activities to the model,  our approach has consistently higher values for both fitness and precision than  ProDiGy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' One might think that this is expected as we extend the set of patterns  used in ProDiGy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' However, note that ProDiGy evaluates every possible synthesis  rules applications while we only focus on a subset of the nodes using log heuris-  tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' There is a trade-off between optimal solution and time in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Nevertheless, the results show that the extended patterns enable us to discover  models with higher quality compared to the existing approach, ProDiGy, while  limiting the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Discovering Sound Free-choice Workflow Nets With Non-block Structures  15  Effects of Ordering Strategy and Search Space Pruning Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 1 shows the  results of experiments 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We observe that the BFS-based ordering strat-  egy performs better than the frequency-based strategy (in terms of F1 score and  time) for four of the five logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We further investigate the reason for the shorter  execution time of BFS-based ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 8, it turns out that the  BFS-ordering strategy is more effective (lower  |V i|  |P i∪T i|) in reducing the search  space  at  the  later  stage  of  the  discovery  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  2  3  4  5  6  7  8  9  10  11  Number of activities added  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='7  Ratio of nodes reduced  |Vi|  |Pi  Ti|  frequency  BFS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 8: Comparison of the to-be-considered  nodes ratio for each iteration between the  two ordering strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  As the model grows, checking the pre-  conditions of an application for the  linear dependent place or transition  rule becomes more expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Reduc-  ing the search space more effectively  at the later stage is more beneficial in  terms of execution time in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  BFS-based ordering achieves this by  considering the closeness of activities  in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In such case, activities  that are closer together are added first  and it is more likely for BFS-based  ordering to focus on a smaller subset  of nodes on the existing net when pruning the search space compared to the  frequency-based one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Table 1: Results about effects of ordering strategy and comparison to IMf  Log  Miner Ordering  Strategy  IMf  filter Fitness Precision  F1  Time (s)  BPI2017A  ours frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='970  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='947  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='958  734  ours  BFS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='989  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='935  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='961  342  IMf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content='967  10  BPI2017O  ours frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content='978  560  ours  BFS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='989  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content='950  7  helpdesk  ours frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content='720  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='801  28  Effects of Non-block Structures Table 1 shows that compared to IMf, the  models discovered by our approach have higher F1 scores for four of the five  logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Note that the fitness values of the models discovered by our approach are  16  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van der Aalst  all higher than the defined threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In general, IMf tends to discover mod-  els with higher fitness values while our approach discovers models with higher  precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In IMf, one can use the filter threshold to balance fitness and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  This is also the case in our approach, the user can set a lower fitness thresh-  old to include more candidate nets that are less fitting but more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 9  shows the discovered models from the two approaches for the hospitalBilling  log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' While the overall structure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 9a is similar to its counterpart in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 9b,  our approach discovered non-block structures at the later stage of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Such construct is not possible to model by IMf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The result shows that our ap-  proach can discover sound free-choice workflow nets with non-block structures  and produce competitive model quality as the state-of-the-art algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  (a) The discovered model using our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Due to the more flexible structure, one can exe-  cute EMPTY, BILLED, or REOPEN after CODE NOK while only BILLED or REOPEN are  executable after CODE OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The construct is not discoverable by IMf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  (b) The discovered model using IMf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Note that activity REOPEN is dropped by the filter of IMf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 9: The models discovered by our approach and IMf for the hospitalBilling log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  6  Conclusion and Future Work  In this paper, we present a discovery algorithm that aims to discover sound free-  choice workflow nets with non-block structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The algorithm utilizes the syn-  thesis rules to incrementally add activities with predefined patterns to discover  models that are guaranteed to be sound and free-choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Moreover, a certain  level of replay fitness is guaranteed by a user-defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  The approach has been implemented and evaluated using various real-life  event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The results show that the process models discovered by our approach  have higher model quality (in terms of both replay fitness and precision) than  the existing approach [12], which also depends on synthesis rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Moreover, our  approach produces competitive model quality compared to the state-of-the-art:  Inductive Miner - infrequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' For future work, we plan to explore more advanced  ordering strategies and investigate their influences on the model quality and  computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' The other direction is to further speed up the approach as  the long execution time is a clear limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' This could be done by exploiting  the log-based heuristics further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  DELETE  CHANGE  IAGN  CODE  REOPEIDELETE  CHANGE  DIAGN  CODE NOK  EMPTY  NEW  CODE OK  End  FIN  RELEASE  BLLEDDiscovering Sound Free-choice Workflow Nets With Non-block Structures  17  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' We thank the Alexander von Humboldt (AvH) Stiftung  for supporting our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content=' Knowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content=' IEEE (2012)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', van der Aalst, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content=' : Prodigy : Human-in-the-loop  process discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In: RCIS 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content=' IEEE (2018)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Dixit, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', Verbeek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', Buijs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', van der Aalst, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' : Interactive  data-driven process model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In: ER 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 11157, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 251–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  Springer (2018)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' van Dongen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', de Medeiros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', Wen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=': Process mining: Overview and  outlook of Petri net discovery algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Petri Nets Other Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Concurr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='  2, 225–242 (2009)  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Leemans, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', Fahland, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', van der Aalst, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' : Scalable process discovery  and conformance checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
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+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 17(2), 599–631 (2018)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Schuster, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', van Zelst, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=', van der Aalst, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' : Incremental discovery of  hierarchical process models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' In: RCIS 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 385, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' 417–433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
+page_content=' Springer (2020)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'}
diff --git a/GdAyT4oBgHgl3EQfSvfs/content/tmp_files/2301.00094v1.pdf.txt b/GdAyT4oBgHgl3EQfSvfs/content/tmp_files/2301.00094v1.pdf.txt
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@@ -0,0 +1,1049 @@
+arXiv:2301.00094v1  [math.CV]  31 Dec 2022
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL
+SHEAVES ON SINGULAR COMPLEX SPACES AND REGULARITY OF
+WEAK KÄHLER-EINSTEIN METRICS
+ZHENQIAN LI
+Abstract. In this article, we will characterize regular points respectively by the local van-
+ishing, positivity of the Ricci curvature and L2-solvability of the ∂-equation together with
+Skoda’s theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubhar-
+monic (psh) functions on any (reduced) complex space of pure dimension. As a by-product,
+we show that any weak Kähler-Einstein metric on singular Q-Fano/Calabi-Yau/general
+type varieties cannot be smooth, and that in general there exists no singular normal Kähler
+complex space such that the Kähler metric is Kähler-Einstein on the regular locus.
+1. Introduction
+Throughout this note, all complex spaces are always assumed to be reduced and para-
+compact unless otherwise mentioned; we mainly refer to [19, 40] for basic references on
+the theory of complex spaces.
+1.1. Local vanishing for multiplier ideals. The local vanishing theorem for the higher
+direct images of sheaves computing multiplier ideals plays an important role in complex
+geometry and algebraic geometry, by which many local/global properties of multiplier
+ideal could be deduced, e.g., the restriction theorem, Nadel vanishing theorem and Skoda’s
+theorem for multiplier ideals and so on (cf. [10, 28], etc.).
+Let (X, ω) be a Hermitian complex space of pure dimension n and ϕ ∈ QPsh(X) be a
+quasi-psh function on X. Then we can define the Nadel-Lebesgue multiplier ideal sheaf
+INL(ϕ) associated to ϕ on X by the integrability with respect to the Lebesgue measure dVω
+(see Definition 2.2), which coincides with the usual multiplier ideal sheaf I (ϕ) introduced
+by Nadel whenever X is smooth. Let π : �X → X be any log resolution of the Jacobian ideal
+JacX of X, then it follows that the Nadel-Lebesgue multiplier ideal sheaf
+INL(ϕ) = π∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+.
+When X is smooth, the Mather discrepancy divisor �K�X/X is nothing but the relative canon-
+ical divisor K�X/X := K�X − π∗KX of �X over X. Then, we have the following local vanishing
+for multiplier ideals (cf. [28, 37])
+Rqπ∗
+�
+O�X(K�X/X) ⊗ I (ϕ ◦ π)
+�
+= 0, ∀q ≥ 1.
+Therefore, it is natural to ask whether we could establish a similar local vanishing result
+in the singular setting, i.e.,
+Rqπ∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+= 0, ∀q ≥ 1.
+In the present note, one of our goals is to study the local vanishing in the context of Nadel-
+Lebesgue multiplier ideals. In particular, based on Skoda’s division for Nadel-Lebesgue
+Date: January 3, 2023.
+2010 Mathematics Subject Classification. 14F18, 32L20, 32Q20, 32S05, 32U05, 32W05.
+Key words. Multiplier ideal sheaves, plurisubharmonic functions, vanishing theorems, ∂-equations, Skoda’s
+L2 division theorem, Kähler-Einstein metrics.
+E-mail: lizhenqian@amss.ac.cn.
+1
+
+2
+ZHENQIAN LI
+multiplier ideals, we will prove that such a local vanishing for Nadel-Lebesgue multiplier
+ideals is in fact equivalent to smoothness of the ambient space in some sense; see Theorem
+1.3 for a detailed statement.
+1.2. Skoda’s ideal generation by L2 estimates for the ∂-equation. In the classical works
+[44, 45], relying on the L2 methods due to [1, 25, 26] in several complex variables, Skoda
+established an analytic criterion on the ideal generation by a given collection of holo-
+morphic functions or sections. In the original proof of Skoda’s ideal generation, as well as
+standard techniques in functional analysis for the argument on a priori estimate and solving
+∂-equation with L2 estimates, he also developed special analytic techniques by restricting
+the domain of the ∂-operator to an appropriate subspace of the usual L2 space and inducing
+an L2 estimate on this new operator.
+As applications, Skoda’s theorem is a crucial ingredient in proving the Briançon-Skoda
+theorem in commutative algebra [6, 27, 35] and an effective version of the Nullstellensatz
+in algebraic geometry [13]. Moreover, a special case of Skoda’s ideal generation also
+played key roles in Siu’s works on the deformation invariance of plurigenera [42] and finite
+generation of the canonical ring [43]. The interaction between several complex variables,
+complex algebraic geometry and partial differential equations has been an attractive area
+for the researchers. For the sake of reader’s convenience, we state a version of Skoda’s L2
+division theorem as below.
+Theorem 1.1. ([45], Théorème 2). Let (X, ω) be an n-dimensional weakly pseudoconvex
+Kähler manifold with ϕ ∈ Psh(X), and g : E → Q be a surjective morphism of Hermitian
+holomorphic vector bundles with rE = rank E and rQ = rank Q. Suppose that E is Nakano
+semi-positive on X and L → X is a Hermitian line bundle such that
+√
+−1Θ(L) − ρ
+√
+−1Θ(det Q) ≥ 0
+for ρ = min{n, rE − rQ} + ε and some ε > 0.
+Then, for every f ∈ H0(X, Q ⊗ KX ⊗ L) satisfying
+�
+X
+⟨�
+gg∗ f, f⟩ · (det gg∗)−ρ−1e−2ϕdVω < +∞,
+there exists h ∈ H0(X, E ⊗ KX ⊗ L) such that f = g · h and
+�
+X
+|h|2 · (det gg∗)−ρe−2ϕdVω ≤ ρ
+ε ·
+�
+X
+⟨�
+gg∗ f, f⟩ · (det gg∗)−ρ−1e−2ϕdVω.
+Due to Theorem 1.1, if we consider the trivial bundles E, Q and L on a pseudoconvex
+domain, then by combining with the strong openness of multiplier ideal sheaves established
+by Guan-Zhou [21], we can reformulate Theorem 1.1 in the language of multiplier ideals
+as follows (cf. also Remark A.3):
+Theorem 1.2. Let X be an n-dimensional complex manifold with ϕ ∈ QPsh(X) a quasi-psh
+function and a ⊂ OX a nonzero ideal sheaf with r (local) generators. Then, it follows that
+I �ϕ + kϕa
+� = a · I �ϕ + (k − 1)ϕa
+�, ∀k ≥ min{n, r},
+where ϕa := 1
+2 log(�
+i |gi|2) and (gi) is any local system of generators of a.
+Motivated by the above reformulation of Theorem 1.1, it is interesting for us to explore
+an analogue to Theorem 1.2 for Nadel-Lebesgue multiplier ideals in the singular setting.
+In order to achieve such a goal, a natural idea is to generalize Skoda’s L2 methods to the
+singular case, i.e., creating an appropriate L2 theory for the ∂-operator on singular complex
+spaces. However, as presented in [15, 16], it seems not to be possible to establish a general
+theory as in the smooth setting to solve the ∂-equation with L2 estimates on complex spaces
+with singularities; one can refer to [17, 38, 39, 41] for some partial results on the related
+topics.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+3
+On the other hand, we can also consider to apply Theorem 1.1 near the singularities
+under some reasonable assumptions on the positivity of curvatures. Fortunately, we could
+show that positivity of the Ricci curvature on the regular locus is in fact equivalent to the
+desired Skoda’s ideal generation and L2-solvability of the ∂-equation. More precisely, we
+state our main result in the following:
+Theorem 1.3. Let X be a (Hermitian) complex space of pure dimension n with x ∈ X a
+normal point and π : �X → X a log resolution of the Jacobian ideal JacX of X. Then, the
+following statements are equivalent:
+(1) For each quasi-psh function ϕ near the point x ∈ X, we have
+Rqπ∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+= 0, ∀q ≥ 1.
+(2) For each quasi-psh function ϕ near the point x ∈ X, we have
+Rqπ∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+= 0, ∀1 ≤ q < n.
+(3) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric
+ω on Ω such that the Ricci curvature Ric(ω) ≥ 0 on the regular locus Ωreg of Ω.
+(4) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric
+ω and a C ∞ differentiable real function ψ on Ω such that Ric(ω) +
+√
+−1∂∂ψ ≥ 0
+on the regular locus Ωreg of Ω.
+(5) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric
+ω and a Hermitian line bundle L on Ω such that for any smooth ϕ ∈ SPsh(Ω) and
+v ∈ L2
+0,q(Ωreg, L) satisfying ∂v = 0 and
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω < +∞
+with the curvature operator Aϕ = [
+√
+−1∂∂ϕ, Λω] on Ωreg, we have u ∈ L2
+0,q−1(Ωreg, L)
+such that ∂u = v and
+�
+Ωreg
+|u|2e−2ϕdVω ≤
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω.
+(6) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric
+ω and a Hermitian line bundle L on Ω such that for any smooth ϕ ∈ SPsh(Ω) and
+v ∈ L2
+0,1(Ωreg, L) satisfying ∂v = 0 and
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω < +∞,
+we have u ∈ L2(Ωreg, L) such that ∂u = v and
+�
+Ωreg
+|u|2e−2ϕdVω ≤
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω.
+(7) The Skoda’s theorem holds for Nadel-Lebesgue multiplier ideals, i.e., for any
+nonzero ideal sheaf a with r generators and quasi-psh function ϕ near the point
+x ∈ X, it holds that
+INL
+�ϕ + kϕa
+� = a · INL
+�ϕ + (k − 1)ϕa
+�, ∀k ≥ min{n, r}.
+(8) For any nonzero ideal sheaf a near the point x ∈ X, it holds that
+INL
+�nϕa
+� = a · INL
+�(n − 1)ϕa
+�.
+(9) For any nonzero ideal sheaf a near the point x ∈ X, it holds that
+INL
+�nϕa
+� ⊂ a.
+(10) The point x ∈ X is a regular point of X.
+
+4
+ZHENQIAN LI
+In the above result, the most interesting and amazing point is that it presents several
+characterizations of regular points by various statements involved Nadel-Lebesgue mul-
+tiplier ideals, which look like almost irrelevant; e.g., (1, 2) in algebraic geometry, (3, 4)
+in differential geometry and (5, 6) in partial differential equations together with (7—9) in
+commutative algebra. The core idea of all arguments originates from the Skoda’s ideal
+generation by the L2 approaches in several complex variables.
+Remark 1.4. Simple examples show that the assumption that x is a normal point of X
+cannot be removed in Theorem 1.3; in particular, any of the statements (1, 2, 7, 8) will not
+imply (10) in that case.
+As a straightforward consequence of Theorem 1.3, we have
+Corollary 1.5. Any normal Kähler space with nonnegative Ricci curvature on the regular
+locus must be non-singular.
+1.3. Kähler-Einstein metrics on singular varieties. Let X be a normal Q-Gorenstein
+Kähler space, that is, a normal Kähler space whose canonical class KX defines a Q-line
+bundle on X. A Kähler current ω ∈ c1(±KX) is called a weak (or singular) Kähler-Einstein
+metric on X if ω has bounded local potentials and is a genuine Kähler-Einstein metric on
+the regular locus Xreg of X (cf. [3, 4, 5, 14], etc.). A weak Kähler-Einstein metric ω on
+X is called a Kähler-Einstein metric if ω is a Kähler metric on X, i.e., ω has smooth local
+potentials. For general expositions on the topic of Kähler-Einstein metrics one can refer to
+[2, 23, 47, 48, 49, 50] and the references therein. In particular, we state some recent results
+as follows.
+Theorem 1.6. ([3, 5, 14, 30, 31, 32, 36], etc.). Let X be a normal Q-Gorenstein complex
+projective variety. Then:
+(1) If X is a Q-Calabi-Yau variety with only log terminal singularities, then X admits
+a weak Kähler-Einstein metric.
+(2) If KX is ample, then X admits a weak Kähler-Einstein metric if and only if X is
+K-stable.
+(3) If −KX is ample, then X admits a weak Kähler-Einstein metric if and only if X is
+K-polystable.
+A basic and widely open problem in Kähler geometry/geometric analysis is understand-
+ing the geometric asymptotic behavior of the weak Kähler-Einstein metric near the singular
+locus Xsing of X. In [24], the authors made a breakthrough with a very precise descrip-
+tion for a class of Calabi-Yau varieties with smoothable isolated singularities, which are
+in further required to be isomorphic to a neighborhood of the vertex in a strongly regular
+Calabi-Yau cone; see also [7, 8, 18] for some recent progress in this direction. In more gen-
+eral situations, by using deep tools in the theory of degenerate complex Monge-Ampère
+equations on singular complex spaces, the continuity of local potentials of weak Kähler-
+Einstein metrics is established for all Q-Fano/Calabi-Yau varieties in [4, 22], but so far
+little is known for the higher order regularity in general and it is desirable to establish one
+for weak Kähler-Einstein potentials. However, relying on Theorem 1.3, we will see that
+too much regularity cannot be expected and in fact any weak Kähler-Einstein potential is at
+most C α (α < 2) differentiable near the singularities. In particular, we obtain the following
+Theorem 1.7. Let X be a normal Q-Gorenstein Kähler space admitting a weak Kähler-
+Einstein metric ω. Then, ω is smooth on X if and only if X is non-singular.
+2. Preliminaries
+Firstly, we introduce the notion of Nadel-Lebesgue multiplier ideal sheaf on any com-
+plex space of pure dimension and then present some useful facts used throughout this note.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+5
+Definition 2.1. Let X be a complex space of pure dimension and ϕ ∈ L1
+loc(Xreg) with
+respect to the Lebesgue measure. Then, the complex space X is said to be a Hermitian
+complex space if there is a Hermitian metric ω on the regular part (may be disconnected)
+Xreg of X such that ω is locally the restriction of a Hermitian metric on some CN for a local
+embedding of X. It follows from the differentiable partition of unity that every complex
+space is a Hermitian complex space as in the smooth case.
+The complex space X is called to be a Kähler space if there is a Hermitian metric ω on X
+such that ω is locally the restriction of a Kähler metric on some CN for a local embedding
+of X. In particular, it admits smooth strictly psh functions as local potentials.
+We say that the function ϕ is quasi-plurisubharmonic (quasi-psh for short) on X if it is
+locally equal to the sum of a psh function and of a smooth function on X. The set of quasi-
+psh (resp. psh and strictly psh) functions on X is denoted by QPsh(X) (resp. Psh(X) and
+SPsh(X)). A quasi-psh function ϕ ∈ QPsh(X) will be said to have analytic singularities on
+X if ϕ can be written locally as
+ϕ = c
+2 log(|f1|2 + · · · + |fN0|2) + O(1),
+where c ∈ R≥0 and (fi) are holomorphic functions.
+Definition 2.2. Let (X, ω) be a Hermitian complex space of pure dimension and ϕ ∈
+L1
+loc(Xreg) with respect to the Lebesgue measure.
+The Nadel-Lebesgue multiplier ideal sheaf associated to ϕ on X is defined to be the
+OX-submodule INL(ϕ) ⊂ MX of germs of meromorphic functions f ∈ MX,x such that
+|f|2e−2ϕ is integrable with respect to the Lebesgue measure dVω near the point x ∈ X. One
+can check that INL(ϕ) is independent of the choice of Hermitian metric ω on X.
+The log canonical threshold (or complex singularity exponent) LCTx(ϕ) of ϕ at a point
+x ∈ X is defined to be
+LCTx(ϕ) := sup �c ≥ 0 | OX,x ⊂ INL(cϕ)x
+� .
+It is convenient to put LCTx(−∞) = 0.
+It is easy to see that INL(ϕ) ⊂ OX is an ideal sheaf when X is a normal complex space
+and ϕ is locally bounded from above on X. In addition, if X is smooth and ϕ ∈ QPsh(X),
+then INL(ϕ) is nothing but the usual multiplier ideal sheaf I (ϕ) introduced by Nadel (see
+[10]).
+Remark 2.3. Since the definition of Nadel-Lebesgue multiplier ideals is local, we can com-
+pute the multiplier ideals by choosing a special Hermitian metric ω for a local embedding
+of X. In particular, if X is an n-dimensional complex subspace of some domain in CN,
+we can take Hermitian metric ω on X to be the inherited standard Kähler metric from CN.
+Then, we have dVω = 1
+n!υn|Xreg, where υ =
+√
+−1
+2
+N�
+k=1
+dzk ∧ d¯zk.
+For the sake of reader’s convenience, we state a basic estimate related to local volume
+of an analytic subset as follows.
+Lemma 2.4. ([20], Lemma 2.3). Let X be a pure n-dimensional analytic subset through
+the origin 0 of some domain in CN (N ≥ 2). Then, there is a Stein neighborhood U ⊂⊂ CN
+of the origin 0 such that for any 0 ≤ ε < 1, we have
+�
+U∩X
+1
+(|z1|2 + · · · + |zN|2)n−1+ε dVω < +∞,
+where dVω = 1
+n!υn|Xreg and υ =
+√
+−1
+2
+N�
+k=1
+dzk∧d¯zk.
+Analogous to the Nadel-Ohsawa multiplier ideal sheaves introduced in [33, 34] (see also
+[9, 12] for the algebro-geometric counterpart), we state some related properties as follows.
+
+6
+ZHENQIAN LI
+Proposition 2.5. (1) Let π : �X → X be a log resolution of the Jacobian ideal JacX of X
+and �K�X/X be the Mather discrepancy divisor. Then, we have the image
+Im
+�
+π∗Ωn
+X ֒→ Ωn
+�X
+�
+= O�X(−�K�X/X) · Ωn
+�X,
+and
+INL(ϕ) = π∗
+�
+O�X(�K�X/X) ⊗ I (ϕ ◦ π)
+�
+.
+Furthermore, we can deduce that INL(ϕ + log |JacX|) = INO(ϕ), the Nadel-Ohsawa
+multiplier ideal sheaf associated to ϕ on X.
+(2) When X is normal and ϕ has analytic singularities, INL(ϕ) coincides with the
+Mather multiplier ideal sheaf defined in [9].
+(3) For any ϕ ∈ QPsh(X), it follows that INL(ϕ) ⊂ MX is a coherent fractional ideal
+sheaf and satisfies the strong openness, i.e., INL(ϕ) = �
+ε>0
+INL
+�(1 + ε)ϕ�.
+For our proof of Theorem 1.3, we need the following L2 estimates for the ∂-equation
+and relative version of Grauert-Riemenschneider vanishing theorem for the higher direct
+images.
+Theorem 2.6. (cf. [10], Theorem 5.2). Let (X, ω) be an n-dimensional Kähler manifold,
+which contains a weakly pseudoconvex Zariski open subset. Let L be a Hermitian line
+bundle on X such that
+√
+−1Θ(L) + Ric(ω) > 0.
+Then, for every smooth ϕ ∈ Psh(X) and v ∈ L2
+0,q(X, L) satisfying ∂v = 0 and
+�
+X
+⟨A−1v, v⟩ e−2ϕdVω < +∞
+with the curvature operator A = [
+√
+−1Θ(L) + Ric(ω) +
+√
+−1∂∂ϕ, Λω] on X, there exists
+u ∈ L2
+0,q−1(X, L) such that ∂u = v and
+�
+X
+|u|2e−2ϕdVω ≤
+�
+X
+⟨A−1v, v⟩ e−2ϕdVω.
+Theorem 2.7. ([11], Theorem 1.1). Let (X, ω) be an n-dimensional Kähler manifold which
+is a Zariski open subset of some Stein space X∗, and L be a Hermitian line bundle on X.
+If for any smooth ϕ ∈ SPsh(X∗) and v ∈ L2
+0,1(X, L) satisfying ∂v = 0 and
+�
+X
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω < +∞
+with the curvature operator Aϕ = [
+√
+−1∂∂ϕ, Λω] on X, there exists u ∈ L2(X, L) such that
+∂u = v and
+�
+X
+|u|2e−2ϕdVω ≤
+�
+X
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω,
+then it follows that L ⊗ K−1
+X is Nakano semi-positive on X.
+Theorem 2.8. ([37], Corollary 1.5). Let π : X → Y be a surjective proper (locally) Kähler
+morphism from a complex manifold X to a complex space Y, and (L, e−ϕL) be a (possibly
+singular) Hermitian line bundle on X with semi-positive curvature. Then, the higher direct
+image sheaf
+Rqπ∗
+�
+KX ⊗ L ⊗ I (ϕL)
+�
+= 0,
+for every q > dim X − dim Y.
+Remark 2.9. Any log resolution π : �X → X of a coherent ideal sheaf I on a complex space
+X is a locally Kähler (proper modification), which is locally a finite sequence of blow-ups
+with smooth centers. Besides, any finite holomorphic mapping between complex spaces is
+(locally) proper Kähler.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+7
+In the remainder of this section, we recall some algebraic properties on the integral
+closure of ideals.
+Definition 2.10. ([46]). Let R be a commutative ring and I an ideal of R. An element
+f ∈ R is said to be integrally dependent on I if it satisfies a relation
+f d + a1 f d−1 + · · · + ad = 0
+(ak ∈ Ik, 1 ≤ k ≤ d).
+The set I consisting of all elements in R which are integrally dependent on I is called
+the integral closure of I in R. I is called integrally closed if I = I. One can prove that I is
+an ideal of R, which is the smallest integrally closed ideal in R containing I.
+Definition 2.11. ([46]). Let R be a commutative ring with identity and let J ⊂ I be ideals in
+R. J is said to be a reduction of I if there exists a nonnegative integer n such that In+1 = JIn.
+A reduction J of I is called minimal if no ideal strictly contained in J is a reduction of
+I. An ideal that has no reduction other than itself is called basic.
+One can prove that minimal reductions do exist in Noetherian local rings and an ideal
+which is a minimal reduction of a given ideal is necessarily basic. Moreover, if R is a
+Noetherian ring, J ⊂ I is a reduction of I if and only if J = I.
+In the analytic setting, we have the following characterization on integral closure and
+reduction of ideals.
+Theorem 2.12. (cf. [29], Théorème 2.1). Let X be a complex space and Y ⊂ X be a
+proper closed complex subspace (may be non-reduced) defined by a coherent OX-ideal I
+with x ∈ Y a point. Let J ⊂ OX be a coherent OX-ideal and I (resp. J) be the germ of
+I (resp. J ) at x. Then, the following conditions are equivalent:
+(1) J ⊂ I.
+(2) For every morphism π : �X → X satisfying: (i) π is a proper and surjective, (ii) �X is
+a normal complex space and (iii) I · O�X is an invertible OX-module, there exists
+an open neighborhood U of x in X such that
+J · O�X|π−1(U) ⊂ I · O�X|π−1(U).
+(3) If V is an open neighborhood of x on which I and J are generated by their
+global sections, then for every system of generators g1, ..., gr ∈ Γ(V, I ) and every
+f ∈ Γ(V, J ), one can find an open neighborhood V′ of x and a constant C > 0
+such that
+|f(y)| ≤ C · sup
+k
+|gk(y)|, ∀y ∈ V′.
+Remark 2.13. Let X be a normal complex space and I ⊂ OX a coherent ideal sheaf. Let
+π : �X → X be any proper modification from a normal complex space �X onto X such that
+I · O�X = O�X(−D) for some effective Cartier divisor D on �X. Then, we have π∗O�X(−D) =
+I , the integral closure of I in OX.
+Lemma 2.14. (cf. Example 9.6.19 in [28]; see also [10], Lemma 11.16). Let X be a
+normal complex space of dimension n and a ⊂ OX a nonzero ideal. Then, there exists
+an open covering {Uα}α∈N of X such that a|Uα has a reduction bα generated by at most n
+elements.
+3. Proofs of the main results
+3.1. Proof of Theorem 1.3. Since all of the statements are local, without loss of gener-
+ality, we may assume that X is an n≥2-dimensional normal (Hermitian) complex subspace
+of some domain in CN with ϕ ∈ QPsh(X) and a = (g1, . . ., gr) · OX an ideal sheaf gen-
+erated by holomorphic functions g1, . . . , gr on X. Moreover, we may also assume that ϕ
+is (locally) a strictly psh function on X if necessary, by adding some smooth strictly psh
+
+8
+ZHENQIAN LI
+function. It is easy to see that the implications (1) =⇒ (2), (3) =⇒ (4), (5) =⇒ (6) and
+(7) =⇒ (8) =⇒ (9) are trivial; in particular, we will present a proof in the following order:
+(1)
+� (2)
+�❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+(10)
+�④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+④
+�
+(9)
+�
+(8)
+�
+(7)
+�
+(3)
+�❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+� (4)
+�
+(5)
+� (6)
+�⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+⑥
+“(2) =⇒ (7)”. By the definition of Nadel-Lebesgue multiplier ideal sheaf, it follows
+that
+a · INL
+�ϕ + (k − 1)ϕa
+� ⊂ INL
+�ϕ + kϕa
+�,
+and so it is sufficient to show the reverse inclusion.
+Case (i). When r ≤ n.
+Let π : �X → X be a common log resolution of JacX and a such that a · O�X = O�X(−F)
+for some effective divisors F on �X. Denote by
+Am := O�X(�K�X/X) ⊗ I (ϕ ◦ π + mϕa ◦ π)
+= O�X(�K�X/X) ⊗ I (ϕ ◦ π) ⊗ O�X(−mF)
+for any m ∈ N, and consider the Koszul complex determined by g1, . . ., gr:
+0 → ΛrV ⊗ O�X(rF) → · · · → Λ2V ⊗ O�X(2F) → V ⊗ O�X(F) → O�X → 0,
+where V is the vector space spanned by g1, . . . , gr. Note that the Koszul complex is locally
+split and its syzygies are locally free, so twisting through by any coherent sheaf will pre-
+serve the exactness. Then, by twisting with Ak (k ≥ r), we obtain the following long exact
+sequence
+0 → ΛrV ⊗ Ak−r → · · · → Λ2V ⊗ Ak−2 → V ⊗ Ak−1 → Ak → 0.
+(⋆)
+On the other hand, for any m ∈ N, by (2) we have the local vanishing of the higher direct
+images Rqπ∗Am = 0 (1 ≤ q < n). Note that
+INL
+�ϕ + mϕa
+� = π∗Am
+by the functoriality property with respect to direct images of sheaves by modifications, and
+then by taking direct images of (⋆) we will deduce the following so-called exact Skoda
+complex (cf. [28], p. 228):
+0 → ΛrV ⊗ INL
+�ϕ + (k − r)ϕa
+� → · · · → V ⊗ INL
+�ϕ + (k − 1)ϕa
+� → INL
+�ϕ + kϕa
+� → 0.
+In particular, the map V ⊗ INL
+�ϕ + (k − 1)ϕa
+� → INL
+�ϕ + kϕa
+� is surjective, by which we
+can infer that INL
+�ϕ + kϕa
+� ⊂ a · INL
+�ϕ + (k − 1)ϕa
+� for any k ≥ r.
+Case (ii). When r > n.
+As the statement is local, then by Lemma 2.14 we may assume that b is a reduction of
+a generated by n elements �g1, ...,�gn. Consider a common log resolution π : �X → X of
+JacX, a and b such that a · O�X = b · O�X = O�X(−F) for some effective divisors F on �X.
+Then, by the same argument as above, we can deduce the following exact Skoda complex:
+0 → ΛnV ⊗ INL
+�ϕ + (k − n)ϕa
+� → · · · → V ⊗ INL
+�ϕ + (k − 1)ϕa
+� → INL
+�ϕ + kϕa
+� → 0.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+9
+for any k ≥ n, where V is the vector space spanned by �g1, ...,�gn. Therefore, it follows that
+INL
+�ϕ + kϕa
+� ⊂ b · INL
+�ϕ + (k − 1)ϕa
+� ⊂ a · INL
+�ϕ + (k − 1)ϕa
+�.
+“(3) =⇒ (5)”. It follows from the assumption that we have a Stein neighborhood Ω ⊂⊂
+X of the point x with a Kähler metric ω such that Ric(ω) ≥ 0 on Ωreg. Let ϕ ∈ SPsh(Ω) be
+any smooth strictly psh function on Ω and L = Ω × C be a trivial bundle equipped with the
+trivial Hermitian metric, which implies that
+√
+−1Θ(L) + Ric(ω) +
+√
+−1∂∂ϕ ≥
+√
+−1∂∂ϕ > 0
+on Ωreg.
+Since Ω is a Stein space, we are able to choose a complex hypersurface Z ⊂ Ω which
+contains the singular locus Ωsing of Ω such that Ω − Z ⊂ Ωreg is a Stein manifold. Then, by
+Theorem 2.6 we obtain that, for any smooth ϕ ∈ SPsh(Ω) and v ∈ L2
+0,q(Ωreg, L) satisfying
+∂v = 0 and
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω < +∞,
+we can find u ∈ L2
+0,q−1(Ωreg, L) such that ∂u = v and
+�
+Ωreg
+|u|2e−2ϕdVω ≤
+�
+Ωreg
+⟨A−1
+ϕ v, v⟩ e−2ϕdVω.
+“(6) =⇒ (4)”. As a straightforward application of Theorem 2.7 on Ωreg, it yields that
+√
+−1Θ(L) + Ric(ω) ≥ 0 on Ωreg. Let Ω′ ⊂ Ω be a small Stein neighborhood of the point
+x such that the Hermitian line bundle L has a smooth potential ψ on Ω′. Therefore, we
+deduce that
+√
+−1∂∂ψ + Ric(ω) =
+√
+−1Θ(L) + Ric(ω) ≥ 0
+on Ω′
+reg.
+“(4) =⇒ (7)”. Due to the definition and Lemma 2.14, it is sufficient to prove
+INL
+�ϕ + kϕa
+� ⊂ a · INL
+�ϕ + (k − 1)ϕa
+�
+for the case r ≤ n near the point x ∈ X. Let f ∈ INL
+�ϕ + kϕa
+�
+x with k ≥ min{n, r} = r,
+then by the strong openness of multiplier ideals there exists small enough ε > 0 such that
+f ∈ INL
+�ϕ + (k + ε)ϕa
+�
+x.
+By the assumption of (4), we let Ω ⊂⊂ X be a Stein neighborhood of the point x with
+a Kähler metric ω and a smooth real function ψ on Ω such that Ric(ω) +
+√
+−1∂∂ψ ≥ 0 on
+Ωreg. After shrinking Ω if necessary, we may assume that the function ψ is bounded on Ω
+and f is holomorphic on Ω such that
+�
+Ω
+|f|2 · |g|−2(r+ε)e−2(ϕ+(k−r)ϕa)dVω < +∞.
+In addition, we also choose a complex hypersurface Z ⊂ Ω which contains the singular
+locus Ωsing of Ω and the common zero-set of holomorphic functions g1, ..., gr such that
+Ω′ := Ω − Z is a Stein manifold.
+Let E = Ω′ × Cr and Q = Ω′ × C be the trivial bundles on Ω′ and L = K−1
+Ω′ be the
+anti-canonical line bundle with the induced metric twisted by a weight e−ψ. The morphism
+g : E → Q determined by holomorphic functions g1, ..., gr is given by
+(h1, ..., hr) �→
+r�
+m=1
+gm · hm = g · h.
+Note that �
+gg∗ = IdQ when rank Q = 1, and on Ω′ we have
+√
+−1Θ(L) − (r − 1 + ε)
+√
+−1Θ(det Q) = Ric(ω) +
+√
+−1∂∂ψ ≥ 0.
+
+10
+ZHENQIAN LI
+Thus, we can apply Theorem 1.1 on Ω′ and then obtain an r-tuple (h1, ..., hr) of holomor-
+phic functions on Ω′ such that f = g · h on Ω′ and
+�
+Ω′ |h|2 · |g|−2(r−1+ε)e−2(ϕ+(k−r)ϕa)dVω =
+�
+Ω′ |h|2e−2(ϕ+(k−1+ε)ϕa)dVω < +∞.
+We can now extend every hm to be a holomorphic function on Ω from the L2 estimate above
+and normality of X, which implies that
+INL
+�ϕ + kϕa
+� ⊂ a · INL
+�ϕ + (k − 1 + ε)ϕa
+� ⊂ a · INL
+�ϕ + (k − 1)ϕa
+�
+on Ω; we finish the argument.
+“(9) =⇒ (10)”. By the assumption, we have INL
+�nϕa
+� ⊂ a. Suppose that x ∈ X
+is a singular point. Then, by the local parametrization for analytic sets, we can find a
+local coordinate system (z′; z′′) = (z1, ..., zn; zn+1, ..., zN) near x such that for some constant
+C > 0, we have |z′′| ≤ C · |z′| for any point z ∈ X near x.
+Let a ⊂ OX be the ideal sheaf generated by holomorphic functions �z1, ...,�zn ∈ OX
+(shrinking X if necessary), where �zk are the residue classes of zk in OX. From the non-
+smoothness of X at the point x, we deduce that the embedding dimension dimC(mX,x/m2
+X,x) ≥
+n + 1 of X at x, which implies that there exists k0 (n + 1 ≤ k0 ≤ N) such that�zk0 � a.
+On the other hand, after shrinking X again, it follows that
+�
+X
+|zk0|2
+|z′|2n dVω ≤ C2(1 + C2)n−1 ·
+�
+X
+|z|−2(n−1)dVω < +∞,
+where the finiteness of the integration follows from Lemma 2.4. Then, we infer that�zk0 ∈
+INL
+�nϕa
+�, but�zk0 � a, which contradicts to the assumption INL
+�nϕa
+� ⊂ a. Thus, we obtain
+that x ∈ X is a regular point.
+“(10) =⇒ (1)”. It is a straightforward consequence of Theorem 2.8.
+“(10) =⇒ (3)”. Since x is a regular point of X, after choosing an appropriate coordinate
+neighborhood of x, we may assume that Ω ∋ x is a Stein domain in Cn. Therefore, we can
+take ω =
+√
+−1
+2
+n�
+k=1
+dzk ∧ d¯zk to be the standard Euclidean metric on Cn and then we have
+Ric(ω) = 0 on Ω; the proof of Theorem 1.3 is concluded.
+□
+Remark 3.1. In addition, we can deduce from the proof of Theorem 1.3 that
+(i) if (1) or (2) holds for each quasi-psh function ϕ with analytic singularities, then x ∈ X
+is a regular point;
+(ii) both of the statements (3) and (4) could be respectively modified to be Ric(ω) ≥ 0
+and Ric(ω) +
+√
+−1∂∂ψ ≥ 0 on a Zariski open subset of Ω contained in Ωreg.
+3.2. Proof of Theorem 1.7. It is sufficient to prove the necessity.
+Let x ∈ X be any point. Since ω is a smooth Kähler metric on X, then ω has smooth
+local potentials, i.e., there exists a Stein neighborhood Ω ⊂ X of x and a smooth strictly psh
+functions ψ on Ω such that ω =
+√
+−1∂∂ψ on Ωreg, which implies that Ric(ω)+
+√
+−1∂∂ψ ≥ 0
+on Ωreg whenever Ric(ω) = ±ω, 0. Thus, it follows from (4) in Theorem 1.3 that x ∈ X is
+a regular point.
+□
+Remark 3.2. The same arguments as in the proof of Theorem 1.3 and 1.7 also imply that
+each local potential of weak Kähler-Einstein metric ω is C 2 differentiable on X if and only
+if X is non-singular, and that there exists no singular normal Kähler space such that the
+Kähler metric is Kähler-Einstein on the regular locus.
+In fact, our method is still available when the weak Kähler-Einstein metric is (locally)
+equivalent to the standard induced Kähler metric by restriction near the singularities; for
+instance, when the weak Kähler-Einstein metric is of locally bounded coefficients.
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+11
+Appendix A. Uniform bounds of powers associated to an L2 division problem
+The ideal membership is an important object to study in commutative algebra, algebraic
+geometry and several complex variables, e.g., the famous Hilbert’s Nullstellensatz and
+Briançon-Skoda theorem and so on. In this part, we are mainly interested in the uniform
+bounds of powers associated to an L2 division problem, a kind of special ideal membership.
+Let X be a Stein manifold of dimension n and a = (g1, . . . , gr)·OX an ideal sheaf generated
+by holomorphic functions g1, . . . , gr on X. In general, the division problem states that,
+given a positive integer k ∈ N and holomorphic function f on X, we wish to determine
+when f is generated by holomorphic functions g1, . . ., gr; more precisely, when we can
+find holomorphic functions h1, . . . , hr ∈ ak−1 on X such that
+f =
+r�
+m=1
+gm · hm.
+Thanks to the Oka-Cartan’s theory on Stein manifolds, the division problem is solvable if
+and only if f ∈ ak.
+Note that the condition f ∈ ak is purely algebraic, and so it is natural to ask whether we
+could find an analytic condition to replace the algebraic one. It is easy to see that f ∈ ak
+implies that |f|e−ϕk is locally bounded on X, or L2
+loc more generally; where ϕk := k log |g|
+and |g|2 := |g1|2 + · · · + |gr|2. On the other hand, local boundedness of |f|e−ϕk is equivalent
+to the fact that f ∈ ak, the integral closure of ak in OX (see Theorem 2.12). Thus, it is
+an interesting question whether we could establish solvability of an L2 analogue of the
+division problem.
+Let ϕ ∈ Psh(X) be a psh function on X and denote by
+A2
+loc(X, ϕ) :=
+�
+f ∈ OX(X)
+��� |f|2e−2ϕ is locally integrable on X
+�
+.
+Then, we raise the following L2 division problem:
+Question A.1. Let X be an n-dimensional Stein manifold with a psh function ϕ ∈ Psh(X),
+and a = (g1, . . . , gr) · OX an ideal sheaf generated by holomorphic functions g1, . . ., gr on
+X. Given positive integer k ∈ N and f ∈ A2
+loc(X, ϕ + ϕk), are there holomorphic functions
+h1, . . ., hr ∈ A2
+loc(X, ϕ + ϕk−1) such that
+f =
+r�
+m=1
+gm · hm?
+A.1. A solution to Question A.1. Unfortunately, the answer of Question A.1 is negative
+for general k (see Example A.2). Motivated by the Skoda’s L2 division theorem (cf. The-
+orem 1.1), it seems to be reasonable to find a uniform integer k0, depending only on n,
+such that Question A.1 is solvable for any k ≥ k0. The goal of this part is to present an
+optimal uniform lower bounds of powers associated to Question A.1. In particular, we will
+establish the following
+Theorem A.1. There exists a uniform integer k0 = min{n, r} such that the solution to
+Question A.1 is positive for any k ≥ k0. In further, the uniform lower bound k0 = min{n, r}
+is optimal.
+In fact, the optimality of uniform integer k0 = min{n, r} is straightforward by the fol-
+lowing:
+Example A.2. Let Bn(0) be the unit ball centered at the origin 0 = (0′, 0′′) in Cr×Cn−r (1 ≤
+r ≤ n) and take g1 = z1, ..., gr = zr, f ≡ 1, ϕ ≡ 0 on Bn(0). Then, for every k < k0 = r, the
+answer of Question A.1 is negative.
+Indeed, by the fact that the log canonical threshold LCT(0′,z′′)(ϕk) = r
+k > 1 of ϕk at any
+point (0′, z′′), one can derive that f ∈ A2
+loc(Bn(0), ϕ + ϕk). Then, we infer from the fact that
+
+12
+ZHENQIAN LI
+f has no zeros in Bn(0) that there exist no holomorphic functions h1, . . . , hr on Bn(0) such
+that f =
+r�
+m=1
+gm · hm.
+Proof of Theorem A.1. It follows from the local vanishing (Theorem 2.8) and the argu-
+ments as in the proof of Theorem 1.3 that for any k ≥ min{n, r}, we have
+I �ϕ + ϕk
+� = a · I �ϕ + ϕk−1
+�.
+Let
+τ : I �ϕ + ϕk−1
+�⊕r −→ I �ϕ + ϕk
+�
+be the sheaf homomorphism defined by
+τ(h1,x, . . . , hr,x) =
+r�
+m=1
+gm · hm,x
+for any germs hm,x ∈ I �ϕ + ϕk−1
+�
+x. Then, we have an exact sequence of sheaves
+I �ϕ + ϕk−1
+�⊕r
+τ
+−→ I �ϕ + ϕk
+� −→ 0.
+It follows from the Oka-Cartan theory on Stein manifolds that the induced sequence of
+sections
+Γ
+�
+X, I �ϕ + ϕk−1
+�⊕r�
+τ∗
+−→ Γ
+�
+X, I �ϕ + ϕk
+��
+−→ 0
+is also exact, which implies that any section f ∈ Γ
+�
+X, I �ϕ + ϕk
+��
+can be written as the
+image f =
+r�
+m=1
+gm · hm for some sections hm ∈ Γ
+�
+X, I �ϕ + ϕk−1
+��
+.
+□
+Remark A.3. (An alternative argument on Theorem A.1). In fact, we could also give an-
+other argument on the proof of Theorem A.1 depending on the strong openness of multi-
+plier ideals established by Guan-Zhou [21] and the Skoda’s L2 division theorem for holo-
+morphic functions (see Theorem 1.1).
+Since the statement is local, it follows from Lemma 2.14 that it is sufficient to prove
+I �ϕ+ϕk
+� ⊂ a·I �ϕ+ϕk−1
+� for the case r ≤ n. Given f ∈ Γ
+�
+X, I �ϕ+ϕk
+��
+, after shrinking
+X, we may assume that X is the unit ball in Cn and
+�
+X
+|f|2e−2(ϕ+ϕk)dλn =
+�
+X
+|f|2 · |g|−2ke−2ϕdλn < +∞.
+Then, for each k ≥ r, by the strong openness of multiplier ideals there exists sufficiently
+small ε > 0 such that
+�
+X
+|f|2e−2(ϕ+(1+ε)ϕk)dλn =
+�
+X
+|f|2 · |g|−2(1+ε)ke−2ϕdλn < +∞,
+shrinking X if necessary. Finally, combining with Theorem 1.1, we deduce the desired
+result.
+A.2. A global L2 version of Question A.1. Let (X, ω) be an n-dimensional Stein manifold
+with a Kähler form ω. Let ϕ ∈ Psh(X) and I = (g1, . . . , gr) · OX an ideal sheaf generated
+by holomorphic functions g1, . . . , gr on X. Denote by
+A2(X, ϕ) :=
+�
+f ∈ OX(X)
+�����
+�
+X
+|f|2e−2ϕdVω < +∞
+�
+.
+Then, we have the following global analogue of Question A.1:
+Question A.2. Can we find a uniform integer k0 such that for each k ≥ k0 and f ∈ A2(X, ϕ+
+ϕk), there exist h1, . . ., hr ∈ A2(X, ϕ + ϕk−1) satisfying
+f =
+r�
+m=1
+gm · hm?
+As an immediate consequence of Theorem 1.1, we obtain the following
+
+ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES
+13
+Theorem A.4. Let X be a pseudoconvex domain in Cn. Then, there exists a uniform integer
+k0 = min{n + 2, r + 1} such that the solution to Question A.2 is positive.
+Remark A.5. (1) More generally, Theorem A.4 also holds for any complete Kähler domain
+in Cn with smooth psh function ϕ ∈ Psh(X).
+(2) In this case, combining with the Example A.2, it follows that the optimal uniform
+lower bound k0 is at least min{n, r}, and at most min{n + 2, r + 1}.
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+
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+page_content='00094v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='CV]  31 Dec 2022  ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL  SHEAVES ON SINGULAR COMPLEX SPACES AND REGULARITY OF  WEAK KÄHLER-EINSTEIN METRICS  ZHENQIAN LI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In this article, we will characterize regular points respectively by the local van-  ishing, positivity of the Ricci curvature and L2-solvability of the ∂-equation together with  Skoda’s theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubhar-  monic (psh) functions on any (reduced) complex space of pure dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' As a by-product,  we show that any weak Kähler-Einstein metric on singular Q-Fano/Calabi-Yau/general  type varieties cannot be smooth, and that in general there exists no singular normal Kähler  complex space such that the Kähler metric is Kähler-Einstein on the regular locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Introduction  Throughout this note, all complex spaces are always assumed to be reduced and para-  compact unless otherwise mentioned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' we mainly refer to [19, 40] for basic references on  the theory of complex spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Local vanishing for multiplier ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' The local vanishing theorem for the higher  direct images of sheaves computing multiplier ideals plays an important role in complex  geometry and algebraic geometry, by which many local/global properties of multiplier  ideal could be deduced, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', the restriction theorem, Nadel vanishing theorem and Skoda’s  theorem for multiplier ideals and so on (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' [10, 28], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Let (X, ω) be a Hermitian complex space of pure dimension n and ϕ ∈ QPsh(X) be a  quasi-psh function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then we can define the Nadel-Lebesgue multiplier ideal sheaf  INL(ϕ) associated to ϕ on X by the integrability with respect to the Lebesgue measure dVω  (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2), which coincides with the usual multiplier ideal sheaf I (ϕ) introduced  by Nadel whenever X is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let π : �X → X be any log resolution of the Jacobian ideal  JacX of X, then it follows that the Nadel-Lebesgue multiplier ideal sheaf  INL(ϕ) = π∗  �  O�X(�K�X/X) ⊗ I (ϕ ◦ π)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  When X is smooth, the Mather discrepancy divisor �K�X/X is nothing but the relative canon-  ical divisor K�X/X := K�X − π∗KX of �X over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, we have the following local vanishing  for multiplier ideals (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' [28, 37])  Rqπ∗  �  O�X(K�X/X) ⊗ I (ϕ ◦ π)  �  = 0, ∀q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Therefore, it is natural to ask whether we could establish a similar local vanishing result  in the singular setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=',  Rqπ∗  �  O�X(�K�X/X) ⊗ I (ϕ ◦ π)  �  = 0, ∀q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  In the present note, one of our goals is to study the local vanishing in the context of Nadel-  Lebesgue multiplier ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In particular, based on Skoda’s division for Nadel-Lebesgue  Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' 14F18, 32L20, 32Q20, 32S05, 32U05, 32W05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Multiplier ideal sheaves, plurisubharmonic functions, vanishing theorems, ∂-equations, Skoda’s  L2 division theorem, Kähler-Einstein metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  E-mail: lizhenqian@amss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  1  2  ZHENQIAN LI  multiplier ideals, we will prove that such a local vanishing for Nadel-Lebesgue multiplier  ideals is in fact equivalent to smoothness of the ambient space in some sense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' see Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3 for a detailed statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Skoda’s ideal generation by L2 estimates for the ∂-equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In the classical works  [44, 45], relying on the L2 methods due to [1, 25, 26] in several complex variables, Skoda  established an analytic criterion on the ideal generation by a given collection of holo-  morphic functions or sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In the original proof of Skoda’s ideal generation, as well as  standard techniques in functional analysis for the argument on a priori estimate and solving  ∂-equation with L2 estimates, he also developed special analytic techniques by restricting  the domain of the ∂-operator to an appropriate subspace of the usual L2 space and inducing  an L2 estimate on this new operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  As applications, Skoda’s theorem is a crucial ingredient in proving the Briançon-Skoda  theorem in commutative algebra [6, 27, 35] and an effective version of the Nullstellensatz  in algebraic geometry [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Moreover, a special case of Skoda’s ideal generation also  played key roles in Siu’s works on the deformation invariance of plurigenera [42] and finite  generation of the canonical ring [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' The interaction between several complex variables,  complex algebraic geometry and partial differential equations has been an attractive area  for the researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' For the sake of reader’s convenience, we state a version of Skoda’s L2  division theorem as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' ([45], Théorème 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let (X, ω) be an n-dimensional weakly pseudoconvex  Kähler manifold with ϕ ∈ Psh(X), and g : E → Q be a surjective morphism of Hermitian  holomorphic vector bundles with rE = rank E and rQ = rank Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Suppose that E is Nakano  semi-positive on X and L → X is a Hermitian line bundle such that  √  −1Θ(L) − ρ  √  −1Θ(det Q) ≥ 0  for ρ = min{n, rE − rQ} + ε and some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Then, for every f ∈ H0(X, Q ⊗ KX ⊗ L) satisfying  �  X  ⟨�  gg∗ f, f⟩ · (det gg∗)−ρ−1e−2ϕdVω < +∞,  there exists h ∈ H0(X, E ⊗ KX ⊗ L) such that f = g · h and  �  X  |h|2 · (det gg∗)−ρe−2ϕdVω ≤ ρ  ε ·  �  X  ⟨�  gg∗ f, f⟩ · (det gg∗)−ρ−1e−2ϕdVω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Due to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1, if we consider the trivial bundles E, Q and L on a pseudoconvex  domain, then by combining with the strong openness of multiplier ideal sheaves established  by Guan-Zhou [21], we can reformulate Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1 in the language of multiplier ideals  as follows (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' also Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3):  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be an n-dimensional complex manifold with ϕ ∈ QPsh(X) a quasi-psh  function and a ⊂ OX a nonzero ideal sheaf with r (local) generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, it follows that  I �ϕ + kϕa  � = a · I �ϕ + (k − 1)ϕa  �, ∀k ≥ min{n, r},  where ϕa := 1  2 log(�  i |gi|2) and (gi) is any local system of generators of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Motivated by the above reformulation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1, it is interesting for us to explore  an analogue to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2 for Nadel-Lebesgue multiplier ideals in the singular setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  In order to achieve such a goal, a natural idea is to generalize Skoda’s L2 methods to the  singular case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', creating an appropriate L2 theory for the ∂-operator on singular complex  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' However, as presented in [15, 16], it seems not to be possible to establish a general  theory as in the smooth setting to solve the ∂-equation with L2 estimates on complex spaces  with singularities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' one can refer to [17, 38, 39, 41] for some partial results on the related  topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES  3  On the other hand, we can also consider to apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1 near the singularities  under some reasonable assumptions on the positivity of curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Fortunately, we could  show that positivity of the Ricci curvature on the regular locus is in fact equivalent to the  desired Skoda’s ideal generation and L2-solvability of the ∂-equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' More precisely, we  state our main result in the following:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a (Hermitian) complex space of pure dimension n with x ∈ X a  normal point and π : �X → X a log resolution of the Jacobian ideal JacX of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, the  following statements are equivalent:  (1) For each quasi-psh function ϕ near the point x ∈ X, we have  Rqπ∗  �  O�X(�K�X/X) ⊗ I (ϕ ◦ π)  �  = 0, ∀q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (2) For each quasi-psh function ϕ near the point x ∈ X, we have  Rqπ∗  �  O�X(�K�X/X) ⊗ I (ϕ ◦ π)  �  = 0, ∀1 ≤ q < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (3) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric  ω on Ω such that the Ricci curvature Ric(ω) ≥ 0 on the regular locus Ωreg of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (4) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric  ω and a C ∞ differentiable real function ψ on Ω such that Ric(ω) +  √  −1∂∂ψ ≥ 0  on the regular locus Ωreg of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (5) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric  ω and a Hermitian line bundle L on Ω such that for any smooth ϕ ∈ SPsh(Ω) and  v ∈ L2  0,q(Ωreg, L) satisfying ∂v = 0 and  �  Ωreg  ⟨A−1  ϕ v, v⟩ e−2ϕdVω < +∞  with the curvature operator Aϕ = [  √  −1∂∂ϕ, Λω] on Ωreg, we have u ∈ L2  0,q−1(Ωreg, L)  such that ∂u = v and  �  Ωreg  |u|2e−2ϕdVω ≤  �  Ωreg  ⟨A−1  ϕ v, v⟩ e−2ϕdVω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (6) For some Stein neighborhood Ω ⊂⊂ X of the point x, there exists a Kähler metric  ω and a Hermitian line bundle L on Ω such that for any smooth ϕ ∈ SPsh(Ω) and  v ∈ L2  0,1(Ωreg, L) satisfying ∂v = 0 and  �  Ωreg  ⟨A−1  ϕ v, v⟩ e−2ϕdVω < +∞,  we have u ∈ L2(Ωreg, L) such that ∂u = v and  �  Ωreg  |u|2e−2ϕdVω ≤  �  Ωreg  ⟨A−1  ϕ v, v⟩ e−2ϕdVω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (7) The Skoda’s theorem holds for Nadel-Lebesgue multiplier ideals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', for any  nonzero ideal sheaf a with r generators and quasi-psh function ϕ near the point  x ∈ X, it holds that  INL  �ϕ + kϕa  � = a · INL  �ϕ + (k − 1)ϕa  �, ∀k ≥ min{n, r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (8) For any nonzero ideal sheaf a near the point x ∈ X, it holds that  INL  �nϕa  � = a · INL  �(n − 1)ϕa  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (9) For any nonzero ideal sheaf a near the point x ∈ X, it holds that  INL  �nϕa  � ⊂ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (10) The point x ∈ X is a regular point of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  4  ZHENQIAN LI  In the above result, the most interesting and amazing point is that it presents several  characterizations of regular points by various statements involved Nadel-Lebesgue mul-  tiplier ideals, which look like almost irrelevant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', (1, 2) in algebraic geometry, (3, 4)  in differential geometry and (5, 6) in partial differential equations together with (7—9) in  commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' The core idea of all arguments originates from the Skoda’s ideal  generation by the L2 approaches in several complex variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Simple examples show that the assumption that x is a normal point of X  cannot be removed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' in particular, any of the statements (1, 2, 7, 8) will not  imply (10) in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  As a straightforward consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3, we have  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Any normal Kähler space with nonnegative Ricci curvature on the regular  locus must be non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Kähler-Einstein metrics on singular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a normal Q-Gorenstein  Kähler space, that is, a normal Kähler space whose canonical class KX defines a Q-line  bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' A Kähler current ω ∈ c1(±KX) is called a weak (or singular) Kähler-Einstein  metric on X if ω has bounded local potentials and is a genuine Kähler-Einstein metric on  the regular locus Xreg of X (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' [3, 4, 5, 14], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' A weak Kähler-Einstein metric ω on  X is called a Kähler-Einstein metric if ω is a Kähler metric on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', ω has smooth local  potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' For general expositions on the topic of Kähler-Einstein metrics one can refer to  [2, 23, 47, 48, 49, 50] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In particular, we state some recent results  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' ([3, 5, 14, 30, 31, 32, 36], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a normal Q-Gorenstein complex  projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then:  (1) If X is a Q-Calabi-Yau variety with only log terminal singularities, then X admits  a weak Kähler-Einstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (2) If KX is ample, then X admits a weak Kähler-Einstein metric if and only if X is  K-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (3) If −KX is ample, then X admits a weak Kähler-Einstein metric if and only if X is  K-polystable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  A basic and widely open problem in Kähler geometry/geometric analysis is understand-  ing the geometric asymptotic behavior of the weak Kähler-Einstein metric near the singular  locus Xsing of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In [24], the authors made a breakthrough with a very precise descrip-  tion for a class of Calabi-Yau varieties with smoothable isolated singularities, which are  in further required to be isomorphic to a neighborhood of the vertex in a strongly regular  Calabi-Yau cone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' see also [7, 8, 18] for some recent progress in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In more gen-  eral situations, by using deep tools in the theory of degenerate complex Monge-Ampère  equations on singular complex spaces, the continuity of local potentials of weak Kähler-  Einstein metrics is established for all Q-Fano/Calabi-Yau varieties in [4, 22], but so far  little is known for the higher order regularity in general and it is desirable to establish one  for weak Kähler-Einstein potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' However, relying on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3, we will see that  too much regularity cannot be expected and in fact any weak Kähler-Einstein potential is at  most C α (α < 2) differentiable near the singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In particular, we obtain the following  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a normal Q-Gorenstein Kähler space admitting a weak Kähler-  Einstein metric ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, ω is smooth on X if and only if X is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Preliminaries  Firstly, we introduce the notion of Nadel-Lebesgue multiplier ideal sheaf on any com-  plex space of pure dimension and then present some useful facts used throughout this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES  5  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a complex space of pure dimension and ϕ ∈ L1  loc(Xreg) with  respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, the complex space X is said to be a Hermitian  complex space if there is a Hermitian metric ω on the regular part (may be disconnected)  Xreg of X such that ω is locally the restriction of a Hermitian metric on some CN for a local  embedding of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' It follows from the differentiable partition of unity that every complex  space is a Hermitian complex space as in the smooth case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  The complex space X is called to be a Kähler space if there is a Hermitian metric ω on X  such that ω is locally the restriction of a Kähler metric on some CN for a local embedding  of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In particular, it admits smooth strictly psh functions as local potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  We say that the function ϕ is quasi-plurisubharmonic (quasi-psh for short) on X if it is  locally equal to the sum of a psh function and of a smooth function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' The set of quasi-  psh (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' psh and strictly psh) functions on X is denoted by QPsh(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Psh(X) and  SPsh(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' A quasi-psh function ϕ ∈ QPsh(X) will be said to have analytic singularities on  X if ϕ can be written locally as  ϕ = c  2 log(|f1|2 + · · · + |fN0|2) + O(1),  where c ∈ R≥0 and (fi) are holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let (X, ω) be a Hermitian complex space of pure dimension and ϕ ∈  L1  loc(Xreg) with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  The Nadel-Lebesgue multiplier ideal sheaf associated to ϕ on X is defined to be the  OX-submodule INL(ϕ) ⊂ MX of germs of meromorphic functions f ∈ MX,x such that  |f|2e−2ϕ is integrable with respect to the Lebesgue measure dVω near the point x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' One  can check that INL(ϕ) is independent of the choice of Hermitian metric ω on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  The log canonical threshold (or complex singularity exponent) LCTx(ϕ) of ϕ at a point  x ∈ X is defined to be  LCTx(ϕ) := sup �c ≥ 0 | OX,x ⊂ INL(cϕ)x  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  It is convenient to put LCTx(−∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  It is easy to see that INL(ϕ) ⊂ OX is an ideal sheaf when X is a normal complex space  and ϕ is locally bounded from above on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In addition, if X is smooth and ϕ ∈ QPsh(X),  then INL(ϕ) is nothing but the usual multiplier ideal sheaf I (ϕ) introduced by Nadel (see  [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Since the definition of Nadel-Lebesgue multiplier ideals is local, we can com-  pute the multiplier ideals by choosing a special Hermitian metric ω for a local embedding  of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In particular, if X is an n-dimensional complex subspace of some domain in CN,  we can take Hermitian metric ω on X to be the inherited standard Kähler metric from CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Then, we have dVω = 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='υn|Xreg, where υ =  √  −1  2  N�  k=1  dzk ∧ d¯zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  For the sake of reader’s convenience, we state a basic estimate related to local volume  of an analytic subset as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' ([20], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a pure n-dimensional analytic subset through  the origin 0 of some domain in CN (N ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, there is a Stein neighborhood U ⊂⊂ CN  of the origin 0 such that for any 0 ≤ ε < 1, we have  �  U∩X  1  (|z1|2 + · · · + |zN|2)n−1+ε dVω < +∞,  where dVω = 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='υn|Xreg and υ =  √  −1  2  N�  k=1  dzk∧d¯zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Analogous to the Nadel-Ohsawa multiplier ideal sheaves introduced in [33, 34] (see also  [9, 12] for the algebro-geometric counterpart), we state some related properties as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  6  ZHENQIAN LI  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' (1) Let π : �X → X be a log resolution of the Jacobian ideal JacX of X  and �K�X/X be the Mather discrepancy divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, we have the image  Im  �  π∗Ωn  X ֒→ Ωn  �X  �  = O�X(−�K�X/X) · Ωn  �X,  and  INL(ϕ) = π∗  �  O�X(�K�X/X) ⊗ I (ϕ ◦ π)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Furthermore, we can deduce that INL(ϕ + log |JacX|) = INO(ϕ), the Nadel-Ohsawa  multiplier ideal sheaf associated to ϕ on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (2) When X is normal and ϕ has analytic singularities, INL(ϕ) coincides with the  Mather multiplier ideal sheaf defined in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (3) For any ϕ ∈ QPsh(X), it follows that INL(ϕ) ⊂ MX is a coherent fractional ideal  sheaf and satisfies the strong openness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', INL(ϕ) = �  ε>0  INL  �(1 + ε)ϕ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  For our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3, we need the following L2 estimates for the ∂-equation  and relative version of Grauert-Riemenschneider vanishing theorem for the higher direct  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' [10], Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let (X, ω) be an n-dimensional Kähler manifold,  which contains a weakly pseudoconvex Zariski open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let L be a Hermitian line  bundle on X such that  √  −1Θ(L) + Ric(ω) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Then, for every smooth ϕ ∈ Psh(X) and v ∈ L2  0,q(X, L) satisfying ∂v = 0 and  �  X  ⟨A−1v, v⟩ e−2ϕdVω < +∞  with the curvature operator A = [  √  −1Θ(L) + Ric(ω) +  √  −1∂∂ϕ, Λω] on X, there exists  u ∈ L2  0,q−1(X, L) such that ∂u = v and  �  X  |u|2e−2ϕdVω ≤  �  X  ⟨A−1v, v⟩ e−2ϕdVω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' ([11], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let (X, ω) be an n-dimensional Kähler manifold which  is a Zariski open subset of some Stein space X∗, and L be a Hermitian line bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  If for any smooth ϕ ∈ SPsh(X∗) and v ∈ L2  0,1(X, L) satisfying ∂v = 0 and  �  X  ⟨A−1  ϕ v, v⟩ e−2ϕdVω < +∞  with the curvature operator Aϕ = [  √  −1∂∂ϕ, Λω] on X, there exists u ∈ L2(X, L) such that  ∂u = v and  �  X  |u|2e−2ϕdVω ≤  �  X  ⟨A−1  ϕ v, v⟩ e−2ϕdVω,  then it follows that L ⊗ K−1  X is Nakano semi-positive on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' ([37], Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let π : X → Y be a surjective proper (locally) Kähler  morphism from a complex manifold X to a complex space Y, and (L, e−ϕL) be a (possibly  singular) Hermitian line bundle on X with semi-positive curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, the higher direct  image sheaf  Rqπ∗  �  KX ⊗ L ⊗ I (ϕL)  �  = 0,  for every q > dim X − dim Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Any log resolution π : �X → X of a coherent ideal sheaf I on a complex space  X is a locally Kähler (proper modification), which is locally a finite sequence of blow-ups  with smooth centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Besides, any finite holomorphic mapping between complex spaces is  (locally) proper Kähler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES  7  In the remainder of this section, we recall some algebraic properties on the integral  closure of ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' ([46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let R be a commutative ring and I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' An element  f ∈ R is said to be integrally dependent on I if it satisfies a relation  f d + a1 f d−1 + · · · + ad = 0  (ak ∈ Ik, 1 ≤ k ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  The set I consisting of all elements in R which are integrally dependent on I is called  the integral closure of I in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' I is called integrally closed if I = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' One can prove that I is  an ideal of R, which is the smallest integrally closed ideal in R containing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' ([46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let R be a commutative ring with identity and let J ⊂ I be ideals in  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' J is said to be a reduction of I if there exists a nonnegative integer n such that In+1 = JIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  A reduction J of I is called minimal if no ideal strictly contained in J is a reduction of  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' An ideal that has no reduction other than itself is called basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  One can prove that minimal reductions do exist in Noetherian local rings and an ideal  which is a minimal reduction of a given ideal is necessarily basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Moreover, if R is a  Noetherian ring, J ⊂ I is a reduction of I if and only if J = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  In the analytic setting, we have the following characterization on integral closure and  reduction of ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' [29], Théorème 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a complex space and Y ⊂ X be a  proper closed complex subspace (may be non-reduced) defined by a coherent OX-ideal I  with x ∈ Y a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let J ⊂ OX be a coherent OX-ideal and I (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' J) be the germ of  I (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' J ) at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, the following conditions are equivalent:  (1) J ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (2) For every morphism π : �X → X satisfying: (i) π is a proper and surjective, (ii) �X is  a normal complex space and (iii) I · O�X is an invertible OX-module, there exists  an open neighborhood U of x in X such that  J · O�X|π−1(U) ⊂ I · O�X|π−1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (3) If V is an open neighborhood of x on which I and J are generated by their  global sections, then for every system of generators g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', gr ∈ Γ(V, I ) and every  f ∈ Γ(V, J ), one can find an open neighborhood V′ of x and a constant C > 0  such that  |f(y)| ≤ C · sup  k  |gk(y)|, ∀y ∈ V′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a normal complex space and I ⊂ OX a coherent ideal sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let  π : �X → X be any proper modification from a normal complex space �X onto X such that  I · O�X = O�X(−D) for some effective Cartier divisor D on �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, we have π∗O�X(−D) =  I , the integral closure of I in OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='19 in [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' see also [10], Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a  normal complex space of dimension n and a ⊂ OX a nonzero ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, there exists  an open covering {Uα}α∈N of X such that a|Uα has a reduction bα generated by at most n  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Proofs of the main results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Since all of the statements are local, without loss of gener-  ality, we may assume that X is an n≥2-dimensional normal (Hermitian) complex subspace  of some domain in CN with ϕ ∈ QPsh(X) and a = (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', gr) · OX an ideal sheaf gen-  erated by holomorphic functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , gr on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Moreover, we may also assume that ϕ  is (locally) a strictly psh function on X if necessary, by adding some smooth strictly psh  8  ZHENQIAN LI  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' It is easy to see that the implications (1) =⇒ (2), (3) =⇒ (4), (5) =⇒ (6) and  (7) =⇒ (8) =⇒ (9) are trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' in particular, we will present a proof in the following order:  (1)  � (2)  �❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  (10)  �④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  ④  �  (9)  �  (8)  �  (7)  �  (3)  �❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  � (4)  �  (5)  � (6)  �⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  “(2) =⇒ (7)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' By the definition of Nadel-Lebesgue multiplier ideal sheaf, it follows  that  a · INL  �ϕ + (k − 1)ϕa  � ⊂ INL  �ϕ + kϕa  �,  and so it is sufficient to show the reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' When r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Let π : �X → X be a common log resolution of JacX and a such that a · O�X = O�X(−F)  for some effective divisors F on �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Denote by  Am := O�X(�K�X/X) ⊗ I (ϕ ◦ π + mϕa ◦ π)  = O�X(�K�X/X) ⊗ I (ϕ ◦ π) ⊗ O�X(−mF)  for any m ∈ N, and consider the Koszul complex determined by g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', gr:  0 → ΛrV ⊗ O�X(rF) → · · · → Λ2V ⊗ O�X(2F) → V ⊗ O�X(F) → O�X → 0,  where V is the vector space spanned by g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Note that the Koszul complex is locally  split and its syzygies are locally free, so twisting through by any coherent sheaf will pre-  serve the exactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, by twisting with Ak (k ≥ r), we obtain the following long exact  sequence  0 → ΛrV ⊗ Ak−r → · · · → Λ2V ⊗ Ak−2 → V ⊗ Ak−1 → Ak → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (⋆)  On the other hand, for any m ∈ N, by (2) we have the local vanishing of the higher direct  images Rqπ∗Am = 0 (1 ≤ q < n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Note that  INL  �ϕ + mϕa  � = π∗Am  by the functoriality property with respect to direct images of sheaves by modifications, and  then by taking direct images of (⋆) we will deduce the following so-called exact Skoda  complex (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' [28], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' 228):  0 → ΛrV ⊗ INL  �ϕ + (k − r)ϕa  � → · · · → V ⊗ INL  �ϕ + (k − 1)ϕa  � → INL  �ϕ + kϕa  � → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  In particular, the map V ⊗ INL  �ϕ + (k − 1)ϕa  � → INL  �ϕ + kϕa  � is surjective, by which we  can infer that INL  �ϕ + kϕa  � ⊂ a · INL  �ϕ + (k − 1)ϕa  � for any k ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' When r > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  As the statement is local, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='14 we may assume that b is a reduction of  a generated by n elements �g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=',�gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Consider a common log resolution π : �X → X of  JacX, a and b such that a · O�X = b · O�X = O�X(−F) for some effective divisors F on �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Then, by the same argument as above, we can deduce the following exact Skoda complex:  0 → ΛnV ⊗ INL  �ϕ + (k − n)ϕa  � → · · · → V ⊗ INL  �ϕ + (k − 1)ϕa  � → INL  �ϕ + kϕa  � → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES  9  for any k ≥ n, where V is the vector space spanned by �g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=',�gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Therefore, it follows that  INL  �ϕ + kϕa  � ⊂ b · INL  �ϕ + (k − 1)ϕa  � ⊂ a · INL  �ϕ + (k − 1)ϕa  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  “(3) =⇒ (5)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' It follows from the assumption that we have a Stein neighborhood Ω ⊂⊂  X of the point x with a Kähler metric ω such that Ric(ω) ≥ 0 on Ωreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let ϕ ∈ SPsh(Ω) be  any smooth strictly psh function on Ω and L = Ω × C be a trivial bundle equipped with the  trivial Hermitian metric, which implies that  √  −1Θ(L) + Ric(ω) +  √  −1∂∂ϕ ≥  √  −1∂∂ϕ > 0  on Ωreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Since Ω is a Stein space, we are able to choose a complex hypersurface Z ⊂ Ω which  contains the singular locus Ωsing of Ω such that Ω − Z ⊂ Ωreg is a Stein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, by  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='6 we obtain that, for any smooth ϕ ∈ SPsh(Ω) and v ∈ L2  0,q(Ωreg, L) satisfying  ∂v = 0 and  �  Ωreg  ⟨A−1  ϕ v, v⟩ e−2ϕdVω < +∞,  we can find u ∈ L2  0,q−1(Ωreg, L) such that ∂u = v and  �  Ωreg  |u|2e−2ϕdVω ≤  �  Ωreg  ⟨A−1  ϕ v, v⟩ e−2ϕdVω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  “(6) =⇒ (4)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' As a straightforward application of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='7 on Ωreg, it yields that  √  −1Θ(L) + Ric(ω) ≥ 0 on Ωreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let Ω′ ⊂ Ω be a small Stein neighborhood of the point  x such that the Hermitian line bundle L has a smooth potential ψ on Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Therefore, we  deduce that  √  −1∂∂ψ + Ric(ω) =  √  −1Θ(L) + Ric(ω) ≥ 0  on Ω′  reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  “(4) =⇒ (7)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Due to the definition and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='14, it is sufficient to prove  INL  �ϕ + kϕa  � ⊂ a · INL  �ϕ + (k − 1)ϕa  �  for the case r ≤ n near the point x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let f ∈ INL  �ϕ + kϕa  �  x with k ≥ min{n, r} = r,  then by the strong openness of multiplier ideals there exists small enough ε > 0 such that  f ∈ INL  �ϕ + (k + ε)ϕa  �  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  By the assumption of (4), we let Ω ⊂⊂ X be a Stein neighborhood of the point x with  a Kähler metric ω and a smooth real function ψ on Ω such that Ric(ω) +  √  −1∂∂ψ ≥ 0 on  Ωreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' After shrinking Ω if necessary, we may assume that the function ψ is bounded on Ω  and f is holomorphic on Ω such that  �  Ω  |f|2 · |g|−2(r+ε)e−2(ϕ+(k−r)ϕa)dVω < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  In addition, we also choose a complex hypersurface Z ⊂ Ω which contains the singular  locus Ωsing of Ω and the common zero-set of holomorphic functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', gr such that  Ω′ := Ω − Z is a Stein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Let E = Ω′ × Cr and Q = Ω′ × C be the trivial bundles on Ω′ and L = K−1  Ω′ be the  anti-canonical line bundle with the induced metric twisted by a weight e−ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' The morphism  g : E → Q determined by holomorphic functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', gr is given by  (h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', hr) �→  r�  m=1  gm · hm = g · h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Note that �  gg∗ = IdQ when rank Q = 1, and on Ω′ we have  √  −1Θ(L) − (r − 1 + ε)  √  −1Θ(det Q) = Ric(ω) +  √  −1∂∂ψ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  10  ZHENQIAN LI  Thus, we can apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1 on Ω′ and then obtain an r-tuple (h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', hr) of holomor-  phic functions on Ω′ such that f = g · h on Ω′ and  �  Ω′ |h|2 · |g|−2(r−1+ε)e−2(ϕ+(k−r)ϕa)dVω =  �  Ω′ |h|2e−2(ϕ+(k−1+ε)ϕa)dVω < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  We can now extend every hm to be a holomorphic function on Ω from the L2 estimate above  and normality of X, which implies that  INL  �ϕ + kϕa  � ⊂ a · INL  �ϕ + (k − 1 + ε)ϕa  � ⊂ a · INL  �ϕ + (k − 1)ϕa  �  on Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' we finish the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  “(9) =⇒ (10)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' By the assumption, we have INL  �nϕa  � ⊂ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Suppose that x ∈ X  is a singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, by the local parametrization for analytic sets, we can find a  local coordinate system (z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' z′′) = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', zn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' zn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', zN) near x such that for some constant  C > 0, we have |z′′| ≤ C · |z′| for any point z ∈ X near x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Let a ⊂ OX be the ideal sheaf generated by holomorphic functions �z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=',�zn ∈ OX  (shrinking X if necessary), where �zk are the residue classes of zk in OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' From the non-  smoothness of X at the point x, we deduce that the embedding dimension dimC(mX,x/m2  X,x) ≥  n + 1 of X at x, which implies that there exists k0 (n + 1 ≤ k0 ≤ N) such that�zk0 � a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  On the other hand, after shrinking X again, it follows that  �  X  |zk0|2  |z′|2n dVω ≤ C2(1 + C2)n−1 ·  �  X  |z|−2(n−1)dVω < +∞,  where the finiteness of the integration follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, we infer that�zk0 ∈  INL  �nϕa  �, but�zk0 � a, which contradicts to the assumption INL  �nϕa  � ⊂ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Thus, we obtain  that x ∈ X is a regular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  “(10) =⇒ (1)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' It is a straightforward consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  “(10) =⇒ (3)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Since x is a regular point of X, after choosing an appropriate coordinate  neighborhood of x, we may assume that Ω ∋ x is a Stein domain in Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Therefore, we can  take ω =  √  −1  2  n�  k=1  dzk ∧ d¯zk to be the standard Euclidean metric on Cn and then we have  Ric(ω) = 0 on Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3 is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In addition, we can deduce from the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3 that  (i) if (1) or (2) holds for each quasi-psh function ϕ with analytic singularities, then x ∈ X  is a regular point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (ii) both of the statements (3) and (4) could be respectively modified to be Ric(ω) ≥ 0  and Ric(ω) +  √  −1∂∂ψ ≥ 0 on a Zariski open subset of Ω contained in Ωreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' It is sufficient to prove the necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Let x ∈ X be any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Since ω is a smooth Kähler metric on X, then ω has smooth  local potentials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', there exists a Stein neighborhood Ω ⊂ X of x and a smooth strictly psh  functions ψ on Ω such that ω =  √  −1∂∂ψ on Ωreg, which implies that Ric(ω)+  √  −1∂∂ψ ≥ 0  on Ωreg whenever Ric(ω) = ±ω, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Thus, it follows from (4) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3 that x ∈ X is  a regular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' The same arguments as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='7 also imply that  each local potential of weak Kähler-Einstein metric ω is C 2 differentiable on X if and only  if X is non-singular, and that there exists no singular normal Kähler space such that the  Kähler metric is Kähler-Einstein on the regular locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  In fact, our method is still available when the weak Kähler-Einstein metric is (locally)  equivalent to the standard induced Kähler metric by restriction near the singularities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' for  instance, when the weak Kähler-Einstein metric is of locally bounded coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES  11  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Uniform bounds of powers associated to an L2 division problem  The ideal membership is an important object to study in commutative algebra, algebraic  geometry and several complex variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', the famous Hilbert’s Nullstellensatz and  Briançon-Skoda theorem and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In this part, we are mainly interested in the uniform  bounds of powers associated to an L2 division problem, a kind of special ideal membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Let X be a Stein manifold of dimension n and a = (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , gr)·OX an ideal sheaf generated  by holomorphic functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , gr on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In general, the division problem states that,  given a positive integer k ∈ N and holomorphic function f on X, we wish to determine  when f is generated by holomorphic functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', gr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' more precisely, when we can  find holomorphic functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , hr ∈ ak−1 on X such that  f =  r�  m=1  gm · hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Thanks to the Oka-Cartan’s theory on Stein manifolds, the division problem is solvable if  and only if f ∈ ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Note that the condition f ∈ ak is purely algebraic, and so it is natural to ask whether we  could find an analytic condition to replace the algebraic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' It is easy to see that f ∈ ak  implies that |f|e−ϕk is locally bounded on X, or L2  loc more generally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' where ϕk := k log |g|  and |g|2 := |g1|2 + · · · + |gr|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' On the other hand, local boundedness of |f|e−ϕk is equivalent  to the fact that f ∈ ak, the integral closure of ak in OX (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Thus, it is  an interesting question whether we could establish solvability of an L2 analogue of the  division problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Let ϕ ∈ Psh(X) be a psh function on X and denote by  A2  loc(X, ϕ) :=  �  f ∈ OX(X)  ��� |f|2e−2ϕ is locally integrable on X  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Then, we raise the following L2 division problem:  Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be an n-dimensional Stein manifold with a psh function ϕ ∈ Psh(X),  and a = (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , gr) · OX an ideal sheaf generated by holomorphic functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', gr on  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Given positive integer k ∈ N and f ∈ A2  loc(X, ϕ + ϕk), are there holomorphic functions  h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', hr ∈ A2  loc(X, ϕ + ϕk−1) such that  f =  r�  m=1  gm · hm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' A solution to Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Unfortunately, the answer of Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1 is negative  for general k (see Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Motivated by the Skoda’s L2 division theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' The-  orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1), it seems to be reasonable to find a uniform integer k0, depending only on n,  such that Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1 is solvable for any k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' The goal of this part is to present an  optimal uniform lower bounds of powers associated to Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In particular, we will  establish the following  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' There exists a uniform integer k0 = min{n, r} such that the solution to  Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1 is positive for any k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In further, the uniform lower bound k0 = min{n, r}  is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  In fact, the optimality of uniform integer k0 = min{n, r} is straightforward by the fol-  lowing:  Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let Bn(0) be the unit ball centered at the origin 0 = (0′, 0′′) in Cr×Cn−r (1 ≤  r ≤ n) and take g1 = z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', gr = zr, f ≡ 1, ϕ ≡ 0 on Bn(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, for every k < k0 = r, the  answer of Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Indeed, by the fact that the log canonical threshold LCT(0′,z′′)(ϕk) = r  k > 1 of ϕk at any  point (0′, z′′), one can derive that f ∈ A2  loc(Bn(0), ϕ + ϕk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, we infer from the fact that  12  ZHENQIAN LI  f has no zeros in Bn(0) that there exist no holomorphic functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , hr on Bn(0) such  that f =  r�  m=1  gm · hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' It follows from the local vanishing (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='8) and the argu-  ments as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3 that for any k ≥ min{n, r}, we have  I �ϕ + ϕk  � = a · I �ϕ + ϕk−1  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Let  τ : I �ϕ + ϕk−1  �⊕r −→ I �ϕ + ϕk  �  be the sheaf homomorphism defined by  τ(h1,x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , hr,x) =  r�  m=1  gm · hm,x  for any germs hm,x ∈ I �ϕ + ϕk−1  �  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, we have an exact sequence of sheaves  I �ϕ + ϕk−1  �⊕r  τ  −→ I �ϕ + ϕk  � −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  It follows from the Oka-Cartan theory on Stein manifolds that the induced sequence of  sections  Γ  �  X, I �ϕ + ϕk−1  �⊕r�  τ∗  −→ Γ  �  X, I �ϕ + ϕk  ��  −→ 0  is also exact, which implies that any section f ∈ Γ  �  X, I �ϕ + ϕk  ��  can be written as the  image f =  r�  m=1  gm · hm for some sections hm ∈ Γ  �  X, I �ϕ + ϕk−1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  □  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' (An alternative argument on Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' In fact, we could also give an-  other argument on the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1 depending on the strong openness of multi-  plier ideals established by Guan-Zhou [21] and the Skoda’s L2 division theorem for holo-  morphic functions (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Since the statement is local, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='14 that it is sufficient to prove  I �ϕ+ϕk  � ⊂ a·I �ϕ+ϕk−1  � for the case r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Given f ∈ Γ  �  X, I �ϕ+ϕk  ��  , after shrinking  X, we may assume that X is the unit ball in Cn and  �  X  |f|2e−2(ϕ+ϕk)dλn =  �  X  |f|2 · |g|−2ke−2ϕdλn < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Then, for each k ≥ r, by the strong openness of multiplier ideals there exists sufficiently  small ε > 0 such that  �  X  |f|2e−2(ϕ+(1+ε)ϕk)dλn =  �  X  |f|2 · |g|−2(1+ε)ke−2ϕdλn < +∞,  shrinking X if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Finally, combining with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1, we deduce the desired  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' A global L2 version of Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let (X, ω) be an n-dimensional Stein manifold  with a Kähler form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let ϕ ∈ Psh(X) and I = (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , gr) · OX an ideal sheaf generated  by holomorphic functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' , gr on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Denote by  A2(X, ϕ) :=  �  f ∈ OX(X)  �����  �  X  |f|2e−2ϕdVω < +∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Then, we have the following global analogue of Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1:  Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Can we find a uniform integer k0 such that for each k ≥ k0 and f ∈ A2(X, ϕ+  ϕk), there exist h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=', hr ∈ A2(X, ϕ + ϕk−1) satisfying  f =  r�  m=1  gm · hm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  As an immediate consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='1, we obtain the following  ON SKODA’S THEOREM FOR NADEL-LEBESGUE MULTIPLIER IDEAL SHEAVES  13  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Let X be a pseudoconvex domain in Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Then, there exists a uniform integer  k0 = min{n + 2, r + 1} such that the solution to Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' (1) More generally, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='4 also holds for any complete Kähler domain  in Cn with smooth psh function ϕ ∈ Psh(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='  (2) In this case, combining with the Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='2, it follows that the optimal uniform  lower bound k0 is at least min{n, r}, and at most min{n + 2, r + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
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+page_content='  [50] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation  I, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
+page_content=' 31 (1978), 339–411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQfSvfs/content/2301.00094v1.pdf'}
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+Fighting the sign problem in a chiral random matrix model with contour deformations
+Matteo Giordano,1 Attila Pásztor,1 Dávid Pesznyák,1 and Zoltán Tulipánt1
+1ELTE Eötvös Loránd University, Institute for Theoretical Physics,
+Pázmány Péter sétány 1/A, H-1117, Budapest, Hungary
+We studied integration contour deformations in the chiral random matrix theory of Stephanov [1]
+with the goal of alleviating the finite-density sign problem. We considered simple ansätze for the
+deformed integration contours, and optimized their parameters.
+We find that optimization of a
+single parameter manages to considerably improve on the severity of the sign problem. We show
+numerical evidence that the improvement achieved is exponential in the degrees of freedom of the
+system, i.e., the size of the random matrix. We also compare the optimization method with contour
+deformations coming from the holomorphic flow equations.
+I.
+INTRODUCTION
+Euclidean quantum field theories at non-zero particle
+density (or chemical potential) generally suffer from a
+complex action problem: the weights in the path integral
+representation are complex, and thus cannot be inter-
+preted as a joint probability density function on the space
+of field configurations (up to a proportionality factor).
+This prevents the use of importance sampling methods
+for the direct simulation of these theories. In QCD, this
+complex action problem severely hampers first-principles
+studies of dense matter in the core of neutron stars, neu-
+tron star mergers, core collapse supernovae, as well as in
+heavy ion collisions at certain collision energies.
+In the presence of a complex action problem one can
+still (in principle) simulate a modified theory with real
+and positive weights, and then use reweighting methods
+to calculate observables in the theory of interest. If the
+target theory has field variables φ, path integral weights
+wt(φ), and partition function Zt =
+�
+Dφ wt(φ), and the
+simulated theory has the same field variables, but dif-
+ferent – real and positive – path integral weights ws(φ)
+and partition function Zs =
+�
+Dφ ws(φ), we can obtain
+expectation values in the target theory via the formula
+⟨O⟩t =
+�
+wt
+ws O
+�
+s
+�
+wt
+ws
+�
+s
+,
+⟨O⟩x = 1
+Zx
+�
+Dφ wx(φ)O(φ) , (1)
+where x may stand for t or s and O(φ) is some phys-
+ical observable of interest. The denominator in Eq. (1)
+gives the ratio of the partition functions in the target and
+simulated theories, i.e.,
+� wt
+ws
+�
+s
+= Zt
+Zs
+.
+(2)
+This ratio is typically exponentially small in the physical
+volume, with the exponent given by the free energy dif-
+ference between the target and simulated theories. This
+ratio is also a rough measure of the numerical difficulty
+of a given reweighting scheme, with a given simulated
+and target theory. In order for reweighting to be effec-
+tive, one wants the target and simulated theories to be as
+close to each other as possible. Ideally, one should find a
+simulated theory with Zs ≈ Zt.
+Two simple choices of a simulated theory are the phase-
+quenched (PQ) theory, with simulated weights propor-
+tional to
+wPQ
+s
+≡ |wt(φ)| ,
+(3)
+or – assuming that the partition function Zt is real –
+the sign-quenched (SQ) theory, with simulated weights
+proportional to
+wSQ
+s
+≡ |Re wt(φ)| .
+(4)
+For the first case (phase reweighting) the reweighting
+factors wt/wPQ
+s
+≡ eiθ are pure phases.
+For the sec-
+ond case (sign reweighting) the reweighting factors are
+wt/wSQ
+s
+= eiθ/ |cos θ|. For certain observables, such as
+manifestly real observables or observables with a con-
+jugation (φ → φ) symmetry, one can substitute wt/wPQ
+s
+with cos θ and wt/wSQ
+s
+with a pure sign cos θ/ |cos θ|. For
+phase or sign reweighting, we can then say that the com-
+plex action problem becomes a sign problem: the cancel-
+lations between contributions with different signs of cosθ
+lead to a small Zt
+Zs ratio, and in turn to small signal-to-
+noise ratios in the expectation values of observables.
+The sign-quenched ensemble always has a less severe
+sign problem, due to the inequality Zt < ZSQ
+s
+< ZPQ
+s
+,
+which is a consequence of cos θ ≤ | cos θ| ≤ 1. However, in
+the limit of a severe sign problem – i.e., as the distribution
+of the argument θ tends to to a uniform distribution on
+[−π, π) – the severity of the sign problem for these two
+reweighting schemes only differs by a constant factor [2],
+given by
+�
+ZPQ
+s
+/ZSQ
+s
+�2 → (π/2)2.
+In QCD and in other (more or less) QCD-like models,
+describing the interactions of several “flavors” of fermions,
+the path integral weights can be written schematically as
+wt(φ) = det M1(φ, µ1) . . . det MNf (φ, µNf )e−SB(φ), (5)
+where the fields φ are real bosonic variables and SB
+is the corresponding bosonic part of the action, Nf is
+the number of fermion flavors in the model, det Mk is
+the fermionic determinant of the kth flavor and µk is
+the corresponding chemical potential, for k = 1, . . . , Nf.
+arXiv:2301.12947v1  [hep-lat]  30 Jan 2023
+
+2
+The source of the sign problem is the fermionic determi-
+nant, which at non-zero µ is generally a complex number.
+Moreover, an important feature of the sign problem in
+QCD and QCD-like theories is that it tends to get much
+worse in the ranges of µ where zeros of the determinant
+in the complex µ plane become dense [3].
+Nonetheless, reweighting from the phase- and sign-
+quenched theories is starting to become feasible even in
+full QCD [2, 4], which has recently led to the calculation
+of the equation of state of a hot-and-dense quark-gluon
+plasma in the region of chemical potentials covered by
+the RHIC Beam Energy Scan [5]. However, the range of
+practical applicability of such an approach is limited both
+in volume and chemical potential by the smallness of the
+ratio Zt/Zs. Lacking a solution of the sign problem, it is
+then desirable to develop methods that at least alleviate
+it, to extend the range of parameters that reweighting
+methods can practically reach.
+One possible route to do this is the use of contour
+deformations in the path integral (see Ref. [6] for a re-
+cent review). If the path integral weights wt(φ) are holo-
+morphic functions of the field variables,1 the multivariate
+Cauchy theorem guarantees that complexified integration
+manifolds in the same homology class as the original one
+yield the same partition function. However, the phase-
+and sign-quenched integrands are not holomorphic, and
+therefore the phase- and sign-quenched partition func-
+tions are not invariant under such deformations. It may
+then be possible to bring the ratios Zt/Zs closer to unity,
+thus making reweighting more effective.
+There are different ways to deform integration con-
+tours. Historically, methods based on Lefschetz thimbles
+appeared first [6, 9–14]. Lefschetz thimbles are the dis-
+joint components of the integration contour defined by
+requiring that the imaginary part of the classical action
+is constant in each component. The thimble structure
+of theories with a fermionic determinant is usually quite
+complicated [15–19]. Simple toy models reveal the follow-
+ing features: i) cancellations between competing thimbles
+are very important for getting the correct results, and
+ii) the thimbles themselves are not smooth at the zeros
+of the fermionic determinant.
+Thus, the use of thim-
+bles might be impractical for such theories.
+However,
+Lefschetz thimbles are, in general, not the numerically
+optimal integration contours [20], i.e., they are not nec-
+essarily the contours with the largest Zt/Zs, so there is
+no need to concentrate solely on them.
+A second class of methods is based on numerical op-
+timization.
+The main idea here is to parametrize the
+integration manifold by a finite number of parameters,
+which are then optimized to make the sign problem
+as mild as possible.
+Such methods were applied to
+a one-dimensional integral [21], the 0+1D scalar the-
+1 A notable exception is lattice QCD with rooted staggered
+fermions [7, 8].
+ory [22], the 0+1D Polyakov-improved Nambu-Jona-
+Lasinio model [23], 0+1D QCD [24], 1+1D scalar field
+theory [25], the 1+1D Thirring model [26], the 2+1D
+Thirring model [27], Bose gases of several dimensions [28],
+1+1D U(1) gauge theory with a complex coupling con-
+stant [29] and the 2+1D XY model at finite density [30].
+Here, we apply contour optimization methods to a
+fermionic toy model that shares relevant technical fea-
+tures with finite chemical potential QCD: the chiral ran-
+dom matrix model proposed by Stephanov in Ref. [1].
+Since it is an exactly solvable model with a sign prob-
+lem, the Stephanov model is a very useful testbed for
+methods aimed at solving or alleviating the sign prob-
+lem.
+This model has been studied with the complex
+Langevin approach [31–33], which fails for this particu-
+lar model [34] even with the introduction of gauge cool-
+ing [33]. There are also preliminary results for this model
+with the tempered Lefschetz thimble method [35] which
+is based on parallel tempering [36] in the flow time of
+the holomorphic flow [11, 37]. This method – similarly
+to other flow-based methods — produces a weaker sign
+problem, albeit at the cost of substantially increasing the
+per-configuration-cost of generating the ensemble com-
+pared to ordinary phase reweighting.
+In this paper we study the Stephanov model with op-
+timization methods. There are, roughly speaking, two
+approaches to such an optimization: one can look for the
+optimum using either a very general ansatz with a large
+number of parameters, or a very specific ansatz tailored
+for the model at hand, and with a small number of pa-
+rameters. The first approach has clearly the potential to
+find a good optimum, e.g., using machine learning tech-
+niques, but it also has some disadvantages. In fact, for
+such a general approach the number of optimization pa-
+rameters has to be increased as one increases the number
+of degrees of freedom of the system. This means that the
+cost of finding good contours might turn out to be pro-
+hibitive, similarly to what happens with methods based
+on Lefschetz thimbles. In this exploratory study we fol-
+low the second, ad hoc approach, and optimize ansätze
+with only few parameters. Moreover, the number of these
+parameters is kept independent of the number of degrees
+of freedom of the system.
+We can then be sure that
+the optimization itself is numerically cheap, and that the
+per-configuration cost of generating the ensembles is es-
+sentially as low as on the original contours. Obviously,
+the drawback of this approach is that to write down an
+ansatz with only a few parameters that produces a sub-
+stantial improvement in the severity of the sign problem,
+some physical or mathematical insight is needed.
+For the toy model studied in this paper, the insight
+required to use the ad hoc approach is available, and so
+we can write down appropriate ansätze.
+We will then
+show that a quite cheap numerical optimization proce-
+dure leads one to contours with a reduced sign problem.
+We will also present numerical evidence that the reduc-
+tion in the severity of the sign problem is exponential:
+while the sign problem on the optimized contours is still
+
+3
+exponential in the number of degrees of freedom, the cor-
+responding exponent is reduced. This conclusion is simi-
+lar to what some of us have shown in Ref. [30] for a purely
+bosonic model (the 2+1 dimensional XY model at non-
+zero chemical potential). Notably, such an exponential
+reduction can be achieved without changing the number
+of optimization parameters with the system size.
+In this work we will only consider phase-quenched sim-
+ulations, for simplicity.
+Similar arguments and meth-
+ods should, however, also apply to the sign-quenched
+case [30].
+The plan of the paper is the following: In Section II
+we introduce the model discussed in this work. In Sec-
+tion III we provide details on the different contour de-
+formation procedures we tested. In Section IV we illus-
+trate the chemical potential and volume dependence of
+the achieved improvement and also compare our results
+with a method based on Lefschetz thimbles: the holo-
+morphic flow of Ref. [11]. We summarize our conclusions
+in Section V.
+II.
+THE CHIRAL RANDOM MATRIX MODEL
+Throughout this paper we will only consider Nf = 2
+with µ1 = µ2 ≡ µ for simplicity.
+The random ma-
+trix model of Stephanov [1] for Nf degenerate flavors of
+quarks is then defined by the partition function
+ZNf
+N
+= eNµ2 �
+dWdW † (det(D + m))Nf e−NTrW W †,
+(6)
+where the massless Dirac matrix is
+D =
+�
+0
+iW + µ
+iW † + µ
+0
+�
+,
+(7)
+m is the quark mass and W is a general N × N complex
+matrix. The model has no concept of physical volume.
+The number of degrees of freedom of the model scales
+with N 2.
+The two observables we will study in this paper are the
+chiral condensate:
+Σ =
+1
+2N
+∂ log ZNf
+N
+∂m
+,
+(8)
+and the quark density
+n =
+1
+2N
+∂ log ZNf
+N
+∂µ
+.
+(9)
+An important feature of the model is that it can be solved
+analytically, both in the N → ∞ limit where the integral
+is dominated by a saddle point, and at finite N where it
+reduces to the calculation of moments of Gaussian inte-
+grals. Thus, in this particular model we will be able to
+compare numerical results with exact analytic solutions.
+The model shares with QCD the feature that the
+phase-quenched theory corresponds to an isospin chemi-
+cal potential, and has an analogue of the pion condensa-
+tion transition at some µ = µPQ
+c
+. For chemical potentials
+exceeding µPQ
+c
+the sign problem of the model is severe.
+From the point of view of the Dirac spectrum, for µ = 0
+the eigenvalues are purely imaginary, while for µ ̸= 0 the
+eigenvalues of D acquire a real part, and are distributed
+inside a strip of width µ2 in the real direction. When the
+quark mass is inside this strip, the model has a severe sign
+problem. This roughly corresponds to the analogue of the
+pion condensed phase in the phase-quenched theory. Due
+to these similarities, this model has been considered sev-
+eral times in the literature as a good toy model for the
+sign problem in QCD [34, 35].
+We will consider the model for Nf = 2 and use the
+same quark chemical potential for both fermion flavors.
+In this model, unlike in QCD, the expectation value
+of the average phase does not always tend to zero in the
+limit of an infinite system. Rather, it only goes to zero
+in a given range of chemical potentials bounded by the
+solutions to the equation [38]:
+0 = 1 − µ2 +
+m2
+µ2 − m2 −
+m2
+4(µ2 − m2)2 .
+(10)
+Using a quark mass of m = 0.2, the two solutions of this
+equation are µ = 0.35 = µPQ
+c
+and 1.02. This is the regime
+where the sign problem in the model is strongest.
+III.
+CONTOUR DEFORMATION METHODS
+A.
+Optimization method
+We will restrict ourselves to ansätze with simple, ana-
+lytically calculable Jacobians with O(N 0) computational
+cost and a small number of parameters, independent of
+the number of degrees of freedom.
+Let A = Re W and B = Im W. These two real matrices
+will be deformed to complex matrices α and β. Thus,
+W = A + iB → X = α + iβ,
+W † = AT − iBT → Y = αT − iβT.
+(11)
+After applying such a deformation X† ̸= Y . After the
+deformation, the severity of the sign problem is given by:
+⟨eiθ⟩ =
+�� det(D + m)detJ
+|det(D + m)detJ |
+�Nf
+e−iNImTrXY
+�
+,
+(12)
+where the Jacobian determinant is
+detJ =
+����
+∂(α, β)
+∂(A, B)
+����.
+(13)
+B.
+Holomorphic flow
+Using the holomorphic flow (or generalized thimble
+method) of Ref. [11] for the complexified action of the
+
+4
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+k2
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+k1
+m = 0.2, µ = 1.0, N = 2, Nf = 2
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+�
+exp(iθ)
+�
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+p2
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+p1
+m = 0.2, µ = 1.0, N = 2, Nf = 2
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+�
+exp(iθ)
+�
+FIG. 1: Left: the average phase with Ansatz-1 as a function of k1 and k2. There is a local minimum at k2 ≈ 0 and k1 > 0.
+Right: the average phase with Ansatz-2 as a function of p1 and p2. There is an apparent saddle parallel to the p1 = 0 line at
+p1 = k1 > 0.
+Stephanov model,
+S = −Nµ2 − Nf log det(D + m) + NTr(XY ),
+(14)
+we deform the integration manifold by evolving the orig-
+inal one with the differential equation
+dYij
+dt
+= ∂S
+∂Yij
+= NXji − Nf[(XG)ji + iµGji],
+(15)
+where the overbar denotes complex conjugation, t is the
+flow parameter and
+G =
+�
+m2 − µ2 − iµ(X + Y ) + Y X
+�−1
+.
+(16)
+Solving this system of equations with initial conditions
+X0 = W, Y0 = W † for a fixed flow time tf we obtain
+a deformed manifold Mtf. We parametrize each point
+on the flowed manifold by the real matrices A and B.
+I.e., we parametrize the flowed manifold by the initial
+conditions of the flow equation.
+The computation of expectation values requires the Ja-
+cobian of the holomorphic flow,
+det J =
+����
+∂(X, Y )
+∂(A, B)
+���� ,
+(17)
+as well. Denoting the Hessian with H, the Jacobian ma-
+trix J is obtained as the solution of the equation
+dJ
+dt = H J,
+(18)
+with initial conditions
+JXij,Aij = 1, JXij,Bij = i, JYij,Aji = 1, JYij,Bji = −i.
+(19)
+Computing the Jacobian directly is numerically expen-
+sive, so we estimate it [39] with
+W = exp
+� � T
+0
+dt Tr H(t)
+�
+.
+(20)
+The difference between W and det J is taken into account
+by reweighting when computing observables,
+⟨O⟩ =
+⟨Oe−∆S⟩S′
+eff
+⟨e−∆S⟩S′
+eff
+,
+(21)
+where S′
+eff = S − ln W, ∆S = Seff − ReS′
+eff and ⟨.⟩S′
+eff is
+the average with respect to e−ReS′
+eff. This way, we needed
+to compute det J exactly only for the configurations used
+for measurements.
+In the large flow time limit, the flowed manifold tends
+towards the Lefschetz thimbles. At smaller flow times,
+it still reduces the sign problem, although less than a
+complete thimble decomposition would.
+IV.
+NUMERICAL RESULTS
+A.
+Simple ansätze
+As a rule, all of our ansätze have been parametrized
+such that the undeformed integration manifold is at value
+zero for all optimizable parameters.
+
+5
+0
+200
+400
+600
+800
+1000
+Nopt
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Contour deformation parameters
+m = 0.2, µ = 1.0, N = 2, Nf = 2
+Re(a)
+Im(a)
+Re(b)
+Im(b)
+Re(c)
+Im(c)
+Re(d)
+Im(d)
+Re(e)
+Im(e)
+Re(f)
+Im(f)
+Re(g)
+Im(g)
+Re(h)
+Im(h)
+Re(j)
+Im(j)
+Re(k)
+Im(k)
+0
+200
+400
+600
+800
+1000
+Nopt
+0.25
+0.30
+0.35
+0.40
+0.45
+�
+exp(iθ)
+�
+m = 0.2, µ = 1.0, N = 2, Nf = 2
+FIG. 2: Left: parameters as a function of the optimization step for Ansatz-3. Right: the average phase as a function of the
+optimization step for Ansatz-3.
+Ansatz-1
+From the definition in Eq. (7) it is easy to see that
+the sign problem can be removed from the quark de-
+terminant by a simple shift of the form α = A + iµ1.
+This, however, introduces a sign problem in the Gaus-
+sian term e−N Tr(XY ). By finding a trade-off between the
+two terms, the severity of the sign problem may be op-
+timized. This motivates our first ansatz, with two real
+parameters k1 and k2 defined by
+α = A + ik11
+(22)
+β = B + ik21.
+(23)
+The Jacobian determinant for this ansatz is simply unity.
+The parameter k2 is introduced on a whim, as the ma-
+trices A and B do not have to be treated symmetrically.
+The results for the average phase in a scan in these two
+parameters for N = 2, m = 0.2 and µ = 1.0 is shown in
+Fig. 1 (left). While there is a clearly non-zero optimal
+value for k1, the optimal value of the k2 parameter is near
+zero. This remains true for all values of the parameters
+N, µ and m we simulated.
+Ansatz-2
+When we introduce a shift A → A + ik1 the argument
+of the Gaussian term changes according to
+Tr(XY ) = Tr(AAT + BBT) − Nk2 + 2ikTrA.
+(24)
+This motivates our second ansatz, with two real param-
+eters p1 and p2 defined by
+α = A + ip11 + p2TrA1,
+(25)
+β = B.
+(26)
+The p1 parameter of this ansatz is identical to the k1
+parameter of the previous ansatz. The Jacobian deter-
+minant for this ansatz is simply detJ = 1 + Np2, i.e.,
+configuration-independent, and can be ignored. The re-
+sults for the average phase in a scan in these two param-
+eters for N = 2, m = 0.2 and µ = 1.0 can be seen in
+Fig. 1 (right). While there is a clearly non-zero optimal
+value for p1 = k1, the p2 parameter only appears to move
+on a saddle.
+Ansatz-3
+We now move on to a more complicated ansatz with
+10 complex (or 20 real) parameters a, b, c, d, e, f, g, h, j, k
+defined by
+α = (a + bTrA + cTrB)1 + (1 + d)A + eB
+(27)
+β = (f + gTrA + hTrB)1 + jA + (1 + k)B
+(28)
+The Jacobian determinant for this ansatz is
+detJ =
+�
+(1 + d)(1 + k) − ej
+�N 2−1×
+��
+(1 + d) + Nb
+��
+(1 + k) + Nh
+�
+−(e + Nc)(j + Ng)
+�
+.
+(29)
+The severity of the sign problem was then optimized via
+the AdaDelta method [40], with the objective function
+− log⟨eiθ⟩ = − log
+Z
+ZPQ
+= − log Z + log ZPQ,
+(30)
+where we suppressed the N and Nf indices for the parti-
+tion function. The gradient with right to the deformation
+parameters is given by
+∇ log ZPQ = −⟨∇SA
+eff⟩,
+(31)
+where
+Sa
+eff = NReTrXY − Nf log |detM| − log |detJ |
+(32)
+with gradient
+∇Sa
+eff =NReTr
+�
+(∇X)Y + X(∇Y )
+�
+− Nf
+2 Tr
+�
+M −1(∇M) + M
+−1(∇M)
+�
+− Re
+�∇detJ
+detJ
+�
+.
+(33)
+Note that for Ansatz-3 the Jacobian is independent of
+the configuration, and the last term can be dropped from
+Eq. (33). For Ansatz-4, to be discused below, the Jaco-
+bian will depend on the configuration, and thus the last
+term is needed. An example of such an optimization run
+is shown in Fig. 2. As with the previous two ansätze,
+only a single parameter emerges k1 = p1 = Ima.
+
+6
+Ansatz-4
+Experiments with the first three ansätze revealed only
+one parameter of interest, which can be thought of as
+a simple one-parameter imaginary shift of the trace of
+the matrix A. One might wonder whether more general
+deformations of the trace could lead to a better improve-
+ment. Thus we look at non-linear deformations of the
+trace τ = TrA of the matrix A with an undeformed B
+matrix. The integral measure is given by
+N
+�
+i,j=1
+dAij = dτ
+N
+�
+i,j=1
+(i,j)̸=(N,N)
+dAij
+= dτ
+N
+�
+i,j=1
+i̸=j
+dAij
+N
+�
+k=1
+d
+�
+Akk − τ
+N
+�
+.
+(34)
+The deformed matrix α is obtained from A as
+A = τ
+N 1 +
+�
+A − τ
+N 1
+�
+= τ
+N 1 + ˜A
+→
+α = τ
+N 1 + ˜A,
+(35)
+where Tr ˜A = 0 and
+τ = t + if(τ; . . . ),
+(36)
+for some function f that depends on τ and possibly other
+parameters. For simplicity, we choose f to be piecewise
+linear,
+f(τ; xk(τ), xk(τ)+1, yk(τ), yk(τ)+1) =
+yk(τ)(xk(τ)+1 − τ)
+xk(τ)+1 − xk(τ)
++ yk(τ)+1(τ − xk(τ))
+xk(τ)+1 − xk(τ)
+.
+(37)
+The parameters to optimize are the yi, while the node
+points xi of the linear interpolation are fixed parameters,
+and chosen with regular spacing, xl+1 − xl = ∆ for all l,
+and
+k(τ) = floor
+�τ − x0
+∆
+�
+.
+(38)
+By numerical experimentation we have found that the
+choice of the node points is not important, as long as the
+full interpolation range is large enough to cover the most
+probable values of TrA on the original contours and ∆ is
+small enough. If these conditions are met, optimal con-
+tours with ansätze with different node points appear to
+be piecewise approximations of the same smooth curve.
+The Jacobian is
+detJ = 1 + iyk(τ)+1 − yk(τ)
+∆
+.
+(39)
+The parameters are then optimized as with Ansatz-3.
+A comparison of the results from this ansatz with the
+6
+4
+2
+0
+2
+4
+6
+ReTrA
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+ImTrA
+piecewise
+α = A + ik  ansatz
+N = 2, Nf = 2, m = 0.2, µ = 1.0
+FIG. 3:
+Ansatz-4 (piecewise optimization of the trace) com-
+pared to Ansatz-1 (imaginary constant shift of A proportional
+to the unit matrix). The two procedures find essentially the
+same contour, as the differing tails are at large values of |TrA|,
+and have small statistical weight.
+constant shift found using ansätze 1 to 3 is shown in
+Fig. 3. For highly probable values of Tr A the two ansätze
+agree, while for the highly improbably values of Tr A,
+the optimization does not move the ansatz away from
+the original contour, as there are no configuration to use
+for the optimization of that part of the contour. These
+two asymptotic regimes are smoothly connected.
+The
+measured sign problem on this contour is identical to the
+one measured with ansätze 1 to 3, up to statistical errors
+– not surprisingly since deviations of f from a constant
+happen on unimportant configurations.
+B.
+Chemical potential and matrix size dependence
+Now that we have discovered a good contour defor-
+mation parameter, let us look at what kind of improve-
+ments can be achieved by such a 1-parameter deforma-
+tion. From here on out we show results with Ansatz-1,
+with k2 set to zero.
+The volume and chemical potential dependence of the
+average phase for the original and optimized contours
+is shown in Fig. 4. The "volume", i.e., matrix size de-
+pendence at a fixed chemical potential in the left panel
+reveals an improvement on the sign problem that is expo-
+nential in the matrix size: while the severity of the sign
+problem is roughly linear on a logarithmic plot for both
+the original and optimized contours, the slopes are quite
+different. The right panel shows the chemical potential
+dependence for several values of N. Apparently, contour
+optimization improves the most on the sign problem in
+the regime where it is the most severe.
+The statistical improvement factor, defined as the
+square of the ratio of the average phase on the deformed
+vs the original contours,
+��
+eiθ�
+orig /
+�
+eiθ�
+def
+�2
+, is shown
+on the left panel of Fig. 5 for N = 2, 4 and 6. For larger
+matrices,
+�
+eiθ�
+was zero within statistical errors on the
+original contours, and this ratio could not be calculated.
+We see that the ratio monotonically increases with N,
+and as a function of µ it is maximal close to the value of
+µ where the sign problem is the strongest. The optimal
+
+7
+2
+4
+6
+8
+10
+N
+10
+-3
+10
+-2
+10
+-1
+�
+exp(iθ)
+�
+m = 0.2, µ = 0.9, Nf = 2
+deformed
+original
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+10
+-3
+10
+-2
+10
+-1
+10
+0
+�
+exp(iθ)
+�
+m = 0.2, Nf = 2
+original, N = 2
+deformed
+original, N = 4
+deformed
+original, N = 6
+deformed
+original, N = 8
+deformed
+FIG. 4: Left: dependence of the average phase on the size of the random matrix for the original and optimized contours. Right:
+dependence of the average phase on the chemical potential for the original and optimized contours.
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+0
+10
+20
+30
+40
+50
+60
+70
+improvement factor
+m = 0.2, Nf = 2
+N = 2
+N = 4
+N = 6
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Im(a) = k1 = p1
+m = 0.2, Nf = 2
+N = 2
+N = 4
+N = 6
+N = 8
+N = 10
+FIG. 5: Left: dependence of the statistical improvement (calculated as the square of the ratio of the average phases on
+the optimized and original contours) achieved by contour optimization as a function of µ for different matrix sizes. Right:
+dependence of the optimal contour parameter k1 = p1 = Ima on µ for different matrix sizes.
+values for the deformation parameter k1 = p1 = Ima for
+different values of µ and N are shown in the right panel
+of Fig. 5.
+As a sanity check, we also calculated the expectation
+value of the chiral condensate and the quark number on
+both the original and the optimized contours, and com-
+pared them to the analytic results, see Fig. 6. They both
+show excellent agreement, but the optimized contours
+have significantly smaller error bars.
+C.
+Comparison with the holomorphic flow
+As experiments with simple ansätze so far revealed
+only a single important contour deformation parameter,
+it is a natural question to ask whether Lefschetz-thimble
+based methods also “find” this deformation or not, and
+whether by utilizing such methods it is possible to im-
+prove the sign problem further compared to such a 1-
+parameter deformation. For this reason, we performed
+the holomorphic flow on our N = 2 random matrices, and
+obtained an estimate of the k1 parameter from the flowed
+variables via: kflow
+1
+= Im ⟨Tr(α(tf) − A)⟩ /N.
+This k1
+can then be substituted back to the 1-parameter ansatz
+α = A + ik11 and the severity of the sign problem can
+be compared with the properly flowed manifold.
+The sign problem as a function of µ is shown on the
+original contour, the optimized contour, the flowed con-
+tour, and on the contour with k1 extracted from the flow
+in Fig. 7. A few observations can be drawn from this
+figure. For small chemical potentials, the flow performs
+better than the optimization, which does not noticeably
+improve the sign problem. For larger chemical potentials,
+optimization vastly outperforms the flow. Of course, this
+is only compared at a fixed flow time, and we do not
+know where the severity of the sign problem would end
+up at infinite flow time (on the thimbles). However, go-
+ing to large flow times gets very expensive already for
+small systems.
+For larger chemical potentials, the 1-parameter ansatz
+with k1 = kflow
+1
+extracted from the flow gives very similar
+results as the full flow. This may be a hint for the possi-
+bility that at larger chemical potentials most of the im-
+provement from the flow comes from this simple deforma-
+tion. Interestingly, while the full flow at small chemical
+potentials gives a slightly weaker sign problem compared
+to the ansatz with kflow
+1
+, at larger chemical potentials the
+situation is reversed: the sign problem is slightly weaker
+with kflow
+1
+than with the solution of the full flow equation.
+While this may be somewhat surprising at first, it is not
+
+8
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Σ
+original, N = 2
+optimized
+analytic
+original, N = 4
+optimized
+analytic
+original, N = 6
+optimized
+analytic
+original, N = 8
+optimized
+analytic
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+1
+0
+1
+2
+3
+4
+5
+n
+original, N = 2
+optimized
+analytic
+original, N = 4
+optimized
+analytic
+original, N = 6
+optimized
+analytic
+original, N = 8
+optimized
+analytic
+FIG. 6: The chiral condensate (left) and the quark number (right) as a function of µ for several values of the matrix size N.
+Analytic results are compared with results from simulations on the original and on the improved contours.
+in contradiction with what we already now about contour
+deformations. The flow goes towards the Lefschetz thim-
+bles, which are not the numerically optimal contours, and
+thus there is no reason for the full flow curve to be always
+above the curve with the simple ansatz with kflow
+1
+.
+V.
+SUMMARY AND DISCUSSION
+We have discussed contour deformations in the chiral
+random matrix model of Stephanov as a way to alleviate
+its sign problem. Using simple ad-hoc ansätze we iden-
+tified a single important deformation parameter, which
+allowed for an exponential reduction in the severity of
+the sign problem as a function of the matrix size.
+Our results are quite encouraging, as they show that a
+simple one-parameter optimization can lead to exponen-
+tially alleviating the sign problem even in a fermionic the-
+ory, where the thimble decomposition is complicated and
+contour deformation approaches based on them might
+not be numerically effective.
+The fermionic nature of
+the matter fields does not appear to be a fundamental
+obstruction in the construction of exponentially better
+contours.
+Furthermore, the phase diagram of the random matrix
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+µ
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+�
+exp(iθ)
+�
+m = 0.2, Nf = 2, N = 2
+original
+deformed: ansatz
+deformed: flow (tf = 0.04)
+deformed: ansatz with k1 from flow
+FIG. 7:
+The severity of the sign problem for N = 2 and m =
+0.2 as a function of µ on the original contours, the optimized
+contours, the flowed contours and the contours where the k1
+parameter of the ansatz is extracted from the flow.
+model is similar to what we expect in full QCD: the chi-
+ral phase transition is “hidden behind” the pion conden-
+sation phase in the phase-quenched theory. Hence, this
+bulk thermodynamic feature – the existence of a phase
+transition in the phase-quenched theory – also does not
+appear to be a fundamental obstruction.
+The results and the ansätze in this paper, however,
+cannot be used directly to construct a good optimization
+ansatz in full QCD, as the toy model studied here and
+QCD differ on an important technical aspect. Concretely,
+in the Stephanov model there are contour deformations
+that can remove the sign problem from the fermion deter-
+minant for a single flavor (so from the full determinant
+when all chemical potentials are equal) – albeit at the
+cost of reintroducing it somewhere else in the Boltzmann
+weights. There are no such deformations in full QCD.
+The complexification of the SU(3) gauge group is the
+SL(3, C) group, which still requires a unit determinant.
+To remove the chemical potential from a single quark de-
+terminant the time-like links would have to be deformed
+to GL(3, C) matrices, with non-unit determinant, which
+lie outside the complexified gauge group.
+Comparison with the holomorphic flow method shows
+that as one goes near the Lefschetz thimbles in this
+model, the bulk (but not all) of the improvement on the
+severity of the sign problem is captured by these types of
+deformations – which have no direct analogue in QCD.
+In the future it will therefore be important to work with
+more realistic toy models of QCD or even full QCD it-
+self, as the choice of a suitable sign-problem improving
+ansatz appears to be strongly dependent on the exact
+symmetries and exact matter content of a given theory.
+Acknowledgements
+This work was supported by the NKFIH grant KKP-
+126769. D.P. is supported by the ÚNKP-22-3 New Na-
+tional Excellence Program of the Ministry for Culture
+and Innovation from the source of the National Research,
+Development and Innovation Fund.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf,len=784
+page_content='Fighting the sign problem in a chiral random matrix model with contour deformations  Matteo Giordano,1 Attila Pásztor,1 Dávid Pesznyák,1 and Zoltán Tulipánt1  1ELTE Eötvös Loránd University, Institute for Theoretical Physics,  Pázmány Péter sétány 1/A, H-1117, Budapest, Hungary  We studied integration contour deformations in the chiral random matrix theory of Stephanov [1]  with the goal of alleviating the finite-density sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' We considered simple ansätze for the  deformed integration contours, and optimized their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  We find that optimization of a  single parameter manages to considerably improve on the severity of the sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' We show  numerical evidence that the improvement achieved is exponential in the degrees of freedom of the  system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=', the size of the random matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' We also compare the optimization method with contour  deformations coming from the holomorphic flow equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  INTRODUCTION  Euclidean quantum field theories at non-zero particle  density (or chemical potential) generally suffer from a  complex action problem: the weights in the path integral  representation are complex, and thus cannot be inter-  preted as a joint probability density function on the space  of field configurations (up to a proportionality factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  This prevents the use of importance sampling methods  for the direct simulation of these theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' In QCD, this  complex action problem severely hampers first-principles  studies of dense matter in the core of neutron stars, neu-  tron star mergers, core collapse supernovae, as well as in  heavy ion collisions at certain collision energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  In the presence of a complex action problem one can  still (in principle) simulate a modified theory with real  and positive weights, and then use reweighting methods  to calculate observables in the theory of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' If the  target theory has field variables φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' path integral weights  wt(φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' and partition function Zt =  �  Dφ wt(φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' and the  simulated theory has the same field variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' but dif-  ferent – real and positive – path integral weights ws(φ)  and partition function Zs =  �  Dφ ws(φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' we can obtain  expectation values in the target theory via the formula  ⟨O⟩t =  �  wt  ws O  �  s  �  wt  ws  �  s  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  ⟨O⟩x = 1  Zx  �  Dφ wx(φ)O(φ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' (1)  where x may stand for t or s and O(φ) is some phys-  ical observable of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' (1)  gives the ratio of the partition functions in the target and  simulated theories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=',  � wt  ws  �  s  = Zt  Zs  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (2)  This ratio is typically exponentially small in the physical  volume, with the exponent given by the free energy dif-  ference between the target and simulated theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This  ratio is also a rough measure of the numerical difficulty  of a given reweighting scheme, with a given simulated  and target theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' In order for reweighting to be effec-  tive, one wants the target and simulated theories to be as  close to each other as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Ideally, one should find a  simulated theory with Zs ≈ Zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Two simple choices of a simulated theory are the phase-  quenched (PQ) theory, with simulated weights propor-  tional to  wPQ  s  ≡ |wt(φ)| ,  (3)  or – assuming that the partition function Zt is real –  the sign-quenched (SQ) theory, with simulated weights  proportional to  wSQ  s  ≡ |Re wt(φ)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (4)  For the first case (phase reweighting) the reweighting  factors wt/wPQ  s  ≡ eiθ are pure phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  For the sec-  ond case (sign reweighting) the reweighting factors are  wt/wSQ  s  = eiθ/ |cos θ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For certain observables, such as  manifestly real observables or observables with a con-  jugation (φ → φ) symmetry, one can substitute wt/wPQ  s  with cos θ and wt/wSQ  s  with a pure sign cos θ/ |cos θ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For  phase or sign reweighting, we can then say that the com-  plex action problem becomes a sign problem: the cancel-  lations between contributions with different signs of cosθ  lead to a small Zt  Zs ratio, and in turn to small signal-to-  noise ratios in the expectation values of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The sign-quenched ensemble always has a less severe  sign problem, due to the inequality Zt < ZSQ  s  < ZPQ  s  ,  which is a consequence of cos θ ≤ | cos θ| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' However, in  the limit of a severe sign problem – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=', as the distribution  of the argument θ tends to to a uniform distribution on  [−π, π) – the severity of the sign problem for these two  reweighting schemes only differs by a constant factor [2],  given by  �  ZPQ  s  /ZSQ  s  �2 → (π/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  In QCD and in other (more or less) QCD-like models,  describing the interactions of several “flavors” of fermions,  the path integral weights can be written schematically as  wt(φ) = det M1(φ, µ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' det MNf (φ, µNf )e−SB(φ), (5)  where the fields φ are real bosonic variables and SB  is the corresponding bosonic part of the action, Nf is  the number of fermion flavors in the model, det Mk is  the fermionic determinant of the kth flavor and µk is  the corresponding chemical potential, for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' , Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='12947v1  [hep-lat]  30 Jan 2023  2  The source of the sign problem is the fermionic determi-  nant, which at non-zero µ is generally a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Moreover, an important feature of the sign problem in  QCD and QCD-like theories is that it tends to get much  worse in the ranges of µ where zeros of the determinant  in the complex µ plane become dense [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Nonetheless, reweighting from the phase- and sign-  quenched theories is starting to become feasible even in  full QCD [2, 4], which has recently led to the calculation  of the equation of state of a hot-and-dense quark-gluon  plasma in the region of chemical potentials covered by  the RHIC Beam Energy Scan [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' However, the range of  practical applicability of such an approach is limited both  in volume and chemical potential by the smallness of the  ratio Zt/Zs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Lacking a solution of the sign problem, it is  then desirable to develop methods that at least alleviate  it, to extend the range of parameters that reweighting  methods can practically reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  One possible route to do this is the use of contour  deformations in the path integral (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' [6] for a re-  cent review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' If the path integral weights wt(φ) are holo-  morphic functions of the field variables,1 the multivariate  Cauchy theorem guarantees that complexified integration  manifolds in the same homology class as the original one  yield the same partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' However, the phase-  and sign-quenched integrands are not holomorphic, and  therefore the phase- and sign-quenched partition func-  tions are not invariant under such deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' It may  then be possible to bring the ratios Zt/Zs closer to unity,  thus making reweighting more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  There are different ways to deform integration con-  tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Historically, methods based on Lefschetz thimbles  appeared first [6, 9–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Lefschetz thimbles are the dis-  joint components of the integration contour defined by  requiring that the imaginary part of the classical action  is constant in each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The thimble structure  of theories with a fermionic determinant is usually quite  complicated [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Simple toy models reveal the follow-  ing features: i) cancellations between competing thimbles  are very important for getting the correct results, and  ii) the thimbles themselves are not smooth at the zeros  of the fermionic determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Thus, the use of thim-  bles might be impractical for such theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  However,  Lefschetz thimbles are, in general, not the numerically  optimal integration contours [20], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=', they are not nec-  essarily the contours with the largest Zt/Zs, so there is  no need to concentrate solely on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  A second class of methods is based on numerical op-  timization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The main idea here is to parametrize the  integration manifold by a finite number of parameters,  which are then optimized to make the sign problem  as mild as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Such methods were applied to  a one-dimensional integral [21], the 0+1D scalar the-  1 A notable exception is lattice QCD with rooted staggered  fermions [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  ory [22], the 0+1D Polyakov-improved Nambu-Jona-  Lasinio model [23], 0+1D QCD [24], 1+1D scalar field  theory [25], the 1+1D Thirring model [26], the 2+1D  Thirring model [27], Bose gases of several dimensions [28],  1+1D U(1) gauge theory with a complex coupling con-  stant [29] and the 2+1D XY model at finite density [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Here, we apply contour optimization methods to a  fermionic toy model that shares relevant technical fea-  tures with finite chemical potential QCD: the chiral ran-  dom matrix model proposed by Stephanov in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Since it is an exactly solvable model with a sign prob-  lem, the Stephanov model is a very useful testbed for  methods aimed at solving or alleviating the sign prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  This model has been studied with the complex  Langevin approach [31–33], which fails for this particu-  lar model [34] even with the introduction of gauge cool-  ing [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' There are also preliminary results for this model  with the tempered Lefschetz thimble method [35] which  is based on parallel tempering [36] in the flow time of  the holomorphic flow [11, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This method – similarly  to other flow-based methods — produces a weaker sign  problem, albeit at the cost of substantially increasing the  per-configuration-cost of generating the ensemble com-  pared to ordinary phase reweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  In this paper we study the Stephanov model with op-  timization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' There are, roughly speaking, two  approaches to such an optimization: one can look for the  optimum using either a very general ansatz with a large  number of parameters, or a very specific ansatz tailored  for the model at hand, and with a small number of pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The first approach has clearly the potential to  find a good optimum, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=', using machine learning tech-  niques, but it also has some disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' In fact, for  such a general approach the number of optimization pa-  rameters has to be increased as one increases the number  of degrees of freedom of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This means that the  cost of finding good contours might turn out to be pro-  hibitive, similarly to what happens with methods based  on Lefschetz thimbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' In this exploratory study we fol-  low the second, ad hoc approach, and optimize ansätze  with only few parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Moreover, the number of these  parameters is kept independent of the number of degrees  of freedom of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  We can then be sure that  the optimization itself is numerically cheap, and that the  per-configuration cost of generating the ensembles is es-  sentially as low as on the original contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Obviously,  the drawback of this approach is that to write down an  ansatz with only a few parameters that produces a sub-  stantial improvement in the severity of the sign problem,  some physical or mathematical insight is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  For the toy model studied in this paper, the insight  required to use the ad hoc approach is available, and so  we can write down appropriate ansätze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  We will then  show that a quite cheap numerical optimization proce-  dure leads one to contours with a reduced sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  We will also present numerical evidence that the reduc-  tion in the severity of the sign problem is exponential:  while the sign problem on the optimized contours is still  3  exponential in the number of degrees of freedom, the cor-  responding exponent is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This conclusion is simi-  lar to what some of us have shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' [30] for a purely  bosonic model (the 2+1 dimensional XY model at non-  zero chemical potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Notably, such an exponential  reduction can be achieved without changing the number  of optimization parameters with the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  In this work we will only consider phase-quenched sim-  ulations, for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Similar arguments and meth-  ods should, however, also apply to the sign-quenched  case [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The plan of the paper is the following: In Section II  we introduce the model discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' In Sec-  tion III we provide details on the different contour de-  formation procedures we tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' In Section IV we illus-  trate the chemical potential and volume dependence of  the achieved improvement and also compare our results  with a method based on Lefschetz thimbles: the holo-  morphic flow of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' We summarize our conclusions  in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  THE CHIRAL RANDOM MATRIX MODEL  Throughout this paper we will only consider Nf = 2  with µ1 = µ2 ≡ µ for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The random ma-  trix model of Stephanov [1] for Nf degenerate flavors of  quarks is then defined by the partition function  ZNf  N  = eNµ2 �  dWdW † (det(D + m))Nf e−NTrW W †,  (6)  where the massless Dirac matrix is  D =  �  0  iW + µ  iW † + µ  0  �  ,  (7)  m is the quark mass and W is a general N × N complex  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The model has no concept of physical volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The number of degrees of freedom of the model scales  with N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The two observables we will study in this paper are the  chiral condensate:  Σ =  1  2N  ∂ log ZNf  N  ∂m  ,  (8)  and the quark density  n =  1  2N  ∂ log ZNf  N  ∂µ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (9)  An important feature of the model is that it can be solved  analytically, both in the N → ∞ limit where the integral  is dominated by a saddle point, and at finite N where it  reduces to the calculation of moments of Gaussian inte-  grals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Thus, in this particular model we will be able to  compare numerical results with exact analytic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The model shares with QCD the feature that the  phase-quenched theory corresponds to an isospin chemi-  cal potential, and has an analogue of the pion condensa-  tion transition at some µ = µPQ  c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For chemical potentials  exceeding µPQ  c  the sign problem of the model is severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  From the point of view of the Dirac spectrum, for µ = 0  the eigenvalues are purely imaginary, while for µ ̸= 0 the  eigenvalues of D acquire a real part, and are distributed  inside a strip of width µ2 in the real direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' When the  quark mass is inside this strip, the model has a severe sign  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This roughly corresponds to the analogue of the  pion condensed phase in the phase-quenched theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Due  to these similarities, this model has been considered sev-  eral times in the literature as a good toy model for the  sign problem in QCD [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  We will consider the model for Nf = 2 and use the  same quark chemical potential for both fermion flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  In this model, unlike in QCD, the expectation value  of the average phase does not always tend to zero in the  limit of an infinite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Rather, it only goes to zero  in a given range of chemical potentials bounded by the  solutions to the equation [38]:  0 = 1 − µ2 +  m2  µ2 − m2 −  m2  4(µ2 − m2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (10)  Using a quark mass of m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, the two solutions of this  equation are µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='35 = µPQ  c  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This is the regime  where the sign problem in the model is strongest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  CONTOUR DEFORMATION METHODS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Optimization method  We will restrict ourselves to ansätze with simple, ana-  lytically calculable Jacobians with O(N 0) computational  cost and a small number of parameters, independent of  the number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Let A = Re W and B = Im W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' These two real matrices  will be deformed to complex matrices α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Thus,  W = A + iB → X = α + iβ,  W † = AT − iBT → Y = αT − iβT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (11)  After applying such a deformation X† ̸= Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' After the  deformation, the severity of the sign problem is given by:  ⟨eiθ⟩ =  �� det(D + m)detJ  |det(D + m)detJ |  �Nf  e−iNImTrXY  �  ,  (12)  where the Jacobian determinant is  detJ =  ����  ∂(α, β)  ∂(A, B)  ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (13)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Holomorphic flow  Using the holomorphic flow (or generalized thimble  method) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' [11] for the complexified action of the  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='6  k2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  k1  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0, N = 2, Nf = 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='40  �  exp(iθ)  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='6  p2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  p1  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0, N = 2, Nf = 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='40  �  exp(iθ)  �  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 1: Left: the average phase with Ansatz-1 as a function of k1 and k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' There is a local minimum at k2 ≈ 0 and k1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Right: the average phase with Ansatz-2 as a function of p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' There is an apparent saddle parallel to the p1 = 0 line at  p1 = k1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Stephanov model,  S = −Nµ2 − Nf log det(D + m) + NTr(XY ),  (14)  we deform the integration manifold by evolving the orig-  inal one with the differential equation  dYij  dt  = ∂S  ∂Yij  = NXji − Nf[(XG)ji + iµGji],  (15)  where the overbar denotes complex conjugation, t is the  flow parameter and  G =  �  m2 − µ2 − iµ(X + Y ) + Y X  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (16)  Solving this system of equations with initial conditions  X0 = W, Y0 = W † for a fixed flow time tf we obtain  a deformed manifold Mtf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' We parametrize each point  on the flowed manifold by the real matrices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=', we parametrize the flowed manifold by the initial  conditions of the flow equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The computation of expectation values requires the Ja-  cobian of the holomorphic flow,  det J =  ����  ∂(X, Y )  ∂(A, B)  ���� ,  (17)  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Denoting the Hessian with H, the Jacobian ma-  trix J is obtained as the solution of the equation  dJ  dt = H J,  (18)  with initial conditions  JXij,Aij = 1, JXij,Bij = i, JYij,Aji = 1, JYij,Bji = −i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (19)  Computing the Jacobian directly is numerically expen-  sive, so we estimate it [39] with  W = exp  � � T  0  dt Tr H(t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (20)  The difference between W and det J is taken into account  by reweighting when computing observables,  ⟨O⟩ =  ⟨Oe−∆S⟩S′  eff  ⟨e−∆S⟩S′  eff  ,  (21)  where S′  eff = S − ln W, ∆S = Seff − ReS′  eff and ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='⟩S′  eff is  the average with respect to e−ReS′  eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This way, we needed  to compute det J exactly only for the configurations used  for measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  In the large flow time limit, the flowed manifold tends  towards the Lefschetz thimbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' At smaller flow times,  it still reduces the sign problem, although less than a  complete thimble decomposition would.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  NUMERICAL RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Simple ansätze  As a rule, all of our ansätze have been parametrized  such that the undeformed integration manifold is at value  zero for all optimizable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  5  0  200  400  600  800  1000  Nopt  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='30  Contour deformation parameters  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0, N = 2, Nf = 2  Re(a)  Im(a)  Re(b)  Im(b)  Re(c)  Im(c)  Re(d)  Im(d)  Re(e)  Im(e)  Re(f)  Im(f)  Re(g)  Im(g)  Re(h)  Im(h)  Re(j)  Im(j)  Re(k)  Im(k)  0  200  400  600  800  1000  Nopt  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='45  �  exp(iθ)  �  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0, N = 2, Nf = 2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 2: Left: parameters as a function of the optimization step for Ansatz-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Right: the average phase as a function of the  optimization step for Ansatz-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Ansatz-1  From the definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' (7) it is easy to see that  the sign problem can be removed from the quark de-  terminant by a simple shift of the form α = A + iµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  This, however, introduces a sign problem in the Gaus-  sian term e−N Tr(XY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' By finding a trade-off between the  two terms, the severity of the sign problem may be op-  timized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This motivates our first ansatz, with two real  parameters k1 and k2 defined by  α = A + ik11  (22)  β = B + ik21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (23)  The Jacobian determinant for this ansatz is simply unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The parameter k2 is introduced on a whim, as the ma-  trices A and B do not have to be treated symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The results for the average phase in a scan in these two  parameters for N = 2, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2 and µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0 is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 1 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' While there is a clearly non-zero optimal  value for k1, the optimal value of the k2 parameter is near  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This remains true for all values of the parameters  N, µ and m we simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Ansatz-2  When we introduce a shift A → A + ik1 the argument  of the Gaussian term changes according to  Tr(XY ) = Tr(AAT + BBT) − Nk2 + 2ikTrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (24)  This motivates our second ansatz, with two real param-  eters p1 and p2 defined by  α = A + ip11 + p2TrA1,  (25)  β = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (26)  The p1 parameter of this ansatz is identical to the k1  parameter of the previous ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The Jacobian deter-  minant for this ansatz is simply detJ = 1 + Np2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=',  configuration-independent, and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The re-  sults for the average phase in a scan in these two param-  eters for N = 2, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2 and µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0 can be seen in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 1 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' While there is a clearly non-zero optimal  value for p1 = k1, the p2 parameter only appears to move  on a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Ansatz-3  We now move on to a more complicated ansatz with  10 complex (or 20 real) parameters a, b, c, d, e, f, g, h, j, k  defined by  α = (a + bTrA + cTrB)1 + (1 + d)A + eB  (27)  β = (f + gTrA + hTrB)1 + jA + (1 + k)B  (28)  The Jacobian determinant for this ansatz is  detJ =  �  (1 + d)(1 + k) − ej  �N 2−1×  ��  (1 + d) + Nb  ��  (1 + k) + Nh  �  −(e + Nc)(j + Ng)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (29)  The severity of the sign problem was then optimized via  the AdaDelta method [40], with the objective function  − log⟨eiθ⟩ = − log  Z  ZPQ  = − log Z + log ZPQ,  (30)  where we suppressed the N and Nf indices for the parti-  tion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The gradient with right to the deformation  parameters is given by  ∇ log ZPQ = −⟨∇SA  eff⟩,  (31)  where  Sa  eff = NReTrXY − Nf log |detM| − log |detJ |  (32)  with gradient  ∇Sa  eff =NReTr  �  (∇X)Y + X(∇Y )  �  − Nf  2 Tr  �  M −1(∇M) + M  −1(∇M)  �  − Re  �∇detJ  detJ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (33)  Note that for Ansatz-3 the Jacobian is independent of  the configuration, and the last term can be dropped from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For Ansatz-4, to be discused below, the Jaco-  bian will depend on the configuration, and thus the last  term is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' An example of such an optimization run  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' As with the previous two ansätze,  only a single parameter emerges k1 = p1 = Ima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  6  Ansatz-4  Experiments with the first three ansätze revealed only  one parameter of interest, which can be thought of as  a simple one-parameter imaginary shift of the trace of  the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' One might wonder whether more general  deformations of the trace could lead to a better improve-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Thus we look at non-linear deformations of the  trace τ = TrA of the matrix A with an undeformed B  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The integral measure is given by  N  �  i,j=1  dAij = dτ  N  �  i,j=1  (i,j)̸=(N,N)  dAij  = dτ  N  �  i,j=1  i̸=j  dAij  N  �  k=1  d  �  Akk − τ  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (34)  The deformed matrix α is obtained from A as  A = τ  N 1 +  �  A − τ  N 1  �  = τ  N 1 + ˜A  →  α = τ  N 1 + ˜A,  (35)  where Tr ˜A = 0 and  τ = t + if(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' ),  (36)  for some function f that depends on τ and possibly other  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For simplicity, we choose f to be piecewise  linear,  f(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' xk(τ), xk(τ)+1, yk(τ), yk(τ)+1) =  yk(τ)(xk(τ)+1 − τ)  xk(τ)+1 − xk(τ)  + yk(τ)+1(τ − xk(τ))  xk(τ)+1 − xk(τ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (37)  The parameters to optimize are the yi, while the node  points xi of the linear interpolation are fixed parameters,  and chosen with regular spacing, xl+1 − xl = ∆ for all l,  and  k(τ) = floor  �τ − x0  ∆  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (38)  By numerical experimentation we have found that the  choice of the node points is not important, as long as the  full interpolation range is large enough to cover the most  probable values of TrA on the original contours and ∆ is  small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' If these conditions are met, optimal con-  tours with ansätze with different node points appear to  be piecewise approximations of the same smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The Jacobian is  detJ = 1 + iyk(τ)+1 − yk(τ)  ∆  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  (39)  The parameters are then optimized as with Ansatz-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  A comparison of the results from this ansatz with the  6  4  2  0  2  4  6  ReTrA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='7  ImTrA  piecewise  α = A + ik  ansatz  N = 2, Nf = 2, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 3:  Ansatz-4 (piecewise optimization of the trace) com-  pared to Ansatz-1 (imaginary constant shift of A proportional  to the unit matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The two procedures find essentially the  same contour, as the differing tails are at large values of |TrA|,  and have small statistical weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  constant shift found using ansätze 1 to 3 is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For highly probable values of Tr A the two ansätze  agree, while for the highly improbably values of Tr A,  the optimization does not move the ansatz away from  the original contour, as there are no configuration to use  for the optimization of that part of the contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' These  two asymptotic regimes are smoothly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The  measured sign problem on this contour is identical to the  one measured with ansätze 1 to 3, up to statistical errors  – not surprisingly since deviations of f from a constant  happen on unimportant configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Chemical potential and matrix size dependence  Now that we have discovered a good contour defor-  mation parameter, let us look at what kind of improve-  ments can be achieved by such a 1-parameter deforma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' From here on out we show results with Ansatz-1,  with k2 set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The volume and chemical potential dependence of the  average phase for the original and optimized contours  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The "volume", i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=', matrix size de-  pendence at a fixed chemical potential in the left panel  reveals an improvement on the sign problem that is expo-  nential in the matrix size: while the severity of the sign  problem is roughly linear on a logarithmic plot for both  the original and optimized contours, the slopes are quite  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The right panel shows the chemical potential  dependence for several values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Apparently, contour  optimization improves the most on the sign problem in  the regime where it is the most severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The statistical improvement factor, defined as the  square of the ratio of the average phase on the deformed  vs the original contours,  ��  eiθ�  orig /  �  eiθ�  def  �2  , is shown  on the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 5 for N = 2, 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For larger  matrices,  �  eiθ�  was zero within statistical errors on the  original contours, and this ratio could not be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  We see that the ratio monotonically increases with N,  and as a function of µ it is maximal close to the value of  µ where the sign problem is the strongest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The optimal  7  2  4  6  8  10  N  10  3  10  2  10  1  �  exp(iθ)  �  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='9, Nf = 2  deformed  original  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  µ  10  3  10  2  10  1  10  0  �  exp(iθ)  �  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, Nf = 2  original, N = 2  deformed  original, N = 4  deformed  original, N = 6  deformed  original, N = 8  deformed  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 4: Left: dependence of the average phase on the size of the random matrix for the original and optimized contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Right:  dependence of the average phase on the chemical potential for the original and optimized contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  µ  0  10  20  30  40  50  60  70  improvement factor  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, Nf = 2  N = 2  N = 4  N = 6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  µ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='7  Im(a) = k1 = p1  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, Nf = 2  N = 2  N = 4  N = 6  N = 8  N = 10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 5: Left: dependence of the statistical improvement (calculated as the square of the ratio of the average phases on  the optimized and original contours) achieved by contour optimization as a function of µ for different matrix sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Right:  dependence of the optimal contour parameter k1 = p1 = Ima on µ for different matrix sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  values for the deformation parameter k1 = p1 = Ima for  different values of µ and N are shown in the right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  As a sanity check, we also calculated the expectation  value of the chiral condensate and the quark number on  both the original and the optimized contours, and com-  pared them to the analytic results, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' They both  show excellent agreement, but the optimized contours  have significantly smaller error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Comparison with the holomorphic flow  As experiments with simple ansätze so far revealed  only a single important contour deformation parameter,  it is a natural question to ask whether Lefschetz-thimble  based methods also “find” this deformation or not, and  whether by utilizing such methods it is possible to im-  prove the sign problem further compared to such a 1-  parameter deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For this reason, we performed  the holomorphic flow on our N = 2 random matrices, and  obtained an estimate of the k1 parameter from the flowed  variables via: kflow  1  = Im ⟨Tr(α(tf) − A)⟩ /N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  This k1  can then be substituted back to the 1-parameter ansatz  α = A + ik11 and the severity of the sign problem can  be compared with the properly flowed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The sign problem as a function of µ is shown on the  original contour, the optimized contour, the flowed con-  tour, and on the contour with k1 extracted from the flow  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' A few observations can be drawn from this  figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For small chemical potentials, the flow performs  better than the optimization, which does not noticeably  improve the sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' For larger chemical potentials,  optimization vastly outperforms the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Of course, this  is only compared at a fixed flow time, and we do not  know where the severity of the sign problem would end  up at infinite flow time (on the thimbles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' However, go-  ing to large flow times gets very expensive already for  small systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  For larger chemical potentials, the 1-parameter ansatz  with k1 = kflow  1  extracted from the flow gives very similar  results as the full flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' This may be a hint for the possi-  bility that at larger chemical potentials most of the im-  provement from the flow comes from this simple deforma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Interestingly, while the full flow at small chemical  potentials gives a slightly weaker sign problem compared  to the ansatz with kflow  1  , at larger chemical potentials the  situation is reversed: the sign problem is slightly weaker  with kflow  1  than with the solution of the full flow equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  While this may be somewhat surprising at first, it is not  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  µ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='5  Σ  original, N = 2  optimized  analytic  original, N = 4  optimized  analytic  original, N = 6  optimized  analytic  original, N = 8  optimized  analytic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  µ  1  0  1  2  3  4  5  n  original, N = 2  optimized  analytic  original, N = 4  optimized  analytic  original, N = 6  optimized  analytic  original, N = 8  optimized  analytic  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 6: The chiral condensate (left) and the quark number (right) as a function of µ for several values of the matrix size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Analytic results are compared with results from simulations on the original and on the improved contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  in contradiction with what we already now about contour  deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' The flow goes towards the Lefschetz thim-  bles, which are not the numerically optimal contours, and  thus there is no reason for the full flow curve to be always  above the curve with the simple ansatz with kflow  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  SUMMARY AND DISCUSSION  We have discussed contour deformations in the chiral  random matrix model of Stephanov as a way to alleviate  its sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Using simple ad-hoc ansätze we iden-  tified a single important deformation parameter, which  allowed for an exponential reduction in the severity of  the sign problem as a function of the matrix size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Our results are quite encouraging, as they show that a  simple one-parameter optimization can lead to exponen-  tially alleviating the sign problem even in a fermionic the-  ory, where the thimble decomposition is complicated and  contour deformation approaches based on them might  not be numerically effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The fermionic nature of  the matter fields does not appear to be a fundamental  obstruction in the construction of exponentially better  contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Furthermore, the phase diagram of the random matrix  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='00  µ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2  �  exp(iθ)  �  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2, Nf = 2, N = 2  original  deformed: ansatz  deformed: flow (tf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='04)  deformed: ansatz with k1 from flow  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' 7:  The severity of the sign problem for N = 2 and m =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='2 as a function of µ on the original contours, the optimized  contours, the flowed contours and the contours where the k1  parameter of the ansatz is extracted from the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  model is similar to what we expect in full QCD: the chi-  ral phase transition is “hidden behind” the pion conden-  sation phase in the phase-quenched theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Hence, this  bulk thermodynamic feature – the existence of a phase  transition in the phase-quenched theory – also does not  appear to be a fundamental obstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The results and the ansätze in this paper, however,  cannot be used directly to construct a good optimization  ansatz in full QCD, as the toy model studied here and  QCD differ on an important technical aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' Concretely,  in the Stephanov model there are contour deformations  that can remove the sign problem from the fermion deter-  minant for a single flavor (so from the full determinant  when all chemical potentials are equal) – albeit at the  cost of reintroducing it somewhere else in the Boltzmann  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' There are no such deformations in full QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  The complexification of the SU(3) gauge group is the  SL(3, C) group, which still requires a unit determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  To remove the chemical potential from a single quark de-  terminant the time-like links would have to be deformed  to GL(3, C) matrices, with non-unit determinant, which  lie outside the complexified gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Comparison with the holomorphic flow method shows  that as one goes near the Lefschetz thimbles in this  model, the bulk (but not all) of the improvement on the  severity of the sign problem is captured by these types of  deformations – which have no direct analogue in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  In the future it will therefore be important to work with  more realistic toy models of QCD or even full QCD it-  self, as the choice of a suitable sign-problem improving  ansatz appears to be strongly dependent on the exact  symmetries and exact matter content of a given theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='  Acknowledgements  This work was supported by the NKFIH grant KKP-  126769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
+page_content=' is supported by the ÚNKP-22-3 New Na-  tional Excellence Program of the Ministry for Culture  and Innovation from the source of the National Research,  Development and Innovation Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
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+page_content=' extrapolations,” 8 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf'}
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diff --git a/UNAyT4oBgHgl3EQfVvd8/content/tmp_files/2301.00149v1.pdf.txt b/UNAyT4oBgHgl3EQfVvd8/content/tmp_files/2301.00149v1.pdf.txt
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+Rethinking Rotation Invariance with Point Cloud Registration
+Jianhui Yu, Chaoyi Zhang, Weidong Cai
+School of Computer Science, University of Sydney, Australia
+{jianhui.yu, chaoyi.zhang, tom.cai}@sydney.edu.au
+Abstract
+Recent investigations on rotation invariance for 3D point
+clouds have been devoted to devising rotation-invariant fea-
+ture descriptors or learning canonical spaces where objects
+are semantically aligned. Examinations of learning frame-
+works for invariance have seldom been looked into. In this
+work, we review rotation invariance in terms of point cloud
+registration and propose an effective framework for rota-
+tion invariance learning via three sequential stages, namely
+rotation-invariant shape encoding, aligned feature integration,
+and deep feature registration. We first encode shape descrip-
+tors constructed with respect to reference frames defined over
+different scales, e.g., local patches and global topology, to
+generate rotation-invariant latent shape codes. Within the in-
+tegration stage, we propose Aligned Integration Transformer
+to produce a discriminative feature representation by inte-
+grating point-wise self- and cross-relations established within
+the shape codes. Meanwhile, we adopt rigid transformations
+between reference frames to align the shape codes for fea-
+ture consistency across different scales. Finally, the deep in-
+tegrated feature is registered to both rotation-invariant shape
+codes to maximize feature similarities, such that rotation in-
+variance of the integrated feature is preserved and shared se-
+mantic information is implicitly extracted from shape codes.
+Experimental results on 3D shape classification, part segmen-
+tation, and retrieval tasks prove the feasibility of our work.
+Our project page is released at: https://rotation3d.github.io/.
+1
+Introduction
+Point cloud analysis has recently drawn much interest from
+researchers. As a common form of 3D representation, the
+growing presence of point cloud data is encouraging the de-
+velopment of many deep learning methods (Qi et al. 2017a;
+Guo et al. 2021; Zhang et al. 2021), showing great success
+for well-aligned point clouds on different tasks. However, it
+is difficult to directly apply 3D models to real data as raw
+3D objects are normally captured at different viewing an-
+gles, resulting in unaligned data samples, which inevitably
+impact the deep learning models which are sensitive to rota-
+tions. Therefore, rotation invariance becomes an important
+research topic in the 3D domain.
+To achieve rotation invariance, a straightforward way is
+to augment training data with massive rotations which, how-
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Aligned Integration
+Transformer
+Registration
+Registered U
+(a)
+(b)
+(c)
+Local & Global Descriptors
+Correspondence
+Mapping
+Local Patches Pℓ Global Shape P"
+Correspondence
+Mapping
+Fℓ
+F"
+U
+RI
+T
+Correspondence
+Mapping
+Shape Descriptors
+(d)
+(e)
+Source Points P# Target Points P$
+T
+Registration
+F#
+F$
+TI
+Encoding
+Integration
+Registration
+Registered P!
+Ours
+PCR
+Figure 1: Frameworks of our design (left) and robust point
+cloud registration (right), where TI and RI are transforma-
+tion invariance and rotation invariance, and T is the rigid
+transformation. The dotted line indicates the computation of
+T between reference frames.
+ever, requires a large memory capacity and exhibits limited
+generalization ability to unseen data (Kim, Park, and Han
+2020). There are attempts to align 3D inputs to a conical
+pose (Jaderberg et al. 2015; Cohen et al. 2018), or to learn
+rotation robust features via equivariance (Deng et al. 2021;
+Luo et al. 2022), while these methods are not rigorously
+rotation-invariant and present noncompetitive performance
+on 3D shape analysis. To maintain consistent model be-
+havior under random rotations, some methods (Zhang et al.
+2019; Chen et al. 2019; Xu et al. 2021) follow Drost et al.
+(2010) to handcraft rotation-invariant point-pair features.
+Others (Zhang et al. 2020; Li et al. 2021a; Zhao et al. 2022)
+design robust features from equivariant orthonormal bases.
+Most of the mentioned works either manipulate model in-
+puts or generate canonical spaces to achieve rotation invari-
+ance (RI). In this work, we review the problem of RI from
+a different aspect: robust point cloud registration (PCR). We
+find that PCR and RI share the same goal: PCR aligns low-
+dimensional point cloud features (e.g., xyz) from the source
+domain to the target domain regardless of transformations,
+arXiv:2301.00149v1  [cs.CV]  31 Dec 2022
+
+while RI can be considered to align high-dimensional la-
+tent features to rotation-invariant features. Specifically, the
+goal of PCR is to explicitly align the source point cloud to
+the target, both representing the same 3D object, and for RI
+learning, we implicitly align the final feature representation
+of a 3D shape to a hidden feature of the same shape, which
+is universally rotation-invariant to any rotations.
+Motivated by this finding, we propose our learning frame-
+work in Fig. 1 with three sequential stages, namely rotation-
+invariant shape encoding, aligned feature integration, and
+deep feature registration. Firstly, we (a) construct and feed
+point pairs with different scales as model inputs, where we
+consider local patches Pℓ with small number of points and
+global shape Pg with the whole 3D points. Hence, the fi-
+nal feature representation can be enriched by information
+from different scales. Low-level rotation-invariant descrip-
+tors are thus built on reference frames and encoded to gener-
+ate latent shape codes Fℓ and Fg following recent PCR work
+(Pan, Cai, and Liu 2022). Secondly, we (b) introduce a vari-
+ant of transformer (Vaswani et al. 2017), Aligned Integration
+Transformer (AIT), to implicitly integrate information from
+both self- and cross-attention branches for effective feature
+integration. In this way, information encoded from different
+point scales is aggregated to represent the same 3D object.
+Moreover, we consider Fℓ and Fg as unaligned since they
+are encoded from unaligned reference frames. To address
+the problem, we follow the evaluation technique proposed
+in PCR (Pan, Cai, and Liu 2022), where we use relative ro-
+tation information (T) with learnable layers to align Fℓ and
+Fg for feature consistency. Finally, to ensure RI of the inte-
+grated feature U, we follow PCR to (c) examine the corre-
+spondence map of (Fg, U) and (Fℓ, U), such that the mu-
+tual information between a local patch of a 3D object and
+the whole 3D object is maximized, and RI is further ensured
+in the final geometric feature.
+The contributions of our work are summarized as follow-
+ing three folds: (1) To our knowledge, we are the first in de-
+veloping a PCR-cored representation learning framework to-
+wards effective RI studies on 3D point clouds. (2) We intro-
+duce Aligned Integration Transformer (AIT), a transformer-
+based architecture to conduct aligned feature integration for
+a comprehensive geometry study from both local and global
+scales. (3) We propose a registration loss to maintain rota-
+tion invariance and discover semantic knowledge shared in
+different parts of the input object. Moreover, the feasibility
+of our proposed framework is successfully demonstrated on
+various 3D tasks.
+2
+Related Work
+Rotation Robust Feature Learning.
+Networks that are
+robust to rotations can be equivariant to rotations. Esteves
+et al. (2018) and Cohen et al. (2018) project 3D data
+into a spherical space for rotation equivariance and per-
+form convolutions in terms of spherical harmonic bases.
+Some (Spezialetti et al. 2020; Sun et al. 2021) learn canon-
+ical spaces to unify the pose of point clouds. Recent
+works (Luo et al. 2022; Deng et al. 2021; Jing et al. 2020)
+vectorize the scalar activations and mapping SO(3) actions
+to a latent space for easy manipulations. Although these
+works present competitive results, they cannot be strictly
+rotation-invariant. Another way for rotation robustness is to
+learn rotation-invariant features. Handcraft point-pair fea-
+tures are rotation-invariant (Zhang et al. 2019; Chen et al.
+2019; Xu et al. 2021), but they focus on local domains
+and ignore the global overview of 3D objects. Others use
+rotation-equivariant local reference frames (LRFs) (Zhang
+et al. 2020; Thomas 2020; Kim, Park, and Han 2020) or
+global reference frames (GRFs) (Li et al. 2021a) as model
+inputs based on principal component analysis (PCA). How-
+ever, they may produce inconsistent features across differ-
+ent reference frames, which would limit the representational
+power. In contrast to abovementioned methods with rotation
+robust model inputs or modules, we examine the relation be-
+tween RI and PCR and propose an effective framework.
+3D Robust Point Cloud Registration.
+Given a pair of Li-
+DAR scans, 3D PCR requires an optimal rigid transforma-
+tion to best align the two scans. Despite the recent emerging
+of ICP-based methods (Besl and McKay 1992; Wang and
+Solomon 2019b), we follow robust correspondence-based
+approaches in our work (Deng, Birdal, and Ilic 2018; Yuan
+et al. 2020; Qin et al. 2022; Pan, Cai, and Liu 2022), where
+RI is widely used to mitigate the impact of geometric trans-
+formations during feature learning. Specifically, both Pan,
+Cai, and Liu (2022) and Qin et al. (2022) analyze the en-
+coding of transformation-robust information and introduce
+a rotation-invariant module with contextual information into
+their registration pipeline. All these methods showing im-
+pressive results are closely related to rotation invariance. We
+hypothesize that the learning framework of RI can be sim-
+ilar to PCR, and we further prove in experiments that our
+network is feasible and able to achieve competitive perfor-
+mance on rotated point clouds.
+Transformers in 3D Point Clouds.
+Transformers (Doso-
+vitskiy et al. 2021; Liu et al. 2021) applied to 2D vision have
+shown great success, and they are gaining prominence in 3D
+point clouds. For example, Zhao et al. (2021) uses vector-
+ized self-attention (Vaswani et al. 2017) and positional em-
+bedding for 3D modeling. Guo et al. (2021) proposes offset
+attention for noise-robust geometric representation learning.
+Cross-attention is widely employed for semantic informa-
+tion exchange (Qin et al. 2022; Yu et al. 2021a), where fea-
+ture relations between the source and target domains are ex-
+plored. Taking advantage of both, we design a simple yet
+effective feature integration module with self and cross re-
+lations. In addition, transformation-related embeddings are
+introduced for consistent feature learning.
+Contrastive Learning with 3D Visual Correspondence.
+Based on visual correspondence, contrastive learning aims
+to train an embedding space where positive samples are
+pushed together whereas negative samples are separated
+away (He et al. 2020). The definition of positivity and neg-
+ativity follows the visual correspondence maps, where pairs
+with high confidence scores are positive otherwise negative.
+Visual correspondence is important in 3D tasks, where se-
+mantic information extracted from matched point pairs im-
+proves the network’s understanding on 3D geometric struc-
+
+tures. For example, PointContrast (Xie et al. 2020) explores
+feature correspondence across multiple views of one 3D
+point cloud with InfoNCE loss (Van den Oord, Li, and
+Vinyals 2018), increasing the model performance for down-
+stream tasks. Info3D (Sanghi 2020) and CrossPoint (Afham
+et al. 2022) minimize the semantic difference of point fea-
+tures under different poses. We follow the same idea by reg-
+istering the deep features to rotation-invariant features at in-
+termediate levels, increasing feature similarities in the em-
+bedding space to ensure rotation invariance.
+3
+Method
+Given a 3D point cloud including Nin points with xyz co-
+ordinates P = {pi ∈ R3}Nin
+i=1, we aim to learn a shape en-
+coder f that is invariant to 3D rotations: f(P) = f(RP),
+where R ∈ SO(3) and SO(3) is the rotation group. RI can
+be investigated and achieved through three stages, namely
+rotation-invariant shape encoding (Section 3.1), aligned fea-
+ture integration (Section 3.2), and deep feature registration
+(Section 3.3).
+3.1
+Rotation-Invariant Shape Encoding
+In this section, we first construct the input point pairs from
+local and global scales based on reference frames, follow-
+ing the idea of Pan, Cai, and Liu (2022) to obtain low-level
+rotation-invariant shape descriptors from LRFs and GRF di-
+rectly. Then we obtain latent shape codes via two set abstrac-
+tion layers as in PointNet++ (Qi et al. 2017b).
+Rotation Invariance for Local Patches.
+To construct
+rotation-invariant features on LRFs, we hope to construct
+an orthonormal basis for each LRF as p ∈ R3×3. Given a
+point pi and its neighbor pj ∈ N(pi), we choose #»
+xiℓ =
+#       »
+pmpi/∥#       »
+pmpi∥2, where pm is the barycenter of the local ge-
+ometry and ∥ · ∥2 is L2-norm. We then define #»
+ziℓ follow-
+ing Tombari, Salti, and Stefano (2010) to have the same di-
+rection as an eigenvector, which corresponds to the smallest
+eigenvalue via eigenvalue decomposition (EVD):
+Σℓ
+i =
+|N (pi)|
+�
+j=1
+αj (#     »
+pipj) (#     »
+pipj)⊤ , αj =
+d − ∥#     »
+pipj∥2
+�|N (pi)|
+j=1
+d − ∥#     »
+pipj∥2
+,
+(1)
+where αj is a weight parameter, allowing nearby pj to have
+large contribution to the covariance matrix, and d is the max-
+imum distance between pi and pj. Finally, we define #»
+yiℓ as
+#»
+ziℓ × #»
+xiℓ. RI is introduced to pi with respect to its neigh-
+bor pj as pℓ
+ij = #     »
+pipj⊤Mℓ
+i. Proofs of the equivariance of Mℓ
+i
+and invariance of pℓ
+ij are shown in the supplementary ma-
+terial. The latent shape code Fℓ ∈ RN×C is obtained via
+PointNet++ and max-pooling.
+Rotation Invariance for Global Shape.
+We apply PCA
+as a practical tool to obtain RI in a global scale. Similar to
+Eq. 1, PCA is performed by
+1
+N0
+�N0
+i=1(#       »
+pmpi)(#       »
+pmpi)⊤ =
+UgΛgUg⊤, where pm is the barycenter of P, Ug
+=
+[# »
+u1g, # »
+u2g, # »
+u3g] and Λg = diag(λg
+1, λg
+2, λg
+3) are eigenvec-
+tor and eigenvalue matrices. We take Ug as the orthonor-
+mal basis Mg = [#»x g, #»y g, #»z g] for GRF. By transform-
+ing point pi with Ug, the shape pose is canonicalized as
+pg
+i = piMg. Proof of the RI of pg
+i is omitted for its sim-
+plicity, and Fg ∈ RN×C is obtained following PointNet++.
+Sign Ambiguity.
+EVD introduces sign ambiguity for
+eigenvectors, which negatively impacts the model perfor-
+mance (Bro, Acar, and Kolda 2008). The description of sign
+ambiguity states that for a random eigenvector #»u, #»u and
+#»u ′, with #»u ′ having an opposite direction to #»u, are both
+acceptable solutions to EVD. To tackle this issue, we sim-
+ply force #»
+ziℓ of LRF to follow the direction of #  »
+opi, with o
+being the origin of the world coordinate. We disambiguate
+basis vectors in Mg by computing an inner product with
+#       »
+pmpi, ∀i ∈ N0. Taking #»x g for example, its direction is con-
+ditioned on the following term:
+#»x g =
+�#»x g,
+if Sx ≥ N0
+2
+#»x ′g,
+otherwise
+,
+Sx =
+N0
+�
+i=1
+1[⟨#»x g, #       »
+pmpi⟩],
+(2)
+where ⟨·, ·⟩ is the inner product, 1[·] is a binary indicator that
+returns 1 if the input argument is positive, otherwise 0. Sx
+denotes the number of points where #»x g and #       »
+pmpi point to
+the same direction. The same rule is applied to disambiguate
+#»y g and #»z g by Sy and Sz. Besides, as mentioned in Li et al.
+(2021a), Mg might be non-rotational (e.g., reflection). To
+ensure Mg a valid rotation, we simply reverse the direction
+of the basis vector whose S value is the smallest. More anal-
+yses on sign ambiguity are in the supplementary material.
+3.2
+Aligned Feature Integration
+Transformer has been widely used in 3D domain to cap-
+ture long-range dependencies (Yu et al. 2021b). In this sec-
+tion, we introduce Aligned Integration Transformer (AIT),
+an effective transformer to align latent shape codes with rel-
+ative rotation angles and integrate information via attention-
+based integration (Cheng et al. 2021). Within each AIT mod-
+ule, we first apply Intra-frame Aligned Self-attention on Fℓ
+and we do not encode Fg, which is treated as supplemen-
+tary information to assist local geometry learning with the
+global shape overview. We discuss that encoding Fg via self-
+attention can increase model overfitting, thus lowering the
+model performance. We will validate our discussion in Sec-
+tion 4.4. Inter-frame Aligned Cross-attention is applied on
+both Fℓ and Fg, and we use Attention-based Feature Inte-
+gration module for information Aggregation.
+Preliminary: Offset Attention.
+AIT utilizes offset atten-
+tion (Guo et al. 2021) for noise robustness. In the follow-
+ing, we use subscripts sa and ca to denote implementations
+related to self- and cross-attention, respectively. We first re-
+view offset attention as follows:
+F = φ(Foa) + Fin, Foa = Fin − ∥softmax(A)∥1v, A = qk⊤,
+(3)
+where q = FinWq, k = FinWk ∈ RN×d, and v =
+FinWv ∈ RN×C are query, key, and value embeddings,
+and Wq, Wk ∈ RC×d, Wv ∈ RC×C are the correspond-
+ing projection matrices. ∥ · ∥1 is L1-norm and φ denotes a
+multi-layer perceptron (MLP). Foa is offset attention-related
+feature and A ∈ RN×N is the attention logits.
+
+𝑾𝒄𝒂
+𝒗
+𝑾𝒄𝒂
+𝒌
+𝑾𝒔𝒂
+𝒒
+𝐞𝒄𝒂
+𝜶 𝑾𝒄𝒂
+𝜶
+𝑭()
+ℓ/,
+𝐹ℓ/,
+𝑨𝒄𝒂
+𝒂𝒕𝒕𝒏: 𝑁×𝑁
+𝑨𝒄𝒂
+𝒓𝒐𝒕: 𝑁×𝑁
+𝑨𝒄𝒂: 𝑁×𝑁
+𝐹,/ℓ
+𝑾𝒔𝒂
+𝒗
+𝑾𝒔𝒂
+𝒌
+𝑾𝒔𝒂
+𝒒
+𝐤𝒔𝒂: 𝑁×𝑑
+𝐪𝒔𝒂: 𝑁×𝑑
+𝐯𝒔𝒂: 𝑁×𝐶
+𝐞𝒔𝒂
+𝜶 𝑾𝒔𝒂
+𝜶
+𝑭()
+ℓ/,
+𝐹ℓ/,
+𝑨𝒔𝒂
+𝒂𝒕𝒕𝒏: 𝑁×𝑁
+𝑨𝒔𝒂
+𝒓𝒐𝒕: 	𝑁×𝑁
+𝑨1): 𝑁×𝑁
+𝑁×𝑁×𝑑
+𝑁×𝑑
+𝐯𝒄𝒂: 𝑁×𝐶
+𝐤𝒔𝒂: 𝑁×𝑑
+𝐪𝒄𝒂: 𝑁×𝑑
+(a) Intra-frame Aligned Self-attention
+(b) Inter-frame Aligned Cross-attention
+𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧
+𝐦𝐚𝐭𝐫𝐢𝐱	𝐩𝐫𝐨𝐝𝐮𝐜𝐭
+𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧
+𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧
+𝐦𝐚𝐭𝐫𝐢𝐱	𝐩𝐫𝐨𝐝𝐮𝐜𝐭
+𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧
+Figure 2: Illustrations of (a) Intra-frame Aligned Self-attention and (b) Inter-frame Aligned Cross-attention modules. Note that
+we only present processes for computing Foa in both modules.
+Intra-frame Aligned Self-attention.
+Point-wise features
+of Fℓ are encoded from unaligned LRFs, so direct imple-
+mentation of self-attention on Fℓ can cause feature inconsis-
+tency during integration. To solve this problem, rigid trans-
+formations between distinct LRFs are considered, which
+are explicitly encoded and injected into point-wise relation
+learning process. We begin by understanding the transfor-
+mation between two LRFs. For any pair of local orthonormal
+bases Mℓ
+i and Mℓ
+j, a rotation can be easily derived ∆Rji =
+Mℓ
+iMℓ
+j
+⊤ and translation is defined as ∆tji = oℓ
+i −oℓ
+j, where
+oℓ
+i/j indicates the origin. In our work, the translation part is
+intentionally ignored, where we show in the supplementary
+material that by keeping both rotation and translation infor-
+mation, the model performance decreases.
+Although ∆Rji is invariant to rotations, we do not di-
+rectly project it into the embedding space, as it is sensitive
+to the order of matrix product: ∆Rji ̸= ∆Rij, giving in-
+consistent rotation information when the product order is not
+maintained. To address this issue, we construct our embed-
+ding via the relative rotation angle ∆αji between Mℓ
+i and
+Mℓ
+j, which is normally used in most PCR works (Yew and
+Lee 2020; Pan, Cai, and Liu 2022) for evaluations. The rel-
+ative rotation angle ∆αji is computed as:
+∆αji = arccos
+�Trace (∆Rji) − 1
+2
+� 180
+π
+∈ [0, π],
+(4)
+where it is easy to see that ∆αji = ∆αij. We further apply
+sinusoidal functions on ∆αji to generate N 2 pairs of angu-
+lar embeddings eα ∈ RN×N×d for all N points as:
+eα
+i,j,2k = sin
+� ∆αji/tα
+100002k/d
+�
+, eα
+i,j,2k+1 = cos
+� ∆αji/tα
+100002k/d
+�
+,
+(5)
+where tα controls the sensitivity to angle variations.
+Finally, we inject eα into offset attention and learn intra-
+frame aligned feature Fℓ
+IAS via self-attention as follows:
+Fℓ
+IAS = φ
+�
+Fℓ
+oa
+�
++ Fℓ, Fℓ
+oa = Fℓ − ∥ softmax(Asa)∥1vsa,
+Asa = Aattn
+sa
++ Arot
+sa , Aattn
+sa
+= qsak⊤
+sa, Arot
+sa = qsa(eα
+saWα
+sa)⊤,
+(6)
+where qsa/ksa/vsa = FℓWq
+sa/FlWk
+sa/FlWv
+sa, Wα
+sa ∈
+Rd×d is a linear projection to refine the learning of eα
+sa, and
+Asa is the attention logits. The same process can be per-
+formed for Fg by swapping the index ℓ and g. Detailed il-
+lustrations are shown in Fig. 2 (a).
+Inter-frame Aligned Cross-attention.
+Semantic infor-
+mation exchange between Fℓ and Fg in the feature space
+is implemented efficiently by cross-attention (Chen, Fan,
+and Panda 2021). Since Fℓ and Fg are learned from differ-
+ent coordinate systems, inter-frame transformations should
+be considered for cross-consistency between Fℓ and Fg.
+An illustration of the cross-attention module is shown in
+Fig. 2 (b). Computation of inter-frame aligned feature Fℓ
+IAC
+via cross-attention follows a similar way as Eq. 6:
+Fℓ
+IAC = φ
+�
+Fℓ
+oa
+�
++ Fℓ, Fℓ
+oa = Fℓ − ∥ softmax(Aca)∥1vca,
+Aca = Aattn
+ca
++ Arot
+ca , Aattn
+ca
+= qcak⊤
+ca, Arot
+ca = qca(eα
+caWα
+ca)⊤,
+(7)
+where qca/kca/vca = FℓWq
+ca/FgWk
+ca/FgWv
+ca. Aca is
+cross-attention logits containing point-wise cross-relations
+over point features defined across local and global scales.
+eα
+ca ∈ RN×d is computed via Eq. 4 and Eq. 5 in terms of the
+transformation between Mℓ
+i and Mg. To this end, the geo-
+metric features learned between local and global reference
+frames can be aligned given eα
+ca, leading to a consistent fea-
+ture representation.
+
+Attention-based Feature Integration.
+Instead of simply
+adding the information from both Fℓ and Fg, we integrate
+information by incrementing attention logits. Specifically,
+we apply self-attention on Fℓ with attention logits Asa
+and cross-attention between Fℓ and Fg with attention log-
+its Aca. We combine Asa and Aca via addition, so that en-
+coded information of all point pairs from a local domain can
+be enriched by the global context of the whole shape. Illus-
+tration is shown in the supplementary material. The whole
+process is formulated as follows:
+U = φ (Foa) + Fℓ,
+Foa = Fℓ − ∥softmax(Asa + Aca)∥1(vsa + vca).
+(8)
+Hence, intra-frame point relations can be compensated by
+inter-frame information communication in a local-to-global
+manner, which enriches the geometric representations.
+3.3
+Deep Feature Registration
+Correspondence mapping (Wang and Solomon 2019a; Pan,
+Cai, and Liu 2022) plays an important role in PCR, and we
+discuss that it is also critical for achieving RI in our design.
+Specifically, although Fℓ and Fg are both rotation-invariant
+by theory, different point sampling methods and the sign
+ambiguity will cause the final feature not strictly rotation-
+invariant. To solve this issue, we first examine the correspon-
+dence map:
+m (X, Y) =
+exp
+�
+Φ1(Y)Φ2(X)⊤/t
+�
+�N
+j=1 exp (Φ1(Y)Φ2(xj)⊤/t)
+,
+(9)
+where Φ1 and Φ2 are MLPs that project latent embeddings
+X and Y to a shared space, and t controls the variation sen-
+sitivity. It can be seen from Eq. 9 that the mapping function
+m reveals feature similarities in the latent space, and it is
+also an essential part for 3D point-level contrastive learning
+in PointContrast (Xie et al. 2020) for the design of InfoNCE
+losses (Van den Oord, Li, and Vinyals 2018), which have
+been proven to be equivalent to maximize the mutual infor-
+mation. Based on this observation, we propose a registration
+loss function Lr = Lℓ
+r + Lg
+r, where Lℓ
+r and Lg
+r represent the
+registration loss of (Fℓ,U) and (Fg,U). Mathematically, Lℓ
+r
+is defined as follows:
+Lℓ
+r = −
+�
+(i,j)∈M
+log
+exp
+�
+Φ1(Uj)Φ2(f ℓ
+i )⊤/t
+�
+�
+(·,k)∈M exp
+�
+Φ1(Uk)Φ2(f ℓ
+i )⊤/t
+�. (10)
+The same rule is followed to compute Lg
+r. Although we fol-
+low the core idea of PointContrast, we differ from it in that
+PointContrast defines positive samples based on feature cor-
+respondences computed at the same layer level, while our
+positive samples are defined across layers.
+The intuition for the loss design is that the 3D shape is
+forced to learn about its local region as it has to distinguish
+it from other parts of different objects. Moreover, we would
+like to maximize the mutual information between different
+poses of the 3D shape, as features encoded from different
+poses should represent the same object, which is very use-
+ful in achieving RI in SO(3). Moreover, the mutual infor-
+mation between Fℓ and Fg is implicitly maximized, such
+Rotation Sensitive
+z/z
+z/SO(3)
+SO(3)/SO(3)
+∆
+PointNet (Qi et al. 2017a)
+89.2
+16.2
+75.5
+59.3
+PoinNet++ (Qi et al. 2017b)
+89.3
+28.6
+85.0
+56.4
+PCT (Guo et al. 2021)
+90.3
+37.2
+88.5
+51.3
+Rotation Robust
+z/z
+z/SO(3)
+SO(3)/SO(3)
+∆
+Spherical CNN* (Esteves et al. 2018)
+88.9
+76.9
+86.9
+10
+SFCNN (Rao, Lu, and Zhou 2019)
+91.4
+84.8
+90.1
+5.3
+RIConv (Zhang et al. 2019)
+86.5
+86.4
+86.4
+0.1
+ClusterNet (Chen et al. 2019)
+87.1
+87.1
+87.1
+0.0
+PR-InvNet (Yu et al. 2020)
+89.2
+89.2
+89.2
+0.0
+RI-GCN (Kim, Park, and Han 2020)
+89.5
+89.5
+89.5
+0.0
+GCAConv (Zhang et al. 2020)
+89.0
+89.1
+89.2
+0.1
+RI-Framework (Li et al. 2021b)
+89.4
+89.4
+89.3
+0.1
+VN-DGCNN (Deng et al. 2021)
+89.5
+89.5
+90.2
+0.7
+SGMNet (Xu et al. 2021)
+90.0
+90.0
+90.0
+0.0
+Li et al. (2021a)
+90.2
+90.2
+90.2
+0.0
+OrientedMP (Luo et al. 2022)
+88.4
+88.4
+88.9
+0.5
+ELGANet (Gu et al. 2022)
+90.3
+90.3
+90.3
+0.0
+Ours
+91.0
+91.0
+91.0
+0.0
+Table 1: Classification results on ModelNet40 under rota-
+tions. * denotes the input type as projected voxels of 2×642,
+while the rest take raw points of 1024×3 as inputs. ∆ is the
+absolute difference between z/SO(3) and SO(3)/SO(3).
+that shared semantic information about geometric structures
+can be learned, leading to a more geometrically accurate and
+discriminative representation. More details about Lℓ
+r can be
+found in the supplementary material.
+4
+Experiments
+We evaluate our model on 3D shape classification, part seg-
+mentation, and retrieval tasks under rotations, and exten-
+sive experiments are conducted to analyze the network de-
+sign. Detailed model architectures for the three tasks are
+shown in the supplementary material. Our evaluating proto-
+cols are the same as (Esteves et al. 2018): training and testing
+the network under azimuthal rotations (z/z); training under
+azimuthal rotations while testing under arbitrary rotations
+(z/SO(3)); and training and testing under arbitrary rotations
+(SO(3)/SO(3)).
+4.1
+3D Object Classification
+Synthetic Dataset.
+We first examine the model perfor-
+mance on the synthetic ModelNet40 (Wu et al. 2015)
+dataset. We sample 1024 points from each data with only
+xyz coordinates as input features. Hyper-parameters for
+training follow the same as (Guo et al. 2021), except
+that points are downsampled in the order of (1024, 512,
+128) with feature dimensions of (3, 128, 256). We report
+and compare our model performance with state-of-the-art
+(SoTA) methods in Table 1. Both rotation sensitive and ro-
+bust methods achieve great performance under z/z. How-
+ever, the former could not generalize well to unseen rota-
+tions. Rotation robust methods like Spherical CNN (Esteves
+et al. 2018) and SFCNN (Rao, Lu, and Zhou 2019) achieve
+competitive results under z/z, but their performance is not
+consistent on z/SO(3) and SO(3)/SO(3) due to the imperfect
+projection from points to voxels when using spherical so-
+lutions. We outperform the recent proposed methods (Luo
+et al. 2022; Xu et al. 2021; Deng et al. 2021) and achieve an
+accuracy of 91.0%, proving the superiority of our framework
+on classification.
+
+Method
+z/SO(3)
+SO(3)/SO(3)
+∆
+PointNet (Qi et al. 2017a)
+16.7
+54.7
+38.0
+PointNet++ (Qi et al. 2017b)
+15.0
+47.4
+32.4
+PCT (Guo et al. 2021)
+28.5
+45.8
+17.3
+RIConv (Zhang et al. 2019)
+78.4
+78.1
+0.3
+RI-GCN (Kim, Park, and Han 2020)
+80.5
+80.6
+0.1
+GCAConv (Zhang et al. 2020)
+80.1
+80.3
+0.2
+RI-Framework (Li et al. 2021b)
+79.8
+79.9
+0.1
+LGR-Net (Zhao et al. 2022)
+81.2
+81.4
+0.2
+VN-DGCNN (Deng et al. 2021)
+79.8
+80.3
+0.5
+OrientedMP (Luo et al. 2022)
+76.7
+77.2
+0.5
+Ours
+86.6
+86.3
+0.3
+Table 2: Classification results on ScanObjectNN OBJ BG
+under z/SO(3) and SO(3)/SO(3).
+GT
+Ours
+RI-GCN
+RIConv
+VN-DGCNN
+Figure 3: Segmentation comparisons on ShapeNetPart,
+where ground truth (GT) samples are shown for refer-
+ence. Red dotted circles indicate obvious failures on certain
+classes, and purple circles denote the slight difference be-
+tween our design and VN-DGCNN.
+Real Dataset.
+Experiments are also conducted on a real-
+scanned dataset. ScanObjectNN (Uy et al. 2019) is a com-
+monly used benchmark to explore the robustness to noisy
+and deformed 3D objects with non-uniform surface density,
+which includes 2,902 incomplete point clouds in 15 classes.
+We use OBJ BG subset with the background noise and sam-
+ple 1,024 points under z/SO(3) and SO(3)/SO(3). Table 2
+shows that our model achieves the highest results with ex-
+cellent consistency with random rotations.
+4.2
+3D Part Segmentation
+Shape part segmentation is a more challenging task than ob-
+ject classification. We use ShapeNetPart (Yi et al. 2016) for
+evaluation, where we sample 2048 points with xyz coordi-
+nates as model inputs. The training strategy is the same as
+the classification task except that the training epoch num-
+ber is 300. Part-averaged IoU (mIoU) is reported in Table
+3, and detailed per-class mIoU values are shown in the sup-
+plementary material. Representative methods such as Point-
+Method
+z/SO(3)
+SO(3)/SO(3)
+∆
+PointNet (Qi et al. 2017a)
+38.0
+62.3
+24.3
+PointNet++ (Qi et al. 2017b)
+48.3
+76.7
+28.4
+PCT (Guo et al. 2021)
+38.5
+75.2
+36.7
+RIConv (Zhang et al. 2019)
+75.3
+75.5
+0.2
+RI-GCN (Kim, Park, and Han 2020)
+77.2
+77.3
+0.1
+RI-Framework (Li et al. 2021b)
+79.2
+79.4
+0.2
+LGR-Net (Zhao et al. 2022)
+80.0
+80.1
+0.1
+VN-DGCNN (Deng et al. 2021)
+81.4
+81.4
+0.0
+OrientedMP (Luo et al. 2022)
+80.1
+80.9
+0.8
+Ours
+80.3
+80.4
+0.1
+Table 3: Segmentation results on ShapeNetPart under
+z/SO(3) and SO(3)/SO(3), where the second best results are
+underlined.
+Method
+micro mAP
+macro mAP
+Score
+Spherical CNN (Esteves et al. 2018)
+0.685
+0.444
+0.565
+SFCNN (Rao, Lu, and Zhou 2019)
+0.705
+0.483
+0.594
+GCAConv (Zhang et al. 2020)
+0.708
+0.490
+0.599
+RI-Framework (Li et al. 2021b)
+0.707
+0.510
+0.609
+Ours
+0.715
+0.510
+0.613
+Table 4: Comparisons of SoTA methods on the 3D shape
+retrieval task.
+Net++ and PCT are vulnerable to rotations. Rotation robust
+methods present competitive results under z/SO(3), where
+we achieve the second best result of 80.3%. We give more
+details of comparison between VN-DGCNN (Deng et al.
+2021) and our work in the supplementary material, where
+our method performs better than VN-DGCNN for several
+classes. Moreover, qualitative results shown in Fig. 3 present
+that we can achieve visually better results than VN-DGCNN
+in certain classes such as the airplane and car. More qualita-
+tive results are shown in the supplementary material.
+4.3
+3D Shape Retrieval
+We further conduct 3D shape retrieval experiments on
+ShapeNetCore55 (Chang et al. 2015), which contains two
+categories of datasets: normal and perturbed. We only use
+the perturbed part to validate our model performance under
+rotations. We combine the training and validation sets and
+validate our method on the testing set following the training
+policy of (Esteves et al. 2018). Experimental results are re-
+ported in Table 4, where the final score is the average value
+of micro and macro mean average of precision (mAP) as
+in (Savva et al. 2017). Similar to the classification task, our
+method achieves SoTA performance.
+4.4
+Ablation Study
+Effectiveness of Transformer Designs.
+We examine the
+effectiveness of our transformer design by conducting clas-
+sification experiments under z/SO(3). We first ablate one or
+both of the angular embeddings and report the results in Ta-
+ble 5 (models A, B, and C). Model B performs better than
+model C by 0.4%, which validates our design of feature in-
+tegration where Mℓ
+i is used as the main source of informa-
+tion. When both angular embeddings are applied, the best
+result is achieved (i.e., 91.0%). Moreover, we validate our
+discussion in Section 3.2 by comparing models D and E. We
+
+.:Model
+eα
+sa
+eα
+ca
+Fg∗
+Asa + Aca
+Lℓ
+r
+Lg
+r
+Acc.
+A
+✓
+✓
+✓
+90.0
+B
+✓
+✓
+✓
+✓
+90.6
+C
+✓
+✓
+✓
+✓
+90.2
+D
+✓
+✓
+✓
+✓
+✓
+✓
+90.2
+E
+✓
+✓
+✓
+✓
+90.4
+F
+✓
+✓
+✓
+90.0
+G
+✓
+✓
+✓
+✓
+90.2
+H
+✓
+✓
+✓
+✓
+90.6
+Ours
+✓
+✓
+✓
+✓
+✓
+91.0
+Table 5: Module analysis of AIT and loss functions. Fg∗
+means encoding Fg via Intra-frame Aligned Self-attention.
+demonstrate in model D that when encoding Fg in the same
+way as Fℓ, the model performance decreases, which indi-
+cates that encoding Fg via self-attention will increase the
+model overfitting. More analyses can be found in the supple-
+mentary material. Finally, we examine the effectiveness of
+our attention logits-based integration scheme by comparing
+our model with the conventional method (model E), which
+applies self- and cross-attention sequentially and repeatedly.
+We observe that our result is better than model E by 0.6%,
+indicating that our design is more effective.
+Registration Loss.
+We sequentially ablate Lg
+r and Lℓ
+r
+(models F, G, and H) to check the effectiveness of our reg-
+istration loss deign. Results in Table 5 demonstrate that we
+can still achieve a satisfactory result of 90.0% without fea-
+ture registration. Individual application of Lg
+r and Lℓ
+r shows
+the improvement when forcing the final representation to be
+close to rotation-invariant features. Moreover, it can be seen
+that model H performs better than model G, which indicates
+that intermediate features learned from the global scale are
+important for shape classification. The best model perfor-
+mance is hence achieved by applying both losses.
+Noise Robustness.
+In real-world applications, raw point
+clouds contain noisy signals. We conduct experiments to
+present the model robustness to noise under z/SO(3). Two
+experiments are conducted: (1) We sample and add Gaus-
+sian noise of zero mean and varying standard deviations
+N(0, σ2) to the input data; (2) We add outliers sampled
+from a unit sphere to each object. As shown in Fig. 4 (left),
+we achieve on par results to RI-Framework when std is low,
+while we perform better while std increases, indicating that
+our model is robust against high levels of noise. Besides,
+as the number of noisy points increases, most methods are
+heavily affected while we can still achieve good results.
+Visualization of Rotation Invariance.
+We further ex-
+amine RI of learned features. Specifically, we use Grad-
+CAM (Selvaraju et al. 2017) to check how the model pays
+attention to different parts of data samples under different
+rotations. Results are reported in Fig. 5 with correspondence
+between gradients and colors shown on the right. RI-GCN
+presents a good result, but its behavior is not consistent over
+some classes (e.g., vase and plant) and it does not pay atten-
+tion to regions that are critical for classification (see toilet),
+showing inferior performance to ours. PointNet++ shows no
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+std of noise
+20
+30
+40
+50
+60
+70
+80
+90
+Accuracy (%)
+RIConv
+RI-GCN
+RI-framework
+SRINet
+ours
+0
+10
+20
+30
+40
+50
+# of noisy points
+30
+40
+50
+60
+70
+80
+90
+Figure 4: Left: Results on Gaussian noise of zero mean and
+variant standard deviation values. Right: Results on differ-
+ent numbers of noisy points.
+high
+low
+high
+low
+airplane
+guitar
+vase
+plant
+toilet
+PointNet++
+RI-GCN
+high
+low
+Ours
+Figure 5: Network attention on PointNet++ (top), RI-GCN
+(mid) and our model (bot).
+resistance to rotations, while our method exhibits a consis-
+tent gradient distribution over different parts with random
+rotations, indicating our network is not affected by rotations.
+5
+Conclusion
+In this work, we rethink and investigate the close relation be-
+tween rotation invariance and point cloud registration, based
+on which we propose a PCR-cored learning framework with
+three stages. With a pair of rotation-invariant shape descrip-
+tors constructed from local and global scales, a comprehen-
+sive learning and feature integration module is proposed,
+Aligned Integration Transformer, to simultaneously effec-
+tively align and integrate shape codes via self- and cross-
+attentions. To further preserve rotation invariance in the fi-
+nal feature representation, a registration loss is proposed to
+align it with intermediate features, where shared semantic
+knowledge of geometric parts is also extracted. Extensive
+experiments demonstrated the superiority and robustness of
+our designs. In future work, we will examine efficient meth-
+ods for invariance learning on large-scale point clouds.
+
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+
+Rethinking Rotation Invariance with Point Cloud Registration
+**Supplementary Material**
+Jianhui Yu, Chaoyi Zhang, Weidong Cai
+School of Computer Science, University of Sydney, Australia
+{jianhui.yu, chaoyi.zhang, tom.cai}@sydney.edu.au
+Mathematical Proofs
+Proof of Rotation Equivariance of Orthonormal Basis
+Mℓ
+i of LRFs.
+We prove the rotation equivariance of Mℓ
+i
+designed for LRFs as mentioned in Section 3.1 of the main
+work. Given a random rotation matrix R ∈ R3×3, it is easy
+to derive that #»
+xiℓ is equivariant to rotations given the rotated
+version #»
+xiℓ
+,rot:
+#»
+xi
+ℓ
+,rot =
+Rpi − Rpm
+∥Rpi − Rpm∥2
+=
+R(pi − pm)
+�
+(R(pi − pm))⊤ R(pi − pm)
+=
+R#       »
+pmpi
+�#       »
+pmpi⊤R⊤R#       »
+pmpi
+= R
+#       »
+pmpi
+∥#       »
+pmpi∥2
+= R#»
+xi
+ℓ,
+(1)
+where the subscript rot represents the axis after rotations.
+Moreover, Σℓ
+i from Eq. (1) of the main work after rotations
+can be represented as follows:
+Σℓ
+i,rot =
+|N (pi)|
+�
+j=1
+αjR#     »
+pipj #     »
+pipj
+⊤R⊤
+= R
+�
+�
+|N (pi)|
+�
+j=1
+αj #     »
+pipj #     »
+pipj
+⊤
+�
+� R⊤ = RΣℓ
+iR⊤.
+(2)
+As mentioned in the main work, eigenvalue decomposition
+can be directly applied to Σℓ
+i, resulting in the following ex-
+pressions:
+RΣℓ
+iR⊤ = RUℓ
+iΛℓ
+iUℓ
+i
+⊤R⊤ =
+�
+RUℓ
+i
+�
+Λℓ
+i
+�
+RUℓ
+i
+�⊤ . (3)
+Since #»
+ziℓ is defined to have the same direction as the eigen-
+vector with the smallest eigenvalue, after rotation, #»
+ziℓ
+,rot =
+R#»
+ziℓ. Thus, the rotated y-axis is:
+#»
+yi
+ℓ
+,rot = #»
+zi
+ℓ
+,rot × #»
+xi
+ℓ
+,rot = R#»
+zi
+ℓ × R#»
+xi
+ℓ
+= det (R)
+�
+R−1�⊤ �#»
+zi
+ℓ × #»
+xi
+ℓ�
+= R
+�#»
+zi
+ℓ × #»
+xi
+ℓ�
+= R#»
+yi
+ℓ.
+(4)
+Since all basis vectors are rotation-equivariant, the local or-
+thonormal basis Mℓ
+i = [#»
+xiℓ, #»
+yiℓ, #»
+ziℓ] is rotation-equivariant.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Proof of Rotation Invariance of Local Shape Descriptors
+pℓ
+ij.
+Here, we show the rotation invariance of local shape
+descriptors pℓ
+ij introduced in Section 3.1 of the main work.
+Based on the proof shown above, it is easy to show the rota-
+tion invariance of pℓ
+ij after rotation R:
+pℓ
+ij,rot = #     »
+pipj
+⊤
+,rotMℓ
+i,rot = (R#     »
+pipj)⊤ �
+RMℓ
+i
+�
+= #     »
+pipj
+⊤Mℓ
+i = pℓ
+ij.
+(5)
+Model Details
+Architectures.
+The model overview for the 3D classifica-
+tion, segmentation, and retrieval tasks is shown in Fig. 1,
+where details of each module design are explained in Sec-
+tion 3 of the main work. The architecture of attention-based
+feature integration module is illustrated in Fig. 2. Please re-
+fer to Section 3.2 of the main work for more details.
+Fg in AIT.
+As mentioned in Section 3.2, to alleviate the
+model overfitting we do not apply self-attention on Fg. We
+argue that since we use global shape information as the sup-
+plementary material to assist local shape learning, this im-
+plementation allow lower-level information to flow effec-
+tively across layers to help the learning of higher-level lo-
+cal shape features, which could reduce the model overfit-
+ting. We encode Fℓ in AIT blocks, as abstracting informa-
+tion from local structures can increase the model’s ability on
+fine-grained pattern recognition and generalizability to com-
+plex scenes (Qi et al. 2017b). Hence, we find that without
+applying learnable modules on Fg is beneficial to our model
+performance.
+Registration Loss.
+In this this part, we first explain the
+benefits of applying the registration loss to preserve the ro-
+tation invariance. We then give more details about Eq. (10)
+in the main work.
+Suppose Uℓ and Ug are local and global part information
+of the final integrated feature U. By maximizing the mutual
+information between (U, Fℓ) and (U, Fg), (Uℓ, Fℓ) and
+(Ug, Fg) are implicitly maximized. In this case, the shared
+geometric information between the local Fℓ/global Fg and
+the integrated domain U are refined, increasing the repre-
+sentation power of U. Besides, the maximized similarities of
+(Uℓ, Fℓ) and (Ug, Fg) also tend to learn rotation invariance
+in an unsupervised manner. Specifically, although Fℓ/Uℓ en-
+codes local patches with different poses (since LRFs are
+arXiv:2301.00149v1  [cs.CV]  31 Dec 2022
+
+airplane
+car
+chair
+…
+toilet
+Rotational 
+Embeddin
+g
+…
+Shape Retrieval
+Classification
+GRF
+LRF
+SA x 2
+SA x 2
+AIT x 4
+U
+𝑳𝒓
+"
+𝑳𝒓ℓ
+MLP
+max
+Fℓ
+F$
+MLP
+Rotation-Invariant 
+Shape Encoding
+Aligned Feature 
+Integration
+Deep Feature
+Registration
+(a) classification/retrieval
+GRF
+LRF
+SA x 2
+SA x 2
+AIT x 4
+U
+𝑳𝒓
+"
+𝑳𝒓ℓ
+MLP
+Fℓ
+F$
+MLP
+FP x 2
+Rotational 
+Embeddin
+g
+Rotation-Invariant 
+Shape Encoding
+Aligned Feature 
+Integration
+Deep Feature
+Registration
+Part Segmentation
+(b) segmentation
+Figure 1: Model overviews for (a) classification / retrieval
+and (b) segmentation. GRF: global reference frame; LRF:
+local reference frame; SA: set abstraction; AIT: Aligned In-
+tegration Transformer; and FP: forward passing.
+𝑾!"
+#
+𝑾!"
+$
+𝑾!"
+%
+𝐤!": 𝑁×𝑑
+𝐪!": 𝑁×𝑑
+𝐯!": 𝑁×𝐶
+𝐤#": 𝑁×𝑑
+𝐯#": 𝑁×𝐶
+𝑾&"
+$
+𝑾&"
+#
+𝑭ℓ
+𝑭(
+𝑨𝒔𝒆𝒍𝒇
+𝒂𝒕𝒕𝒏: 𝑁×𝑁
+𝑨𝒔𝒆𝒍𝒇
+𝑨𝒄𝒓𝒐𝒔𝒔
+𝒓𝒐𝒕
+: 𝑁×𝑁
+𝑨𝒄𝒓𝒐𝒔𝒔
+𝑨𝒔𝒉𝒂𝒓𝒆𝒅: 𝑁×𝑁
+𝑨𝒔𝒆𝒍𝒇
+𝒓𝒐𝒕 : 𝑁×𝑁 𝑨𝒄𝒓𝒐𝒔𝒔
+𝒂𝒕𝒕𝒏 : 𝑁×𝑁
+𝑭0"
+𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧
+𝐦𝐚𝐭𝐫𝐢𝐱	𝐩𝐫𝐨𝐝𝐮𝐜𝐭
+𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧
+Figure 2: Illustrations of attention-based feature integra-
+tion, where blue and green boxes indicate self- and cross-
+attention. Brown, gold and purple colored components cor-
+respond to v, k and q implementations.
+unaligned), and Fg/Ug encodes the whole 3D object in a
+canonical pose, their feature similarity should be enforced
+to be similar as they represent the same 3D object, no matter
+what their poses are. Moreover, the mutual information be-
+tween a local scale of a 3D object and a global scale of the
+same object is maximized, which embeds U with more ac-
+curate geometric information to distinguish it from objects
+in different classes.
+We then explain the symbols in Eq. (10). M stands for
+the set of all B × N pairs of positive samples across mini-
+batches, i.e., M = {(0, 0), ..., (B ×N, B ×N)}, where B is
+the number of batch size and N is the number of points. The
+point feature Uk is the set of negative keys where (·, k) ∈ M
+and k ̸= j. Note that since the registration loss function is
+applied to both Fℓ and Fg, the point features of the same
+point encoded from the local and global scales are pushed
+close to each other, where the mutual information is in-
+creased such that shared geometric information can be dis-
+covered across the local and global scales.
+Training Details.
+For all three tasks, we set the batch
+size to 32 for training and 16 for testing. We use farthest
+point sampling to re-sample the points from the initial 10k
+points to 1024 points for classification and retrieval and 2048
+points for segmentation. Random point translation within
+[−0.2, 0.2] and rescaling within [0.67, 1.5] were adopted for
+augmentation. We trained the model for 250 epochs with
+tα = 15 and t = 0.017. SGD is adopted as the optimizer,
+where the learning rate was set to 1e-2 with momentum of
+0.9 and weight decay of 1e-4. Cosine annealing was applied
+to reschedule the learning rate for each epoch. For classi-
+fication and retrieval, we used one RTX2080Ti GPU with
+PyTorch for model implementation, and we used two GPUs
+for the segmentation task. The normal vector information is
+ignored for all experiments.
+More Analysis Experiments
+Influence of Randomness.
+We report the variance and
+mean values of each model in Table 5 of the main work to
+derive a more accurate and reliable estimate of our model
+performance. We hence report the variance and mean values
+of performance of each model in Table 5 on ModelNet40
+with 5 training rounds. As shown in Table 1, we can see
+that even for our model weights with the lowest performance
+90.6% (among our five repeated runs), it still surpasses the
+highest performance among models from A to H.
+Model
+A
+B
+C
+Acc. (%)
+89.8±0.2
+90.4±0.2
+90.1±0.1
+Model
+D
+E
+F
+Acc. (%)
+89.8±0.4
+90.1± 0.3
+89.6±0.4
+Model
+G
+H
+Best
+Acc. (%)
+90.0±0.2
+90.3±0.3
+90.8±0.2
+Table 1: Variance and Mean values of different model per-
+formances on ModelNet40 with z/SO(3).
+Point Re-sampling and Down-sampling.
+We examine
+different point re-sampling strategies from the initial 10k
+
+input points down to 1024 input points. Experimental re-
+sults of applying different sampling techniques on Model-
+Net40 are shown in Table 2, where we use random sam-
+pling (RS), farthest point sampling (FPS), uniform sampling
+(US), and inverse density importance sampling (IDIS) from
+(Groh, Wieschollek, and Lensch 2018) to examine the im-
+pact of different sampling methods on rotation invariance.
+Note that point sampling affects both LRF and GRF con-
+structions in our design, therefore we can only give analysis
+when considering both reference frames. We can see that
+random sampling gives the lowest model performance with
+89.7%, with 1.3% absolute performance drop compared to
+our method using FPS. Inverse density importance sampling
+can achieve a comparable result as our method, while it is
+not strictly invariant to rotations. We argue that due to the
+information compensation between features encoded from
+LRFs and GRF, different sampling strategies will not affect
+our model performance quite much.
+Sampling Method
+RS
+FPS (ours)
+US
+IDIS
+z/z
+89.7
+91.0
+90.2
+90.6
+z/SO(3)
+89.7
+91.0
+90.2
+90.6
+SO(3)/SO(3)
+89.7
+91.0
+90.2
+90.6
+Table 2: Classification results (%) on ModelNet40 with dif-
+ferent re-sampling techniques.
+Visualization of U.
+To better present the discriminabil-
+ity of the learned features, we summarize the shape fea-
+ture representation U by maxpooling and visualize it via t-
+SNE (Van der Maaten and Hinton 2008). Experiments are
+conducted on object classification under z/z and z/SO(3).
+Only the first 16 classes are selected for a clear represen-
+tation purpose as shown in Fig. 3. Although it is difficult to
+correctly separate all categories, we can see that some shape
+classes can be perfectly predicted, and the overall represen-
+tation ability of U under different testing protocols is satis-
+factory and consistent.
+Figure 3: t-SNE of the aggregated U with z/SO(3) (Left) and
+SO(3)/SO(3) (Right). Clusters indicate good predictions in
+object classification.
+Constructions of pℓ
+ij.
+We examine the model performance
+when using different methods to construct the local rotation-
+invariant feature pℓ
+ij. Specifically, in addition to the proposed
+Method
+PPFs
+LRFs (Ours)
+Acc. (%)
+89.3
+91.0
+Table 3: Classification results (%) on ModelNet40 with
+z/SO(3).
+method that builds pℓ
+ij based on LRFs, we examine point-
+pair features (PPFs) to build pℓ
+ij following (Deng, Birdal,
+and Ilic 2018). As reported in Table 3, we find that the model
+performance of using PPFs is lower than our LRF-based
+method. The reason is that point positions, which provide
+information of exact shape of the 3D objects, are important
+for shape learning. However, point-pair features give infor-
+mation about the topology of a 3D shape, and different 3D
+shapes can have the same topology, which introduces diffi-
+culties for exact 3D shape learning.
+Model Complexity.
+Inference model sizes of different
+methods along with the corresponding construction time
+for LRFs and inference speed are reported in Table 4.
+The construction time measured in seconds (s) shows time
+cost for different models generating their low-level rotation-
+invariant shape features, where we record the total time
+for local and global representation constructions of RI-
+Framework and our work. VN-DGCNN does not compute
+the rotation-invariant shape features, therefore no result can
+be reported. The inference speed with the unit of number of
+instances evaluated within one second (ins./s) is measured
+for each method with a batch size of 1. When computing the
+inference speed, the amount of time for low-level rotation-
+invariant feature construction of methods (Li et al. 2021b;
+Kim, Park, and Han 2020; Zhang et al. 2019; Li et al. 2021a)
+is also considered. Table 4 shows that our method only needs
+a relatively short construction time for both LRFs and GRF.
+Meanwhile, the trade-off between the accuracy and infer-
+ence speed is hard to balance. We will investigate the model
+design for a much high accuracy and faster speeds in the
+future work.
+Sign Ambiguity.
+As mentioned in the main work, we pro-
+pose simple techniques to address the sign ambiguity issue
+introduced by eigenvalue decomposition when computing
+the LRFs and GRF. We thus examine the model performance
+with no sign disambiguation techniques applied, of which
+the results are reported in Table 5. It can be seen that sign
+ambiguity negatively affects the model performance, where
+Method
+Params (M)
+Times (s)
+Speed (ins./s)
+Acc (%)
+RIConv
+0.68
+0.041
+396.4
+86.4
+RI-GCN
+4.19
+0.057
+139.1
+89.5
+RI-Framework
+2.36
+0.134
+43.1
+89.4
+VN-DGCNN
+2.77
+-
+77.3
+89.5
+Li et al. (2021a)
+2.76
+0.047
+35.8
+90.2
+Ours
+3.11
+0.043
+205.3
+91.0
+Table 4: Model complexity construction time for LRFs, and
+inference speed on ModelNet40 with z/SO(3), where Li
+et al. (2021a) is considered without test time augmentation.
+
+performances drop by 0.7% and 0.9% when uncertainty of
+vector directions is introduced to the model training. With
+our proposed solutions, the model behavior can be stabilized
+hence the classification accuracy increases.
+Method
+no@Mℓ
+no@Mg
+no@Mℓ and Mg
+Acc. (%)
+90.3
+90.1
+89.8
+Table 5: Classification results (%) on ModelNet40 with
+z/SO(3), where “no@” denotes no sign disambiguation tech-
+nique applied.
+Rotational Effect of Mg.
+As mentioned in (Li et al.
+2021a), different ways to ensure Mg is a valid rotation ma-
+trix would result in different model performances. In this
+part, we examine four different methods to ensure Mg is a
+valid rotation as follows: (a) we randomly permute two basis
+vectors regardless of the S value; (b) we randomly negate the
+value of a basis vector regardless of the S value; (c) we per-
+mute two basis vectors of S values being the smallest two;
+and (d) we simply reverse the direction of the basis vector
+whose S value is the smallest, which is the proposed method
+in our implementation. We can see from Table 6 that our
+simple design achieves the highest value, while all the others
+decrease the model performance, which shows the effective-
+ness of our proposed method.
+3D Semantic Segmentation.
+To check our model’s effec-
+tiveness on real-world large scenes, additional experiments
+are conducted on S3DIS dataset (Armeni et al. 2016), which
+includes six indoor areas of three different buildings. Each
+point is labeled by one of the 13 categories (e.g., ceiling,
+chair or clutter). Following the same pre-processing steps as
+(Qi et al. 2017b; Wang et al. 2019), each room is divided into
+1m×1m blocks and for each block 4096 points are sampled
+during training process. We use area-5 for testing and all the
+other areas for training. The quantitative results are shown
+in Table 7 following (Zhao et al. 2022), where it shows that
+under random rotations, our model outperforms LGR-Net
+by 7.8%, showing a more effective way to process large in-
+door scenes. For a more intuitive understanding of our model
+performance, qualitative results are shown in Fig. 4 for ref-
+erence.
+3D Part Segmentation.
+For visualization purposes (see
+Fig. 4 in the main work) as well as a detailed analysis of
+model behavior for each category, we report the per-class
+mIoU accuracies under z/SO(3) and SO(3)/SO(3) in Ta-
+bles 8 and 9, where bold numbers indicate the best results
+for each category. As per-class mIoU scores are not reported
+in VN-DGCNN (Deng et al. 2021), we follow the official
+implementation1 and report per-class mIoU results in both
+tables. However, the reproduced results of VN-DGCNN
+are much lower than the ones reported in their work, and
+our model achieves better segmentation results than VN-
+DGCNN (Deng et al. 2021) for most categories. Our model
+also outperforms the state-of-the-art methods (Zhao et al.
+1https://github.com/FlyingGiraffe/vnn-pc
+Method
+a
+b
+c
+d (ours)
+Acc. (%)
+90.1
+90.1
+90.5
+91.0
+Table 6: Classification results (%) on ModelNet40 with
+z/SO(3).
+Method
+z/z
+z/SO(3)
+SO(3)/SO(3)
+PointNet (Qi et al. 2017a)
+41.1
+4.1
+29.3
+DGCNN (Wang et al. 2019)
+48.4
+3.6
+34.3
+RIConv (Zhang et al. 2019)
+22.0
+22.0
+22.0
+LRG-Net (Zhao et al. 2022)
+43.4
+43.4
+43.4
+Ours
+51.2
+51.2
+51.2
+Table 7: Semantic segmentation results (mIoU) on S3DIS
+area-5.
+2022; Luo et al. 2022) in several classes (e.g., airplane, chair,
+and table) under different testing conditions. Furthermore,
+it is also obvious that our model performance is consistent
+across all categories when tested under different rotations.
+In addition, we present more qualitative examples in Fig. 5,
+which includes all 16 classes. For each class, we show two
+pairs of ground truth and our predicted samples. We can see
+that although errors occur when the boundary between the
+different parts have marginal difference, our model achieves
+great performance for most classes.
+References
+Armeni, I.; Sener, O.; Zamir, A. R.; Jiang, H.; Brilakis, I.;
+Fischer, M.; and Savarese, S. 2016. 3D semantic parsing of
+large-scale indoor spaces. In CVPR.
+Deng, C.; Litany, O.; Duan, Y.; Poulenard, A.; Tagliasac-
+chi, A.; and Guibas, L. J. 2021. Vector neurons: A general
+framework for SO (3)-equivariant networks. In ICCV.
+Deng, H.; Birdal, T.; and Ilic, S. 2018. PPFNet: Global con-
+text aware local features for robust 3d point matching. In
+CVPR.
+Groh, F.; Wieschollek, P.; and Lensch, H. 2018.
+Flex-
+convolution. In ACCV.
+Guo, M.-H.; Cai, J.-X.; Liu, Z.-N.; Mu, T.-J.; Martin, R. R.;
+and Hu, S.-M. 2021. PCT: Point cloud transformer. In CVM.
+Kim, S.; Park, J.; and Han, B. 2020.
+Rotation-Invariant
+Local-to-Global Representation Learning for 3D Point
+Cloud. In NeurIPS.
+Li, F.; Fujiwara, K.; Okura, F.; and Matsushita, Y. 2021a. A
+Closer Look at Rotation-Invariant Deep Point Cloud Analy-
+sis. In ICCV.
+Li, X.; Li, R.; Chen, G.; Fu, C.-W.; Cohen-Or, D.; and Heng,
+P.-A. 2021b. A rotation-invariant framework for deep point
+cloud analysis. In TVCG.
+Luo, S.; Li, J.; Guan, J.; Su, Y.; Cheng, C.; Peng, J.; and
+Ma, J. 2022. Equivariant Point Cloud Analysis via Learning
+Orientations for Message Passing. In CVPR.
+Qi, C. R.; Su, H.; Mo, K.; and Guibas, L. J. 2017a. Point-
+Net: Deep learning on point sets for 3D classification and
+segmentation. In CVPR.
+
+Method
+mIoU
+air
+bag
+cap
+car chair
+ear
+guitar knife lamp laptop motor mug pistol rocket skate table
+plane
+phone
+bike
+board
+PointNet (Qi et al. 2017a)
+37.8
+40.4 48.1 46.3 24.5 45.1
+39.4
+29.2
+42.6 52.7
+36.7
+21.2
+55.0 29.7
+26.6
+32.1
+35.8
+PointNet++ (Qi et al. 2017b)
+48.3
+51.3 66.0 50.8 25.2 66.7
+27.7
+29.7
+65.6 59.7
+70.1
+17.2
+67.3 49.9
+23.4
+43.8
+57.6
+PCT (Guo et al. 2021)
+38.5
+32.2 44.8 36.3 26.1 36.2
+40.2
+48.1
+42.1 54.0
+40.9
+18.7
+50.5 25.6
+27.7
+44.7
+47.6
+RIConv (Zhang et al. 2019)
+75.3
+80.6 80.0 70.8 68.8 86.8
+70.3
+87.3
+84.7 77.8
+80.6
+57.4
+91.2 71.5
+52.3
+66.5
+78.4
+GCAConv (Zhang et al. 2020)
+77.2
+80.9 82.6 81.0 70.2 88.4
+70.6
+87.1
+87.2 81.8
+78.9
+58.7
+91.0 77.9
+52.3
+66.8
+80.3
+RI-Framework (Li et al. 2021b)
+79.2
+81.4 82.3 86.3 75.3 88.5
+72.8
+90.3
+82.1 81.3
+81.9
+67.5
+92.6 75.5
+54.8
+75.1
+78.9
+LGR-Net (Zhao et al. 2022)
+80.0
+81.5 80.5 81.4 75.5 87.4
+72.6
+88.7
+83.4 83.1
+86.8
+66.2
+92.9 76.8
+62.9
+80.0
+80.0
+VN-DGCNN⋆ (Deng et al. 2021)
+75.3
+81.1 74.8 72.9 73.8 87.8
+55.9
+91.4
+83.8 80.2
+84.4
+44.5
+92.8 74.6
+57.2
+70.2
+78.9
+OrientedMP (Luo et al. 2022)
+80.1
+81.7 79.0 85.0 78.1 89.7
+76.5
+91.6
+85.9 81.6
+82.1
+67.6
+95.0 79.6
+64.4
+76.9
+80.7
+Ours
+80.3
+84.5 82.7 83.9 76.6 90.2
+76.1
+91.6
+86.6 83.5
+84.6
+50.1
+94.4 81.9
+60.3
+75.3
+81.8
+Table 8: Segmentation results of class-wise and averaged mIoU on ShapeNetPart under z/SO(3), where ⋆ means our reproduced
+results of VN-DGCNN using the official code.
+Method
+mIoU
+air
+bag
+cap
+car chair
+ear
+guitar knife lamp laptop motor mug pistol rocket skate table
+plane
+phone
+bike
+board
+PointNet (Qi et al. 2017a)
+74.4
+81.6 68.7 74.0 70.3 87.6
+68.5
+88.9
+80.0 74.9
+83.6
+56.5
+77.6 75.2
+53.9
+69.4
+79.9
+PointNet++ (Qi et al. 2017b)
+76.7
+79.5 71.6 87.7 70.7 88.8
+64.9
+88.8
+78.1 79.2
+94.9
+54.3
+92.0 76.4
+50.3
+68.4
+81.0
+PCT (Wang et al. 2019)
+75.2
+80.1 69.0 82.5 66.8 88.4
+69.4
+90.4
+85.3 81.8
+79.6
+39.9
+89.2 76.5
+51.8
+72.6
+80.0
+RIConv (Zhang et al. 2019)
+75.5
+80.6 80.2 70.7 68.8 86.8
+70.4
+87.2
+84.3 78.0
+80.1
+57.3
+91.2 71.3
+52.1
+66.6
+78.5
+GCAConv (Zhang et al. 2020)
+77.3
+81.2 82.6 81.6 70.2 88.6
+70.6
+86.2
+86.6 81.6
+79.6
+58.9
+90.8 76.8
+53.2
+67.2
+81.6
+RI-Framework (Li et al. 2021b)
+79.4
+81.4 84.5 85.1 75.0 88.2
+72.4
+90.7
+84.4 80.3
+84.0
+68.8
+92.6 76.1
+52.1
+74.1
+80.0
+LGR-Net (Zhao et al. 2022)
+80.1
+81.7 78.1 82.5 75.1 87.6
+74.5
+89.4
+86.1 83.0
+86.4
+65.3
+92.6 75.2
+64.1
+79.8
+80.5
+VN-DGCNN⋆ (Deng et al. 2021)
+74.7
+80.0 79.4 79.1 71.5 89.2
+66.1
+89.0
+83.5 80.6
+82.0
+29.3
+91.4 73.4
+51.5
+67.8
+81.0
+OrientedMP (Luo et al. 2022)
+80.9
+81.8 78.8 85.4 78.0 89.6
+76.7
+91.6
+85.7 81.7
+82.1
+67.6
+95.0 79.1
+63.5
+76.5
+81.0
+Ours
+80.4
+84.3 82.2 84.6 77.9 89.9
+76.6
+91.3
+86.7 84.1
+84.3
+50.1
+93.4 79.0
+63.7
+75.3
+82.3
+Table 9: Segmentation results of class-wise and averaged mIoU on ShapeNetPart under SO(3)/SO(3).
+Qi, C. R.; Yi, L.; Su, H.; and Guibas, L. J. 2017b. Point-
+Net++: Deep hierarchical feature learning on point sets in a
+metric space. In NeurIPS.
+Van der Maaten, L.; and Hinton, G. 2008. Visualizing data
+using t-SNE. In JMLR.
+Wang, Y.; Sun, Y.; Liu, Z.; Sarma, S. E.; Bronstein, M. M.;
+and Solomon, J. M. 2019. Dynamic graph CNN for learning
+on point clouds. In ACM ToG.
+Zhang, Z.; Hua, B.-S.; Chen, W.; Tian, Y.; and Yeung, S.-K.
+2020. Global context aware convolutions for 3D point cloud
+understanding. In 3DV.
+Zhang, Z.; Hua, B.-S.; Rosen, D. W.; and Yeung, S.-K. 2019.
+Rotation invariant convolutions for 3D point clouds deep
+learning. In 3DV.
+Zhao, C.; Yang, J.; Xiong, X.; Zhu, A.; Cao, Z.; and Li, X.
+2022. Rotation invariant point cloud analysis: Where local
+geometry meets global topology. In Pattern Recognition.
+
+Input
+GT
+Ours
+ceiling
+floor
+wall
+beam
+column
+window
+door
+table
+chair
+sofa
+bookcase
+board
+clutter
+Figure 4: Visualization of semantic segmentation results on S3DIS area-5. The first row is the original inputs, the second row
+is the ground truth (GT) samples and the last row is our predicted results.
+
+GT
+Ours
+GT
+Ours
+GT
+Ours
+GT
+Ours
+Figure 5: Segmentation comparisons between the ground truth (GT) and our model on ShapeNetPart dataset under z/SO(3).
+
+1111
\ No newline at end of file
diff --git a/UNAyT4oBgHgl3EQfVvd8/content/tmp_files/load_file.txt b/UNAyT4oBgHgl3EQfVvd8/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf,len=2180
+page_content='Rethinking Rotation Invariance with Point Cloud Registration  Jianhui Yu, Chaoyi Zhang, Weidong Cai  School of Computer Science, University of Sydney, Australia  {jianhui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='yu, chaoyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='zhang, tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='cai}@sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='au  Abstract  Recent investigations on rotation invariance for 3D point  clouds have been devoted to devising rotation-invariant fea-  ture descriptors or learning canonical spaces where objects  are semantically aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Examinations of learning frame-  works for invariance have seldom been looked into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In this  work, we review rotation invariance in terms of point cloud  registration and propose an effective framework for rota-  tion invariance learning via three sequential stages, namely  rotation-invariant shape encoding, aligned feature integration,  and deep feature registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We first encode shape descrip-  tors constructed with respect to reference frames defined over  different scales, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', local patches and global topology, to  generate rotation-invariant latent shape codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Within the in-  tegration stage, we propose Aligned Integration Transformer  to produce a discriminative feature representation by inte-  grating point-wise self- and cross-relations established within  the shape codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Meanwhile, we adopt rigid transformations  between reference frames to align the shape codes for fea-  ture consistency across different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Finally, the deep in-  tegrated feature is registered to both rotation-invariant shape  codes to maximize feature similarities, such that rotation in-  variance of the integrated feature is preserved and shared se-  mantic information is implicitly extracted from shape codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Experimental results on 3D shape classification, part segmen-  tation, and retrieval tasks prove the feasibility of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Our project page is released at: https://rotation3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  1  Introduction  Point cloud analysis has recently drawn much interest from  researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' As a common form of 3D representation, the  growing presence of point cloud data is encouraging the de-  velopment of many deep learning methods (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021), showing great success  for well-aligned point clouds on different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' However, it  is difficult to directly apply 3D models to real data as raw  3D objects are normally captured at different viewing an-  gles, resulting in unaligned data samples, which inevitably  impact the deep learning models which are sensitive to rota-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Therefore, rotation invariance becomes an important  research topic in the 3D domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  To achieve rotation invariance, a straightforward way is  to augment training data with massive rotations which, how-  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Aligned Integration  Transformer  Registration  Registered U  (a)  (b)  (c)  Local & Global Descriptors  Correspondence  Mapping  Local Patches Pℓ Global Shape P"  Correspondence  Mapping  Fℓ  F"  U  RI  T  Correspondence  Mapping  Shape Descriptors  (d)  (e)  Source Points P# Target Points P$  T  Registration  F#  F$  TI  Encoding  Integration  Registration  Registered P!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Ours  PCR  Figure 1: Frameworks of our design (left) and robust point  cloud registration (right), where TI and RI are transforma-  tion invariance and rotation invariance, and T is the rigid  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The dotted line indicates the computation of  T between reference frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  ever, requires a large memory capacity and exhibits limited  generalization ability to unseen data (Kim, Park, and Han  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' There are attempts to align 3D inputs to a conical  pose (Jaderberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2018), or to learn  rotation robust features via equivariance (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022), while these methods are not rigorously  rotation-invariant and present noncompetitive performance  on 3D shape analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To maintain consistent model be-  havior under random rotations, some methods (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021) follow Drost et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  (2010) to handcraft rotation-invariant point-pair features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Others (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022)  design robust features from equivariant orthonormal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Most of the mentioned works either manipulate model in-  puts or generate canonical spaces to achieve rotation invari-  ance (RI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In this work, we review the problem of RI from  a different aspect: robust point cloud registration (PCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We  find that PCR and RI share the same goal: PCR aligns low-  dimensional point cloud features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', xyz) from the source  domain to the target domain regardless of transformations,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='00149v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='CV]  31 Dec 2022  while RI can be considered to align high-dimensional la-  tent features to rotation-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Specifically, the  goal of PCR is to explicitly align the source point cloud to  the target, both representing the same 3D object, and for RI  learning, we implicitly align the final feature representation  of a 3D shape to a hidden feature of the same shape, which  is universally rotation-invariant to any rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Motivated by this finding, we propose our learning frame-  work in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 1 with three sequential stages, namely rotation-  invariant shape encoding, aligned feature integration, and  deep feature registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Firstly, we (a) construct and feed  point pairs with different scales as model inputs, where we  consider local patches Pℓ with small number of points and  global shape Pg with the whole 3D points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Hence, the fi-  nal feature representation can be enriched by information  from different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Low-level rotation-invariant descrip-  tors are thus built on reference frames and encoded to gener-  ate latent shape codes Fℓ and Fg following recent PCR work  (Pan, Cai, and Liu 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Secondly, we (b) introduce a vari-  ant of transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017), Aligned Integration  Transformer (AIT), to implicitly integrate information from  both self- and cross-attention branches for effective feature  integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In this way, information encoded from different  point scales is aggregated to represent the same 3D object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Moreover, we consider Fℓ and Fg as unaligned since they  are encoded from unaligned reference frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To address  the problem, we follow the evaluation technique proposed  in PCR (Pan, Cai, and Liu 2022), where we use relative ro-  tation information (T) with learnable layers to align Fℓ and  Fg for feature consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Finally, to ensure RI of the inte-  grated feature U, we follow PCR to (c) examine the corre-  spondence map of (Fg, U) and (Fℓ, U), such that the mu-  tual information between a local patch of a 3D object and  the whole 3D object is maximized, and RI is further ensured  in the final geometric feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  The contributions of our work are summarized as follow-  ing three folds: (1) To our knowledge, we are the first in de-  veloping a PCR-cored representation learning framework to-  wards effective RI studies on 3D point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2) We intro-  duce Aligned Integration Transformer (AIT), a transformer-  based architecture to conduct aligned feature integration for  a comprehensive geometry study from both local and global  scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (3) We propose a registration loss to maintain rota-  tion invariance and discover semantic knowledge shared in  different parts of the input object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Moreover, the feasibility  of our proposed framework is successfully demonstrated on  various 3D tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2  Related Work  Rotation Robust Feature Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Networks that are  robust to rotations can be equivariant to rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Esteves  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2018) and Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2018) project 3D data  into a spherical space for rotation equivariance and per-  form convolutions in terms of spherical harmonic bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Some (Spezialetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021) learn canon-  ical spaces to unify the pose of point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Recent  works (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020)  vectorize the scalar activations and mapping SO(3) actions  to a latent space for easy manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Although these  works present competitive results, they cannot be strictly  rotation-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Another way for rotation robustness is to  learn rotation-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Handcraft point-pair fea-  tures are rotation-invariant (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021), but they focus on local domains  and ignore the global overview of 3D objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Others use  rotation-equivariant local reference frames (LRFs) (Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Thomas 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Kim, Park, and Han 2020) or  global reference frames (GRFs) (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021a) as model  inputs based on principal component analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' How-  ever, they may produce inconsistent features across differ-  ent reference frames, which would limit the representational  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In contrast to abovementioned methods with rotation  robust model inputs or modules, we examine the relation be-  tween RI and PCR and propose an effective framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  3D Robust Point Cloud Registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Given a pair of Li-  DAR scans, 3D PCR requires an optimal rigid transforma-  tion to best align the two scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Despite the recent emerging  of ICP-based methods (Besl and McKay 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Wang and  Solomon 2019b), we follow robust correspondence-based  approaches in our work (Deng, Birdal, and Ilic 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yuan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Pan, Cai, and Liu 2022), where  RI is widely used to mitigate the impact of geometric trans-  formations during feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Specifically, both Pan,  Cai, and Liu (2022) and Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2022) analyze the en-  coding of transformation-robust information and introduce  a rotation-invariant module with contextual information into  their registration pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' All these methods showing im-  pressive results are closely related to rotation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We  hypothesize that the learning framework of RI can be sim-  ilar to PCR, and we further prove in experiments that our  network is feasible and able to achieve competitive perfor-  mance on rotated point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Transformers in 3D Point Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Transformers (Doso-  vitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021) applied to 2D vision have  shown great success, and they are gaining prominence in 3D  point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' For example, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2021) uses vector-  ized self-attention (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017) and positional em-  bedding for 3D modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2021) proposes offset  attention for noise-robust geometric representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Cross-attention is widely employed for semantic informa-  tion exchange (Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021a), where fea-  ture relations between the source and target domains are ex-  plored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Taking advantage of both, we design a simple yet  effective feature integration module with self and cross re-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In addition, transformation-related embeddings are  introduced for consistent feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Contrastive Learning with 3D Visual Correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Based on visual correspondence, contrastive learning aims  to train an embedding space where positive samples are  pushed together whereas negative samples are separated  away (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The definition of positivity and neg-  ativity follows the visual correspondence maps, where pairs  with high confidence scores are positive otherwise negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Visual correspondence is important in 3D tasks, where se-  mantic information extracted from matched point pairs im-  proves the network’s understanding on 3D geometric struc-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' For example, PointContrast (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020) explores  feature correspondence across multiple views of one 3D  point cloud with InfoNCE loss (Van den Oord, Li, and  Vinyals 2018), increasing the model performance for down-  stream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Info3D (Sanghi 2020) and CrossPoint (Afham  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022) minimize the semantic difference of point fea-  tures under different poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We follow the same idea by reg-  istering the deep features to rotation-invariant features at in-  termediate levels, increasing feature similarities in the em-  bedding space to ensure rotation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  3  Method  Given a 3D point cloud including Nin points with xyz co-  ordinates P = {pi ∈ R3}Nin  i=1, we aim to learn a shape en-  coder f that is invariant to 3D rotations: f(P) = f(RP),  where R ∈ SO(3) and SO(3) is the rotation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' RI can  be investigated and achieved through three stages, namely  rotation-invariant shape encoding (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1), aligned fea-  ture integration (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2), and deep feature registration  (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  Rotation-Invariant Shape Encoding  In this section, we first construct the input point pairs from  local and global scales based on reference frames, follow-  ing the idea of Pan, Cai, and Liu (2022) to obtain low-level  rotation-invariant shape descriptors from LRFs and GRF di-  rectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Then we obtain latent shape codes via two set abstrac-  tion layers as in PointNet++ (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Rotation Invariance for Local Patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  To construct  rotation-invariant features on LRFs, we hope to construct  an orthonormal basis for each LRF as p ∈ R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Given a  point pi and its neighbor pj ∈ N(pi), we choose #»  xiℓ =  #       »  pmpi/∥#       »  pmpi∥2, where pm is the barycenter of the local ge-  ometry and ∥ · ∥2 is L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We then define #»  ziℓ follow-  ing Tombari, Salti, and Stefano (2010) to have the same di-  rection as an eigenvector, which corresponds to the smallest  eigenvalue via eigenvalue decomposition (EVD):  Σℓ  i =  |N (pi)|  �  j=1  αj (#     »  pipj) (#     »  pipj)⊤ , αj =  d − ∥#     »  pipj∥2  �|N (pi)|  j=1  d − ∥#     »  pipj∥2  ,  (1)  where αj is a weight parameter, allowing nearby pj to have  large contribution to the covariance matrix, and d is the max-  imum distance between pi and pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Finally, we define #»  yiℓ as  #»  ziℓ × #»  xiℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' RI is introduced to pi with respect to its neigh-  bor pj as pℓ  ij = #     »  pipj⊤Mℓ  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Proofs of the equivariance of Mℓ  i  and invariance of pℓ  ij are shown in the supplementary ma-  terial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The latent shape code Fℓ ∈ RN×C is obtained via  PointNet++ and max-pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Rotation Invariance for Global Shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We apply PCA  as a practical tool to obtain RI in a global scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Similar to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 1, PCA is performed by  1  N0  �N0  i=1(#       »  pmpi)(#       »  pmpi)⊤ =  UgΛgUg⊤, where pm is the barycenter of P, Ug  =  [# »  u1g, # »  u2g, # »  u3g] and Λg = diag(λg  1, λg  2, λg  3) are eigenvec-  tor and eigenvalue matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We take Ug as the orthonor-  mal basis Mg = [#»x g, #»y g, #»z g] for GRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' By transform-  ing point pi with Ug, the shape pose is canonicalized as  pg  i = piMg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Proof of the RI of pg  i is omitted for its sim-  plicity, and Fg ∈ RN×C is obtained following PointNet++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Sign Ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  EVD introduces sign ambiguity for  eigenvectors, which negatively impacts the model perfor-  mance (Bro, Acar, and Kolda 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The description of sign  ambiguity states that for a random eigenvector #»u, #»u and  #»u ′, with #»u ′ having an opposite direction to #»u, are both  acceptable solutions to EVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To tackle this issue, we sim-  ply force #»  ziℓ of LRF to follow the direction of #  »  opi, with o  being the origin of the world coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We disambiguate  basis vectors in Mg by computing an inner product with  #       »  pmpi, ∀i ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Taking #»x g for example, its direction is con-  ditioned on the following term:  #»x g =  �#»x g,  if Sx ≥ N0  2  #»x ′g,  otherwise  ,  Sx =  N0  �  i=1  1[⟨#»x g, #       »  pmpi⟩],  (2)  where ⟨·, ·⟩ is the inner product, 1[·] is a binary indicator that  returns 1 if the input argument is positive, otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Sx  denotes the number of points where #»x g and #       »  pmpi point to  the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The same rule is applied to disambiguate  #»y g and #»z g by Sy and Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Besides, as mentioned in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  (2021a), Mg might be non-rotational (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', reflection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To  ensure Mg a valid rotation, we simply reverse the direction  of the basis vector whose S value is the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' More anal-  yses on sign ambiguity are in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  Aligned Feature Integration  Transformer has been widely used in 3D domain to cap-  ture long-range dependencies (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In this sec-  tion, we introduce Aligned Integration Transformer (AIT),  an effective transformer to align latent shape codes with rel-  ative rotation angles and integrate information via attention-  based integration (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Within each AIT mod-  ule, we first apply Intra-frame Aligned Self-attention on Fℓ  and we do not encode Fg, which is treated as supplemen-  tary information to assist local geometry learning with the  global shape overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We discuss that encoding Fg via self-  attention can increase model overfitting, thus lowering the  model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We will validate our discussion in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Inter-frame Aligned Cross-attention is applied on  both Fℓ and Fg, and we use Attention-based Feature Inte-  gration module for information Aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Preliminary: Offset Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  AIT utilizes offset atten-  tion (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021) for noise robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In the follow-  ing, we use subscripts sa and ca to denote implementations  related to self- and cross-attention, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We first re-  view offset attention as follows:  F = φ(Foa) + Fin, Foa = Fin − ∥softmax(A)∥1v, A = qk⊤,  (3)  where q = FinWq, k = FinWk ∈ RN×d, and v =  FinWv ∈ RN×C are query, key, and value embeddings,  and Wq, Wk ∈ RC×d, Wv ∈ RC×C are the correspond-  ing projection matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ∥ · ∥1 is L1-norm and φ denotes a  multi-layer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Foa is offset attention-related  feature and A ∈ RN×N is the attention logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  𝑾𝒄𝒂  𝒗  𝑾𝒄𝒂  𝒌  𝑾𝒔𝒂  𝒒  𝐞𝒄𝒂  𝜶 𝑾𝒄𝒂  𝜶  𝑭()  ℓ/,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  𝐹ℓ/,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  𝑨𝒄𝒂  𝒂𝒕𝒕𝒏: 𝑁×𝑁  𝑨𝒄𝒂  𝒓𝒐𝒕: 𝑁×𝑁  𝑨𝒄𝒂: 𝑁×𝑁  𝐹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='/ℓ  𝑾𝒔𝒂  𝒗  𝑾𝒔𝒂  𝒌  𝑾𝒔𝒂  𝒒  𝐤𝒔𝒂: 𝑁×𝑑  𝐪𝒔𝒂: 𝑁×𝑑  𝐯𝒔𝒂: 𝑁×𝐶  𝐞𝒔𝒂  𝜶 𝑾𝒔𝒂  𝜶  𝑭()  ℓ/,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  𝐹ℓ/,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  𝑨𝒔𝒂  𝒂𝒕𝒕𝒏: 𝑁×𝑁  𝑨𝒔𝒂  𝒓𝒐𝒕:  𝑁×𝑁  𝑨1): 𝑁×𝑁  𝑁×𝑁×𝑑  𝑁×𝑑  𝐯𝒄𝒂: 𝑁×𝐶  𝐤𝒔𝒂: 𝑁×𝑑  𝐪𝒄𝒂: 𝑁×𝑑  (a) Intra-frame Aligned Self-attention  (b) Inter-frame Aligned Cross-attention  𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧  𝐦𝐚𝐭𝐫𝐢𝐱 𝐩𝐫𝐨𝐝𝐮𝐜𝐭  𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧  𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧  𝐦𝐚𝐭𝐫𝐢𝐱 𝐩𝐫𝐨𝐝𝐮𝐜𝐭  𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧  Figure 2: Illustrations of (a) Intra-frame Aligned Self-attention and (b) Inter-frame Aligned Cross-attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Note that  we only present processes for computing Foa in both modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Intra-frame Aligned Self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Point-wise features  of Fℓ are encoded from unaligned LRFs, so direct imple-  mentation of self-attention on Fℓ can cause feature inconsis-  tency during integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To solve this problem, rigid trans-  formations between distinct LRFs are considered, which  are explicitly encoded and injected into point-wise relation  learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We begin by understanding the transfor-  mation between two LRFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' For any pair of local orthonormal  bases Mℓ  i and Mℓ  j, a rotation can be easily derived ∆Rji =  Mℓ  iMℓ  j  ⊤ and translation is defined as ∆tji = oℓ  i −oℓ  j, where  oℓ  i/j indicates the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In our work, the translation part is  intentionally ignored, where we show in the supplementary  material that by keeping both rotation and translation infor-  mation, the model performance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Although ∆Rji is invariant to rotations, we do not di-  rectly project it into the embedding space, as it is sensitive  to the order of matrix product: ∆Rji ̸= ∆Rij, giving in-  consistent rotation information when the product order is not  maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To address this issue, we construct our embed-  ding via the relative rotation angle ∆αji between Mℓ  i and  Mℓ  j, which is normally used in most PCR works (Yew and  Lee 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Pan, Cai, and Liu 2022) for evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The rel-  ative rotation angle ∆αji is computed as:  ∆αji = arccos  �Trace (∆Rji) − 1  2  � 180  π  ∈ [0, π],  (4)  where it is easy to see that ∆αji = ∆αij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We further apply  sinusoidal functions on ∆αji to generate N 2 pairs of angu-  lar embeddings eα ∈ RN×N×d for all N points as:  eα  i,j,2k = sin  � ∆αji/tα  100002k/d  �  , eα  i,j,2k+1 = cos  � ∆αji/tα  100002k/d  �  ,  (5)  where tα controls the sensitivity to angle variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Finally, we inject eα into offset attention and learn intra-  frame aligned feature Fℓ  IAS via self-attention as follows:  Fℓ  IAS = φ  �  Fℓ  oa  �  + Fℓ, Fℓ  oa = Fℓ − ∥ softmax(Asa)∥1vsa,  Asa = Aattn  sa  + Arot  sa , Aattn  sa  = qsak⊤  sa, Arot  sa = qsa(eα  saWα  sa)⊤,  (6)  where qsa/ksa/vsa = FℓWq  sa/FlWk  sa/FlWv  sa, Wα  sa ∈  Rd×d is a linear projection to refine the learning of eα  sa, and  Asa is the attention logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The same process can be per-  formed for Fg by swapping the index ℓ and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Detailed il-  lustrations are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Inter-frame Aligned Cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Semantic infor-  mation exchange between Fℓ and Fg in the feature space  is implemented efficiently by cross-attention (Chen, Fan,  and Panda 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Since Fℓ and Fg are learned from differ-  ent coordinate systems, inter-frame transformations should  be considered for cross-consistency between Fℓ and Fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  An illustration of the cross-attention module is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Computation of inter-frame aligned feature Fℓ  IAC  via cross-attention follows a similar way as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 6:  Fℓ  IAC = φ  �  Fℓ  oa  �  + Fℓ, Fℓ  oa = Fℓ − ∥ softmax(Aca)∥1vca,  Aca = Aattn  ca  + Arot  ca , Aattn  ca  = qcak⊤  ca, Arot  ca = qca(eα  caWα  ca)⊤,  (7)  where qca/kca/vca = FℓWq  ca/FgWk  ca/FgWv  ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Aca is  cross-attention logits containing point-wise cross-relations  over point features defined across local and global scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  eα  ca ∈ RN×d is computed via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 5 in terms of the  transformation between Mℓ  i and Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To this end, the geo-  metric features learned between local and global reference  frames can be aligned given eα  ca, leading to a consistent fea-  ture representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Attention-based Feature Integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Instead of simply  adding the information from both Fℓ and Fg, we integrate  information by incrementing attention logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Specifically,  we apply self-attention on Fℓ with attention logits Asa  and cross-attention between Fℓ and Fg with attention log-  its Aca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We combine Asa and Aca via addition, so that en-  coded information of all point pairs from a local domain can  be enriched by the global context of the whole shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Illus-  tration is shown in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The whole  process is formulated as follows:  U = φ (Foa) + Fℓ,  Foa = Fℓ − ∥softmax(Asa + Aca)∥1(vsa + vca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  (8)  Hence, intra-frame point relations can be compensated by  inter-frame information communication in a local-to-global  manner, which enriches the geometric representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  Deep Feature Registration  Correspondence mapping (Wang and Solomon 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Pan,  Cai, and Liu 2022) plays an important role in PCR, and we  discuss that it is also critical for achieving RI in our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Specifically, although Fℓ and Fg are both rotation-invariant  by theory, different point sampling methods and the sign  ambiguity will cause the final feature not strictly rotation-  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To solve this issue, we first examine the correspon-  dence map:  m (X, Y) =  exp  �  Φ1(Y)Φ2(X)⊤/t  �  �N  j=1 exp (Φ1(Y)Φ2(xj)⊤/t)  ,  (9)  where Φ1 and Φ2 are MLPs that project latent embeddings  X and Y to a shared space, and t controls the variation sen-  sitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' It can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 9 that the mapping function  m reveals feature similarities in the latent space, and it is  also an essential part for 3D point-level contrastive learning  in PointContrast (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020) for the design of InfoNCE  losses (Van den Oord, Li, and Vinyals 2018), which have  been proven to be equivalent to maximize the mutual infor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Based on this observation, we propose a registration  loss function Lr = Lℓ  r + Lg  r, where Lℓ  r and Lg  r represent the  registration loss of (Fℓ,U) and (Fg,U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Mathematically, Lℓ  r  is defined as follows:  Lℓ  r = −  �  (i,j)∈M  log  exp  �  Φ1(Uj)Φ2(f ℓ  i )⊤/t  �  �  (·,k)∈M exp  �  Φ1(Uk)Φ2(f ℓ  i )⊤/t  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (10)  The same rule is followed to compute Lg  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Although we fol-  low the core idea of PointContrast, we differ from it in that  PointContrast defines positive samples based on feature cor-  respondences computed at the same layer level, while our  positive samples are defined across layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  The intuition for the loss design is that the 3D shape is  forced to learn about its local region as it has to distinguish  it from other parts of different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Moreover, we would  like to maximize the mutual information between different  poses of the 3D shape, as features encoded from different  poses should represent the same object, which is very use-  ful in achieving RI in SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Moreover, the mutual infor-  mation between Fℓ and Fg is implicitly maximized, such  Rotation Sensitive  z/z  z/SO(3)  SO(3)/SO(3)  ∆  PointNet (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017a)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  PoinNet++ (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017b)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  PCT (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021)  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  Rotation Robust  z/z  z/SO(3)  SO(3)/SO(3)  ∆  Spherical CNN* (Esteves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2018)  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='9  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='9  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='9  10  SFCNN (Rao, Lu, and Zhou 2019)  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='8  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  RIConv (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019)  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  ClusterNet (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  PR-InvNet (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='0  Table 1: Classification results on ModelNet40 under rota-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' * denotes the input type as projected voxels of 2×642,  while the rest take raw points of 1024×3 as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ∆ is the  absolute difference between z/SO(3) and SO(3)/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  that shared semantic information about geometric structures  can be learned, leading to a more geometrically accurate and  discriminative representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' More details about Lℓ  r can be  found in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  4  Experiments  We evaluate our model on 3D shape classification, part seg-  mentation, and retrieval tasks under rotations, and exten-  sive experiments are conducted to analyze the network de-  sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Detailed model architectures for the three tasks are  shown in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Our evaluating proto-  cols are the same as (Esteves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2018): training and testing  the network under azimuthal rotations (z/z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' training under  azimuthal rotations while testing under arbitrary rotations  (z/SO(3));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and training and testing under arbitrary rotations  (SO(3)/SO(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  3D Object Classification  Synthetic Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We first examine the model perfor-  mance on the synthetic ModelNet40 (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2015)  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We sample 1024 points from each data with only  xyz coordinates as input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Hyper-parameters for  training follow the same as (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021), except  that points are downsampled in the order of (1024, 512,  128) with feature dimensions of (3, 128, 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We report  and compare our model performance with state-of-the-art  (SoTA) methods in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Both rotation sensitive and ro-  bust methods achieve great performance under z/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' How-  ever, the former could not generalize well to unseen rota-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Rotation robust methods like Spherical CNN (Esteves  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2018) and SFCNN (Rao, Lu, and Zhou 2019) achieve  competitive results under z/z, but their performance is not  consistent on z/SO(3) and SO(3)/SO(3) due to the imperfect  projection from points to voxels when using spherical so-  lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We outperform the recent proposed methods (Luo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021) and achieve an  accuracy of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0%, proving the superiority of our framework  on classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Method  z/SO(3)  SO(3)/SO(3)  ∆  PointNet (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017a)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' 2022)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' 2021)  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='6  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  Table 2: Classification results on ScanObjectNN OBJ BG  under z/SO(3) and SO(3)/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  GT  Ours  RI-GCN  RIConv  VN-DGCNN  Figure 3: Segmentation comparisons on ShapeNetPart,  where ground truth (GT) samples are shown for refer-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Red dotted circles indicate obvious failures on certain  classes, and purple circles denote the slight difference be-  tween our design and VN-DGCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Real Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Experiments are also conducted on a real-  scanned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ScanObjectNN (Uy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019) is a com-  monly used benchmark to explore the robustness to noisy  and deformed 3D objects with non-uniform surface density,  which includes 2,902 incomplete point clouds in 15 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We use OBJ BG subset with the background noise and sam-  ple 1,024 points under z/SO(3) and SO(3)/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Table 2  shows that our model achieves the highest results with ex-  cellent consistency with random rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  3D Part Segmentation  Shape part segmentation is a more challenging task than ob-  ject classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We use ShapeNetPart (Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2016) for  evaluation, where we sample 2048 points with xyz coordi-  nates as model inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The training strategy is the same as  the classification task except that the training epoch num-  ber is 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Part-averaged IoU (mIoU) is reported in Table  3, and detailed per-class mIoU values are shown in the sup-  plementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Representative methods such as Point-  Method  z/SO(3)  SO(3)/SO(3)  ∆  PointNet (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017a)  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  PointNet++ (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017b)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='7  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  PCT (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021)  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='7  RIConv (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019)  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  RI-GCN (Kim, Park, and Han 2020)  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  RI-Framework (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021b)  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  LGR-Net (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  VN-DGCNN (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  OrientedMP (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='8  Ours  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  Table 3: Segmentation results on ShapeNetPart under  z/SO(3) and SO(3)/SO(3), where the second best results are  underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Method  micro mAP  macro mAP  Score  Spherical CNN (Esteves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2018)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='685  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='444  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='565  SFCNN (Rao, Lu, and Zhou 2019)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='705  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='483  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='594  GCAConv (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='708  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='490  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='599  RI-Framework (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='707  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='510  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='609  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='715  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='510  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='613  Table 4: Comparisons of SoTA methods on the 3D shape  retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Net++ and PCT are vulnerable to rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Rotation robust  methods present competitive results under z/SO(3), where  we achieve the second best result of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We give more  details of comparison between VN-DGCNN (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2021) and our work in the supplementary material, where  our method performs better than VN-DGCNN for several  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Moreover, qualitative results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 3 present  that we can achieve visually better results than VN-DGCNN  in certain classes such as the airplane and car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' More qualita-  tive results are shown in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  3D Shape Retrieval  We further conduct 3D shape retrieval experiments on  ShapeNetCore55 (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2015), which contains two  categories of datasets: normal and perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We only use  the perturbed part to validate our model performance under  rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We combine the training and validation sets and  validate our method on the testing set following the training  policy of (Esteves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Experimental results are re-  ported in Table 4, where the final score is the average value  of micro and macro mean average of precision (mAP) as  in (Savva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Similar to the classification task, our  method achieves SoTA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  Ablation Study  Effectiveness of Transformer Designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We examine the  effectiveness of our transformer design by conducting clas-  sification experiments under z/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We first ablate one or  both of the angular embeddings and report the results in Ta-  ble 5 (models A, B, and C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Model B performs better than  model C by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4%, which validates our design of feature in-  tegration where Mℓ  i is used as the main source of informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' When both angular embeddings are applied, the best  result is achieved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Moreover, we validate our  discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2 by comparing models D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' :Model  eα  sa  eα  ca  Fg∗  Asa + Aca  Lℓ  r  Lg  r  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  A  ✓  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  B  ✓  ✓  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6  C  ✓  ✓  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  D  ✓  ✓  ✓  ✓  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  E  ✓  ✓  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  F  ✓  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  G  ✓  ✓  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  H  ✓  ✓  ✓  ✓  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6  Ours  ✓  ✓  ✓  ✓  ✓  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  Table 5: Module analysis of AIT and loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Fg∗  means encoding Fg via Intra-frame Aligned Self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  demonstrate in model D that when encoding Fg in the same  way as Fℓ, the model performance decreases, which indi-  cates that encoding Fg via self-attention will increase the  model overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' More analyses can be found in the supple-  mentary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Finally, we examine the effectiveness of  our attention logits-based integration scheme by comparing  our model with the conventional method (model E), which  applies self- and cross-attention sequentially and repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We observe that our result is better than model E by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6%,  indicating that our design is more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Registration Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We sequentially ablate Lg  r and Lℓ  r  (models F, G, and H) to check the effectiveness of our reg-  istration loss deign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Results in Table 5 demonstrate that we  can still achieve a satisfactory result of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0% without fea-  ture registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Individual application of Lg  r and Lℓ  r shows  the improvement when forcing the final representation to be  close to rotation-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Moreover, it can be seen  that model H performs better than model G, which indicates  that intermediate features learned from the global scale are  important for shape classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The best model perfor-  mance is hence achieved by applying both losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Noise Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  In real-world applications, raw point  clouds contain noisy signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We conduct experiments to  present the model robustness to noise under z/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Two  experiments are conducted: (1) We sample and add Gaus-  sian noise of zero mean and varying standard deviations  N(0, σ2) to the input data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2) We add outliers sampled  from a unit sphere to each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 4 (left),  we achieve on par results to RI-Framework when std is low,  while we perform better while std increases, indicating that  our model is robust against high levels of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Besides,  as the number of noisy points increases, most methods are  heavily affected while we can still achieve good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Visualization of Rotation Invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We further ex-  amine RI of learned features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Specifically, we use Grad-  CAM (Selvaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017) to check how the model pays  attention to different parts of data samples under different  rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 5 with correspondence  between gradients and colors shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' RI-GCN  presents a good result, but its behavior is not consistent over  some classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', vase and plant) and it does not pay atten-  tion to regions that are critical for classification (see toilet),  showing inferior performance to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' PointNet++ shows no  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='05  std of noise  20  30  40  50  60  70  80  90  Accuracy (%)  RIConv  RI-GCN  RI-framework  SRINet  ours  0  10  20  30  40  50  # of noisy points  30  40  50  60  70  80  90  Figure 4: Left: Results on Gaussian noise of zero mean and  variant standard deviation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Right: Results on differ-  ent numbers of noisy points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  high  low  high  low  airplane  guitar  vase  plant  toilet  PointNet++  RI-GCN  high  low  Ours  Figure 5: Network attention on PointNet++ (top), RI-GCN  (mid) and our model (bot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  resistance to rotations, while our method exhibits a consis-  tent gradient distribution over different parts with random  rotations, indicating our network is not affected by rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  5  Conclusion  In this work, we rethink and investigate the close relation be-  tween rotation invariance and point cloud registration, based  on which we propose a PCR-cored learning framework with  three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' With a pair of rotation-invariant shape descrip-  tors constructed from local and global scales, a comprehen-  sive learning and feature integration module is proposed,  Aligned Integration Transformer, to simultaneously effec-  tively align and integrate shape codes via self- and cross-  attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' To further preserve rotation invariance in the fi-  nal feature representation, a registration loss is proposed to  align it with intermediate features, where shared semantic  knowledge of geometric parts is also extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Extensive  experiments demonstrated the superiority and robustness of  our designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In future work, we will examine efficient meth-  ods for invariance learning on large-scale point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='.:References  Afham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Thilakarathna, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' and Rodrigo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Crosspoint:  Self-supervised cross-modal contrastive learning for 3D  point cloud understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Method for registration  of 3-D shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Acar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Kolda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Resolving the sign  ambiguity in the singular value decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In JoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Chang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Funkhouser, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Guibas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Hanrahan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Huang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Savarese, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Savva, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Su,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Shapenet: An information-rich 3D model  repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In arXiv preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Mu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In CVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Fan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Momentum contrast for unsupervised visual representation  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Jaderberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Simonyan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Jing, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Eismann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Suriana, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Townshend, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Learning from protein structure with geo-  metric vector perceptrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Han, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Rotation-Invariant  Local-to-Global Representation Learning for 3D Point  Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Fujiwara, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Okura, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Matsushita, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Fu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Cohen-Or, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Heng,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In TVCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Lin,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Swin transformer: Hierarchical vision  transformer using shifted windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Luo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Guan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Peng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Pan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Cai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Mo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='  Qi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Su, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Guibas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Qin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Peng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Xu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Geometric Transformer for Fast and Robust Point  Cloud Registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Rao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Spherical fractal con-  volutional neural networks for point cloud recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In  CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Sanghi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Info3d: Representation learning on 3D ob-  jects using mutual information maximization and contrastive  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In ECCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Savva, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Su, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Kanezaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Furuya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Ohbuchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Bai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='  Large-scale 3D shape retrieval from ShapeNet Core55:  SHREC’17 track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In workshop of 3DOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Selvaraju, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Cogswell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Das, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Vedantam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Parikh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Batra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='  In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Spezialetti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Stella, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Marcon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Silva, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Salti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  and Di Stefano, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Learning to orient surfaces by self-  supervised spherical cnns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Tagliasacchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Deng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Sabour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yazdani,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Salti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Hua, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Nguyen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Van den Oord, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In ECCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='  Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Hua, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Tian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' and Yeung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' In 3DV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Hua, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='  Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Xiong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Cao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Rotation invariant point cloud analysis: Where local  geometry meets global topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Jia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Torr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Koltun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Point transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Rethinking Rotation Invariance with Point Cloud Registration  **Supplementary Material**  Jianhui Yu, Chaoyi Zhang, Weidong Cai  School of Computer Science, University of Sydney, Australia  {jianhui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='yu, chaoyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='zhang, tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='cai}@sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='au  Mathematical Proofs  Proof of Rotation Equivariance of Orthonormal Basis  Mℓ  i of LRFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We prove the rotation equivariance of Mℓ  i  designed for LRFs as mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1 of the main  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Given a random rotation matrix R ∈ R3×3, it is easy  to derive that #»  xiℓ is equivariant to rotations given the rotated  version #»  xiℓ  ,rot:  #»  xi  ℓ  ,rot =  Rpi − Rpm  ∥Rpi − Rpm∥2  =  R(pi − pm)  �  (R(pi − pm))⊤ R(pi − pm)  =  R#       »  pmpi  �#       »  pmpi⊤R⊤R#       »  pmpi  = R  #       »  pmpi  ∥#       »  pmpi∥2  = R#»  xi  ℓ,  (1)  where the subscript rot represents the axis after rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Moreover, Σℓ  i from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (1) of the main work after rotations  can be represented as follows:  Σℓ  i,rot =  |N (pi)|  �  j=1  αjR#     »  pipj #     »  pipj  ⊤R⊤  = R  �  �  |N (pi)|  �  j=1  αj #     »  pipj #     »  pipj  ⊤  �  � R⊤ = RΣℓ  iR⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  (2)  As mentioned in the main work, eigenvalue decomposition  can be directly applied to Σℓ  i, resulting in the following ex-  pressions:  RΣℓ  iR⊤ = RUℓ  iΛℓ  iUℓ  i  ⊤R⊤ =  �  RUℓ  i  �  Λℓ  i  �  RUℓ  i  �⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (3)  Since #»  ziℓ is defined to have the same direction as the eigen-  vector with the smallest eigenvalue, after rotation, #»  ziℓ  ,rot =  R#»  ziℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Thus, the rotated y-axis is:  #»  yi  ℓ  ,rot = #»  zi  ℓ  ,rot × #»  xi  ℓ  ,rot = R#»  zi  ℓ × R#»  xi  ℓ  = det (R)  �  R−1�⊤ �#»  zi  ℓ × #»  xi  ℓ�  = R  �#»  zi  ℓ × #»  xi  ℓ�  = R#»  yi  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  (4)  Since all basis vectors are rotation-equivariant, the local or-  thonormal basis Mℓ  i = [#»  xiℓ, #»  yiℓ, #»  ziℓ] is rotation-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Proof of Rotation Invariance of Local Shape Descriptors  pℓ  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Here, we show the rotation invariance of local shape  descriptors pℓ  ij introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1 of the main work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Based on the proof shown above, it is easy to show the rota-  tion invariance of pℓ  ij after rotation R:  pℓ  ij,rot = #     »  pipj  ⊤  ,rotMℓ  i,rot = (R#     »  pipj)⊤ �  RMℓ  i  �  = #     »  pipj  ⊤Mℓ  i = pℓ  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  (5)  Model Details  Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  The model overview for the 3D classifica-  tion, segmentation, and retrieval tasks is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 1,  where details of each module design are explained in Sec-  tion 3 of the main work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The architecture of attention-based  feature integration module is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Please re-  fer to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2 of the main work for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Fg in AIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2, to alleviate the  model overfitting we do not apply self-attention on Fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We  argue that since we use global shape information as the sup-  plementary material to assist local shape learning, this im-  plementation allow lower-level information to flow effec-  tively across layers to help the learning of higher-level lo-  cal shape features, which could reduce the model overfit-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We encode Fℓ in AIT blocks, as abstracting informa-  tion from local structures can increase the model’s ability on  fine-grained pattern recognition and generalizability to com-  plex scenes (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Hence, we find that without  applying learnable modules on Fg is beneficial to our model  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Registration Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  In this this part, we first explain the  benefits of applying the registration loss to preserve the ro-  tation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We then give more details about Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (10)  in the main work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Suppose Uℓ and Ug are local and global part information  of the final integrated feature U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' By maximizing the mutual  information between (U, Fℓ) and (U, Fg), (Uℓ, Fℓ) and  (Ug, Fg) are implicitly maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In this case, the shared  geometric information between the local Fℓ/global Fg and  the integrated domain U are refined, increasing the repre-  sentation power of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Besides, the maximized similarities of  (Uℓ, Fℓ) and (Ug, Fg) also tend to learn rotation invariance  in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Specifically, although Fℓ/Uℓ en-  codes local patches with different poses (since LRFs are  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='00149v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='CV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='31 Dec 2022  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='airplane  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='car  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='chair  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='toilet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Rotational  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Embeddin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='g  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Shape Retrieval  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Classification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='GRF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='LRF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='SA x 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='SA x 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='AIT x 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='U  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='𝑳𝒓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='𝑳𝒓ℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Fℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='F$  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Rotation-Invariant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Shape Encoding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Aligned Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Integration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Deep Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Registration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='(a) classification/retrieval  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='GRF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='LRF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='SA x 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='SA x 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='AIT x 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='U  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='𝑳𝒓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='𝑳𝒓ℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Fℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='F$  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='FP x 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Rotational  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Embeddin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='g  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Rotation-Invariant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Shape Encoding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Aligned Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Integration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Deep Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Registration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Part Segmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='(b) segmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='Figure 1: Model overviews for (a) classification / retrieval  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='and (b) segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' GRF: global reference frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' LRF:  local reference frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' SA: set abstraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' AIT: Aligned In-  tegration Transformer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and FP: forward passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='"  #  𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='"  $  𝑾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='"  %  𝐤!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ": 𝑁×𝑑  𝐪!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ": 𝑁×𝑑  𝐯!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ": 𝑁×𝐶  𝐤#": 𝑁×𝑑  𝐯#": 𝑁×𝐶  𝑾&"  $  𝑾&"  #  𝑭ℓ  𝑭(  𝑨𝒔𝒆𝒍𝒇  𝒂𝒕𝒕𝒏: 𝑁×𝑁  𝑨𝒔𝒆𝒍𝒇  𝑨𝒄𝒓𝒐𝒔𝒔  𝒓𝒐𝒕  : 𝑁×𝑁  𝑨𝒄𝒓𝒐𝒔𝒔  𝑨𝒔𝒉𝒂𝒓𝒆𝒅: 𝑁×𝑁  𝑨𝒔𝒆𝒍𝒇  𝒓𝒐𝒕 : 𝑁×𝑁 𝑨𝒄𝒓𝒐𝒔𝒔  𝒂𝒕𝒕𝒏 : 𝑁×𝑁  𝑭0"  𝐚𝐝𝐝𝐢𝐭𝐢𝐨𝐧  𝐦𝐚𝐭𝐫𝐢𝐱 𝐩𝐫𝐨𝐝𝐮𝐜𝐭  𝐬𝐮𝐛𝐭𝐫𝐚𝐜𝐭𝐢𝐨𝐧  Figure 2: Illustrations of attention-based feature integra-  tion, where blue and green boxes indicate self- and cross-  attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Brown, gold and purple colored components cor-  respond to v, k and q implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  unaligned), and Fg/Ug encodes the whole 3D object in a  canonical pose, their feature similarity should be enforced  to be similar as they represent the same 3D object, no matter  what their poses are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Moreover, the mutual information be-  tween a local scale of a 3D object and a global scale of the  same object is maximized, which embeds U with more ac-  curate geometric information to distinguish it from objects  in different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We then explain the symbols in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' M stands for  the set of all B × N pairs of positive samples across mini-  batches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', M = {(0, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', (B ×N, B ×N)}, where B is  the number of batch size and N is the number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The  point feature Uk is the set of negative keys where (·, k) ∈ M  and k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Note that since the registration loss function is  applied to both Fℓ and Fg, the point features of the same  point encoded from the local and global scales are pushed  close to each other, where the mutual information is in-  creased such that shared geometric information can be dis-  covered across the local and global scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Training Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  For all three tasks, we set the batch  size to 32 for training and 16 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We use farthest  point sampling to re-sample the points from the initial 10k  points to 1024 points for classification and retrieval and 2048  points for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Random point translation within  [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2] and rescaling within [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='67, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5] were adopted for  augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We trained the model for 250 epochs with  tα = 15 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' SGD is adopted as the optimizer,  where the learning rate was set to 1e-2 with momentum of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='9 and weight decay of 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Cosine annealing was applied  to reschedule the learning rate for each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' For classi-  fication and retrieval, we used one RTX2080Ti GPU with  PyTorch for model implementation, and we used two GPUs  for the segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The normal vector information is  ignored for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  More Analysis Experiments  Influence of Randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We report the variance and  mean values of each model in Table 5 of the main work to  derive a more accurate and reliable estimate of our model  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We hence report the variance and mean values  of performance of each model in Table 5 on ModelNet40  with 5 training rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' As shown in Table 1, we can see  that even for our model weights with the lowest performance  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6% (among our five repeated runs), it still surpasses the  highest performance among models from A to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Model  A  B  C  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (%)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  Model  D  E  F  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (%)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  Model  G  H  Best  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (%)  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  Table 1: Variance and Mean values of different model per-  formances on ModelNet40 with z/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Point Re-sampling and Down-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We examine  different point re-sampling strategies from the initial 10k  input points down to 1024 input points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Experimental re-  sults of applying different sampling techniques on Model-  Net40 are shown in Table 2, where we use random sam-  pling (RS), farthest point sampling (FPS), uniform sampling  (US), and inverse density importance sampling (IDIS) from  (Groh, Wieschollek, and Lensch 2018) to examine the im-  pact of different sampling methods on rotation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Note that point sampling affects both LRF and GRF con-  structions in our design, therefore we can only give analysis  when considering both reference frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We can see that  random sampling gives the lowest model performance with  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='7%, with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3% absolute performance drop compared to  our method using FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Inverse density importance sampling  can achieve a comparable result as our method, while it is  not strictly invariant to rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We argue that due to the  information compensation between features encoded from  LRFs and GRF, different sampling strategies will not affect  our model performance quite much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Sampling Method  RS  FPS (ours)  US  IDIS  z/z  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='7  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6  z/SO(3)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='7  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6  SO(3)/SO(3)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='7  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6  Table 2: Classification results (%) on ModelNet40 with dif-  ferent re-sampling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Visualization of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  To better present the discriminabil-  ity of the learned features, we summarize the shape fea-  ture representation U by maxpooling and visualize it via t-  SNE (Van der Maaten and Hinton 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Experiments are  conducted on object classification under z/z and z/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Only the first 16 classes are selected for a clear represen-  tation purpose as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Although it is difficult to  correctly separate all categories, we can see that some shape  classes can be perfectly predicted, and the overall represen-  tation ability of U under different testing protocols is satis-  factory and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Figure 3: t-SNE of the aggregated U with z/SO(3) (Left) and  SO(3)/SO(3) (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Clusters indicate good predictions in  object classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Constructions of pℓ  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  We examine the model performance  when using different methods to construct the local rotation-  invariant feature pℓ  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Specifically, in addition to the proposed  Method  PPFs  LRFs (Ours)  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (%)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  Table 3: Classification results (%) on ModelNet40 with  z/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  method that builds pℓ  ij based on LRFs, we examine point-  pair features (PPFs) to build pℓ  ij following (Deng, Birdal,  and Ilic 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' As reported in Table 3, we find that the model  performance of using PPFs is lower than our LRF-based  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The reason is that point positions, which provide  information of exact shape of the 3D objects, are important  for shape learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' However, point-pair features give infor-  mation about the topology of a 3D shape, and different 3D  shapes can have the same topology, which introduces diffi-  culties for exact 3D shape learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Model Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Inference model sizes of different  methods along with the corresponding construction time  for LRFs and inference speed are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  The construction time measured in seconds (s) shows time  cost for different models generating their low-level rotation-  invariant shape features, where we record the total time  for local and global representation constructions of RI-  Framework and our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' VN-DGCNN does not compute  the rotation-invariant shape features, therefore no result can  be reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The inference speed with the unit of number of  instances evaluated within one second (ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='/s) is measured  for each method with a batch size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' When computing the  inference speed, the amount of time for low-level rotation-  invariant feature construction of methods (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Kim, Park, and Han 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021a)  is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Table 4 shows that our method only needs  a relatively short construction time for both LRFs and GRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Meanwhile, the trade-off between the accuracy and infer-  ence speed is hard to balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We will investigate the model  design for a much high accuracy and faster speeds in the  future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Sign Ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  As mentioned in the main work, we pro-  pose simple techniques to address the sign ambiguity issue  introduced by eigenvalue decomposition when computing  the LRFs and GRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We thus examine the model performance  with no sign disambiguation techniques applied, of which  the results are reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' It can be seen that sign  ambiguity negatively affects the model performance, where  Method  Params (M)  Times (s)  Speed (ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='/s)  Acc (%)  RIConv  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='041  396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  RI-GCN  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='057  139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5  RI-Framework  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='134  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  VN-DGCNN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='77 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5  Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2021a)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='047  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='8  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  Ours  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='043  205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  Table 4: Model complexity construction time for LRFs, and  inference speed on ModelNet40 with z/SO(3), where Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (2021a) is considered without test time augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  performances drop by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='7% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='9% when uncertainty of  vector directions is introduced to the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' With  our proposed solutions, the model behavior can be stabilized  hence the classification accuracy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Method  no@Mℓ  no@Mg  no@Mℓ and Mg  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (%)  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='8  Table 5: Classification results (%) on ModelNet40 with  z/SO(3), where “no@” denotes no sign disambiguation tech-  nique applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Rotational Effect of Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  As mentioned in (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2021a), different ways to ensure Mg is a valid rotation ma-  trix would result in different model performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In this  part, we examine four different methods to ensure Mg is a  valid rotation as follows: (a) we randomly permute two basis  vectors regardless of the S value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (b) we randomly negate the  value of a basis vector regardless of the S value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (c) we per-  mute two basis vectors of S values being the smallest two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  and (d) we simply reverse the direction of the basis vector  whose S value is the smallest, which is the proposed method  in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We can see from Table 6 that our  simple design achieves the highest value, while all the others  decrease the model performance, which shows the effective-  ness of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  3D Semantic Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  To check our model’s effec-  tiveness on real-world large scenes, additional experiments  are conducted on S3DIS dataset (Armeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2016), which  includes six indoor areas of three different buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Each  point is labeled by one of the 13 categories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', ceiling,  chair or clutter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Following the same pre-processing steps as  (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019), each room is divided into  1m×1m blocks and for each block 4096 points are sampled  during training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We use area-5 for testing and all the  other areas for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The quantitative results are shown  in Table 7 following (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022), where it shows that  under random rotations, our model outperforms LGR-Net  by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='8%, showing a more effective way to process large in-  door scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' For a more intuitive understanding of our model  performance, qualitative results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 4 for ref-  erence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  3D Part Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  For visualization purposes (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 4 in the main work) as well as a detailed analysis of  model behavior for each category, we report the per-class  mIoU accuracies under z/SO(3) and SO(3)/SO(3) in Ta-  bles 8 and 9, where bold numbers indicate the best results  for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' As per-class mIoU scores are not reported  in VN-DGCNN (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021), we follow the official  implementation1 and report per-class mIoU results in both  tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' However, the reproduced results of VN-DGCNN  are much lower than the ones reported in their work, and  our model achieves better segmentation results than VN-  DGCNN (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021) for most categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Our model  also outperforms the state-of-the-art methods (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='com/FlyingGiraffe/vnn-pc  Method  a  b  c  d (ours)  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' (%)  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='5  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  Table 6: Classification results (%) on ModelNet40 with  z/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Method  z/z  z/SO(3)  SO(3)/SO(3)  PointNet (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017a)  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='1  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  DGCNN (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='6  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='3  RIConv (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019)  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='0  LRG-Net (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022)  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='4  Ours  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='2  Table 7: Semantic segmentation results (mIoU) on S3DIS  area-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2022) in several classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=', airplane, chair,  and table) under different testing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Furthermore,  it is also obvious that our model performance is consistent  across all categories when tested under different rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  In addition, we present more qualitative examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 5,  which includes all 16 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' For each class, we show two  pairs of ground truth and our predicted samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' We can see  that although errors occur when the boundary between the  different parts have marginal difference, our model achieves  great performance for most classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  References  Armeni, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Sener, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Zamir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Brilakis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='  Fischer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Savarese, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' 3D semantic parsing of  large-scale indoor spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Deng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Duan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Poulenard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Tagliasac-  chi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Guibas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Vector neurons: A general  framework for SO (3)-equivariant networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Deng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Birdal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Ilic, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' PPFNet: Global con-  text aware local features for robust 3d point matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In  CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Groh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Wieschollek, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Lensch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Flex-  convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In ACCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Mu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Martin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  and Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' PCT: Point cloud transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In CVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Han, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Rotation-Invariant  Local-to-Global Representation Learning for 3D Point  Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='3  Table 9: Segmentation results of class-wise and averaged mIoU on ShapeNetPart under SO(3)/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Qi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' Su, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Guibas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2017b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Point-  Net++: Deep hierarchical feature learning on point sets in a  metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content='  Van der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Visualizing data  using t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Sarma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Bronstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  and Solomon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Dynamic graph CNN for learning  on point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In ACM ToG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Hua, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Tian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Yeung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Global context aware convolutions for 3D point cloud  understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In 3DV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Hua, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Rosen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Yeung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Rotation invariant convolutions for 3D point clouds deep  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In 3DV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Xiong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Cao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' and Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' Rotation invariant point cloud analysis: Where local  geometry meets global topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' In Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  Input  GT  Ours  ceiling  floor  wall  beam  column  window  door  table  chair  sofa  bookcase  board  clutter  Figure 4: Visualization of semantic segmentation results on S3DIS area-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content=' The first row is the original inputs, the second row  is the ground truth (GT) samples and the last row is our predicted results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  GT  Ours  GT  Ours  GT  Ours  GT  Ours  Figure 5: Segmentation comparisons between the ground truth (GT) and our model on ShapeNetPart dataset under z/SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
+page_content='  1111' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf'}
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+arXiv:2301.05381v1  [math.AT]  13 Jan 2023
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH
+Z2 COEFFICIENTS
+KATE POIRIER AND THOMAS TRADLER
+Abstract. Luc Menichi showed that the BV algebras on H●(LS2;Z2)[−2] coming from
+string topology and the one on HH●(H●(S2;Z2),H●(S2;Z2)) using Poincar´e duality on
+H●(S2;Z2) are not isomorphic.
+In this note we show how one can obtain the string
+topology BV algebra on Hochschild cohomology using a Poincar´e duality structure with
+higher homotopies.
+This Poincar´e duality (with higher homotopies) on cohomology is
+induced by a local Poincar´e duality (with higher homotopies) on the cochain level.
+1. Main Statements
+1.1. BV algebras for the 2-sphere with Z2 coefficients. In [M, Theorem 24] Luc
+Menichi calculated the string topology BV algebra (defined by Moira Chas and Dennis
+Sullivan in [CS]) for the 2-sphere M = S2 with Z2 = Z/2Z coefficients to be
+H●(LS2;Z2) ≅ Λa ⊗ Z2[u] ≅ ⊕
+k≥0
+Z2.αk ⊕ ⊕
+k≥0
+Z2.βk,
+(1.2)
+where αk ⋅ αℓ = αk+ℓ,βk ⋅ βℓ = 0, and αk ⋅ βℓ = βℓ ⋅ αk = βk+ℓ,
+and ∆ST(αk) = 0 and ∆ST(βk) = k ⋅ αk−1 + k ⋅ βk+1.
+Here, the degrees are given by ∣a∣ = −2, ∣u∣ = 1, and thus, setting αk = 1⊗uk and βk = a⊗uk,
+we have ∣αk∣ = k, ∣βk∣ = k − 2.
+Moreover, in [M, Proposition 20], Menichi calculated the BV algebra for Hochschild
+cohomology of the cohomology of S2 with Z2 coefficients to be
+HH●(H●(S2;Z2);H●(S2;Z2)) ≅ Λg ⊗ Z2[f] ≅ ⊕
+k≥0
+Z2.φk ⊕ ⊕
+k≥0
+Z2.ψk,
+(1.3)
+where φk ⋅ φℓ = φk+ℓ,ψk ⋅ ψℓ = 0, and φk ⋅ ψℓ = ψℓ ⋅ φk = ψk+ℓ,
+and ∆(φk) = 0 and ∆(ψk) = k ⋅ φk−1.
+There degrees are similarly given by ∣g∣ = −2, ∣f∣ = 1, and so, setting φk = 1 ⊗ f k and
+ψk = g ⊗ f k, we have ∣φk∣ = k, ∣ψk∣ = k − 2.
+Note, that these BV algebras differ in their ∆ operators, and, in fact, Menichi obtained
+the following result.
+Theorem 1.4. [M, Corollary 30] There is no isomorphism of BV algebras between
+(H●(LS2;Z2),⋅,∆ST) and (HH●(H●(S2;Z2);H●(S2;Z2)),⋅,∆).
+It is worth noting though, that the induced Gerstenhaber algebras on H●(LS2;Z2) and
+HH●(H●(S2;Z2);H●(S2;Z2)) are isomorphic; cf. [M, Corollary 23].
+2010 Mathematics Subject Classification. 55P50, 16E40 (primary), 57P10, 08A65 (secondary).
+Key words and phrases. String topology, Hochschild cohomology, BV algebra, Poincar´e duality.
+1
+
+2
+K. POIRIER AND T. TRADLER
+1.5. The BV algebra coming from local Poincar´e duality. A crucial ingredient for de-
+termining the ∆ operator on Hochschild cohomology (1.3) comes from a choice of Poincar´e
+duality structure given as a bimodule isomorphism F ∶ H●(S2;Z2)
+≅
+�→ H●(S2;Z2). For the
+(Z2-)cohomology H●(S2;Z2) and homology H●(S2;Z2), an obvious choice is to define the
+degree +2 isomorphism F as follows:
+degree −2
+degree −1
+degree 0
+degree 1
+degree 2
+H●(S2;Z2)
+= ...
+Z2
+0
+Z2
+0
+0
+...
+H●(S2;Z2)
+= ...
+0
+0
+Z2
+0
+Z2
+...
+F
+F
+F
+Indeed, this choice of F, which is the map given by capping with the Z2-fundamental class
+of S2, leads to the BV algebra in (1.3) as we will confirm in theorem 1.6 (together with
+5.3) below.
+Note that on the cochain level, capping with a fundamental cycle is not a bimodule
+map; see observation 4.4. Using a generalization of bimodule maps, which, in addition
+to F, allows for higher homotopies, we show in this paper that one can obtain the string
+topology BV algebra (1.2) on Hochschild cohomology. The precise definition of bimodule
+maps with higher homotopies was studied in [T1] and will be reviewed in section 2 below.
+Applying the concept of bimodule maps with higher homotopies, we will construct a spe-
+cific example of such a map for the case of a local cochain model of the 2-sphere C●(S2;Z2)
+in example 4.20. Then we pull this map back to obtain a bimodule map with homotopies
+̃F for cohomology H●(S2;Z2) (see proposition 4.23). The resulting map ̃F, recorded in
+example 4.2, has precisely one higher homotopy, when compared to F (see (4.3)). Now,
+using this ̃F, we obtain the following main theorem.
+Theorem 1.6. The BV algebra on HH●(H●(S2;Z2);H●(S2;Z2)) induced by ̃F (coming
+from a local bimodule map with higher homotopies transferred to cohomology) is isomorphic
+to the string topology BV algebra (1.2) on H●(LS2;Z2).
+Proof. We compute both BV algebras coming from F and ̃F, respectively.
+Denote by
+H● ∶= H●(S2;Z2) and H● ∶= H●(S2;Z2). Then the bimodule map F and the bimodule map
+̃F (with higher homotopies) induce respective (graded module) isomorphisms F and ̃
+F of
+Hochschild cohomologies (see 5.3 and 5.4)
+(1.7)
+F, ̃
+F ∶ HH●(H●,H●)
+≅
+�→ HH●(H●,H●).
+Next, we use the explicit description (as reviewed in 5.1 and 5.2) stating that
+HH●(H●,H●) ≅ ⊕
+k≥0
+Z2.φk ⊕ ⊕
+k≥0
+Z2.ψk,
+where ∣φk∣ = k, ∣ψk∣ = k − 2,
+HH●(H●,H●) ≅ ⊕
+k≥0
+Z2.θk ⊕ ⊕
+k≥0
+Z2.χk,
+where ∣θk∣ = k + 2, ∣χk∣ = k.
+With this notation, F induces in (1.7) the map F(φk) = θk,F(ψk) = χk (see 5.3), while ̃F
+induces the map ̃
+F(φk) = θk + χk+2, ̃
+F(ψk) = χk (computed in 5.4).
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+3
+The BV ∆ operator on HH●(H●,H●) is given by transferring Connes’ B-operator from
+HH●(H●,H●) to HH●(H●,H●) via the given Poincar´e duality bimodule isomorphism,
+where, Connes’ B-operator is the map (computed in 5.2):
+B ∶ HH●(H●,H●) → HH●(H●,H●),
+B(θk) = 0,B(χk) = k ⋅ θk−1.
+Thus the induced operators ∆ = F −1○B○F and ̃∆ = ̃
+F −1○B○ ̃
+F from F and ̃F, respectively,
+are:
+∆(φk) = 0,
+∆(ψk) = F −1(k ⋅ θk−1) = k ⋅ φk−1,
+(1.8)
+̃∆(φk−2) = ̃∆(ψk) = ̃
+F −1(k ⋅ θk−1) = k ⋅ (φk−1 + ψk+1),
+(1.9)
+where we used that ̃
+F −1 is given by ̃
+F −1(θk) = φk + ψk+2 and ̃
+F −1(χk) = ψk.
+Note that the ∆ operator in (1.8) coincides with ∆ from (1.3). To see that the BV algebra
+with ̃∆ is isomorphic to the string topology BV algebra (1.2), we use an isomorphism which
+was essentially described in [M, last paragraph of Lemma 21]: the map Θ ∶ HH●(H●,H●) →
+H●(LS2;Z2) with Θ(φk) = αk +k ⋅βk+2 and Θ(ψk) = βk is an algebra isomorphism such that
+∆ST ○ Θ = Θ ○ ̃∆.
+□
+1.10. Organization of the paper. In section 2, we review basics about Hochschild
+cochain complexes and morphisms between them coming from bimodule maps up to higher
+homotopies. In section 3 we review the induced BV algebra coming from a chosen Poincar´e
+duality structure. In section 4 we give various computations of bimodule maps (with and
+without higher homotopies). In particular, the construction of a bimodule map (up to
+certain controlled structures) for the 2-simplex coming from locality (see example 4.9) con-
+stitutes the main computational aspect of this paper, with its proof being spelled out in
+appendix A. In section 5 we compute the BV algebras coming from the Poincar´e duality
+structures on cohomology of the 2-sphere from section 4, and with this complete the proof
+of theorem 1.6.
+Acknowledgments. We would like to thank Mahmoud Zeinalian for discussions about
+this topic. The second author was partially supported by a PSC-CUNY research award.
+2. Bimodule maps with higher homotopies
+In this section we review bimodule maps, bimodule maps up to higher homotopies,
+and homotopy inner products; see also [T1]. Moreover we review the induced maps on
+Hochschild cohomology.
+2.1. Basic setup. Denote by R a commutative ring with unit; our main example of interest
+is R = Z2. Let A be a unital dg-algebra over R, where we note that all the differentials in
+this paper will always go down, i.e., d ∶ Aj → Aj−1. We denote by A the space A shifted up
+by one, i.e., Aj ∶= Aj−1.
+Let M be a dg-bimodule over A. If we want to emphasize the corresponding algebra A,
+then we will also write M/A instead of M. From now on, all modules as well as module
+maps will be written in bold.
+Note that if M is a dg-bimodule over A, then so is its dual space M∗ with M∗
+j ∶= (M−j)∗ =
+HomR(M−j,R) with differential d ∶ M∗
+j → M∗
+j−1,d(n)(m) ∶= (−1)∣n∣+1n(d(m)), and module
+maps (n.a)(m) ∶= n(a.m) and (a.n)(m) ∶= (−1)∣a∣⋅(∣n∣+∣m∣)n(m.a) for n ∈ M∗,m ∈ M,a ∈ A,
+
+4
+K. POIRIER AND T. TRADLER
+where ∣.∣ denotes the degree. Moreover, the dg-algebra A is itself a dg-module A ∶= A over
+A (with module structure given by the algebra multiplication), and thus A∗ is a dg-module
+over A as well.
+Define the Hochschild cochain complex of A with values in M to be given by CH●(A,M) ∶=
+∏r≥0 Hom(A⊗r,M), where the differential D = D0 + D1, with D2 = 0, is defined for ϕ ∈
+Hom(A⊗r,M) by setting
+D0(ϕ)(a1,... ,ar) ∶= d(ϕ(a1,... ,ar)) +
+r
+∑
+j=1
+(−1)∣ϕ∣+∣a1∣+⋅⋅⋅+∣aj−1∣ ⋅ ϕ(a1,... ,daj,... ,ar),
+D1(ϕ)(a1,... ,ar+1) ∶= (−1)∣ϕ∣⋅∣a1∣ ⋅ a1.ϕ(a2,... ,ar+1)
++
+r
+∑
+j=1
+(−1)∣ϕ∣+∣a1∣+⋅⋅⋅+∣aj∣ ⋅ ϕ(a1,... ,aj ⋅ aj+1,... ,ar+1) + (−1)∣ϕ∣+1+∣a1∣+⋅⋅⋅+∣ar∣ ⋅ ϕ(a1,... ,ar).ar+1.
+We will mainly use the normalized Hochschild cochain complex CH
+●(A,M), which is the
+subcomplex of CH●(A,M) consisting of those ϕ ∈ CH●(A,M) which vanish when any of the
+inputs is the unit 1 ∈ A. The inclusion CH
+●(A,M) ↪ CH●(A,M) is a quasi-isomorphism;
+see [L, 1.5.7].
+2.2. Inner products and homotopy inner products. Let M and N be dg-bimodules
+over A, and let F ∶ M → N be a dg-bimodule map. Then, there is an induced cochain map
+CH(F) ∶ CH
+●(A,M) → CH
+●(A,N), ϕ ↦ F ○ ϕ. An inner product on M is a dg-bimodule
+map F ∶ M → M∗.
+We will need a more general notion of inner product F ∶ M → M∗ which also allows
+for higher homotopies. To this end, consider a sequence of maps F = {Fp,q ∶ A⊗p ⊗ M ⊗
+A⊗q → M∗}p,q≥0, and define its differential by setting DF = {(DF)p,q}p,q≥0 to be (DF)p,q =
+(D0F)p,q + (D1F)p,q, where ∀a1,... ,ap,b1,... ,bq,∈ A and m,n ∈ M:
+(D0F)p,q(a1,... ,ap;m;b1,... ,bq)(n)
+(2.3)
+∶=
+p
+∑
+j=1
+(−1)∣F∣+∣a1∣+⋅⋅⋅+∣aj−1∣ ⋅ Fp,q(a1,... ,daj,... ,ap;m;b1,... ,bq)(n)
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+1 ⋅ Fp,q(a1,... ,ap;dm;b1,... ,bq)(n)
++
+q
+∑
+j=1
+(−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bj−1∣ ⋅ Fp,q(a1,... ,ap;m;b1,... ,dbj,... ,bq)(n)
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq∣+1 ⋅ Fp,q(a1,... ,ap;m;b1,... ,bq)(dn)
+and, using the notation F−1,q = Fp,−1 = 0:
+(D1F)p,q(a1,... ,ap;m;b1,... ,bq)(n)
+(2.4)
+∶= (−1)∣F∣+∣a1∣⋅(∣a2∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq∣+∣n∣) ⋅ Fp−1,q(a2,... ,ap;m;b1,... ,bq)(n.a1)
++
+p−1
+∑
+j=1
+(−1)∣F∣+∣a1∣+⋅⋅⋅+∣aj∣ ⋅ Fp−1,q(a1,... ,aj ⋅ aj+1,... ,ap;m;b1,... ,bq)(n)
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap−1∣+1 ⋅ Fp−1,q(a1,... ,ap−1;ap.m;b1,... ,bq)(n)
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+5
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣ ⋅ Fp,q−1(a1,... ,ap;m.b1;b2,... ,bq)(n)
++
+q−1
+∑
+j=1
+(−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bj∣ ⋅ Fp,q−1(a1,... ,ap;m;b1,... ,bj ⋅ bj+1,... ,bq)(n)
++ (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq−1∣+1 ⋅ Fp,q−1(a1,... ,ap;m;b1,... ,bq−1)(bq.n)
+Then we call F a homotopy inner product for M if DF = 0. By slight abuse of notation,
+we will still use the notation F ∶ M → M∗ for F = {Fp,q}p,q with all its homotopies.
+We will sometimes depict evaluations of F as follows:
+(2.5)
+Fp,q(a1,... ,ap;m;b1,... ,bq)(n)
+a1
+a2
+...
+ap
+m
+b1
+b2
+...
+bq
+n
+Note, that a homotopy inner product F = {Fp,q}p,q with Fp,q = 0 for p + q > 0 is precisely
+an inner product, i.e., a dg-bimodule map F0,0 ∶ M → M∗.
+Let F be a homotopy inner product for M. We will always assume that F vanishes
+when any of the algebra inputs is the unit 1 ∈ A. Then, there is an induced map F ∶=
+CH(F) ∶ CH
+●(A,M) → CH
+●(A,M∗) by setting F = CH(F) = ∑p,q≥0 CH(F)p,q, where
+CH(F)p,q ∶ Hom(A⊗r,M) → Hom(A⊗p+r+q,M∗) is
+(2.6)
+(CH(F)p,q(ϕ))(a1,... ,ap+r+q)
+∶= (−1)∣ϕ∣⋅(∣a1∣+⋅⋅⋅+∣ap∣) ⋅ Fp,q(a1,... ,ap;ϕ(ap+1,... ,ap+r);ap+r+1,... ,ap+r+q).
+a1
+a2
+...
+ap
+ap+r+1
+...
+ap+r+q
+ϕ
+ap+1
+ap+r
+...
+A direct but lengthy calculation shows that
+(2.7)
+CH(DF)(ϕ) = D ○ CH(F)(ϕ) − (−1)∣F∣ ⋅ CH(F) ○ D(ϕ),
+∀ϕ ∈ CH●(A,M).
+Corollary 2.8. If F is a homotopy inner product, i.e., DF = 0, then F = CH(F) ∶
+CH
+●(A,M) → CH
+●(A,M∗) is a cochain map. By abuse of notation, we often write F
+for the induced map on Hochschild cohomology. If F = DF′ for some F′, then the induced
+map on Hochschild cohomology vanishes, i.e., HH(F) = 0.
+2.9. Pullback under a dg-morphism. Let A and B be two unital dg-algebras, and
+let f ∶ B → A be a dg-algebra map (which is necessarily of degree 0).
+Then any dg-
+bimodule M = M/A over A induces a dg-bimodule M/B ∶= M over B via the module structure
+b.m ∶= f(b).m and m.b ∶= m.f(b) for any m ∈ M,b ∈ B. Moreover, any dg-bimodule map
+F/A ∶ M/A → N/A also induces a dg-bimodule map F/B ∶ M/B → N/B. Moreover, a homotopy
+inner product F/A for M/A also induces a homotopy inner product for M/B via
+(F/B)p,q(b1,... ,bp;m;c1,... ,cq)(n) ∶= (F/A)p,q(f(b1),... ,f(bp);m;f(c1),... ,f(cq))(n)
+
+6
+K. POIRIER AND T. TRADLER
+for all b1,... ,bp,c1,... ,cq,∈ B and m,n ∈ M/B.
+Moreover, for a dg-algebra map f ∶ B → A, we have the dg-modules B/B and B∗
+/B, and,
+from A/A and A∗
+/A, we also get A/B and A∗
+/B. The dg-algebra map f then induces a dg-
+bimodule map f/B ∶ B/B → A/B and, by dualizing, the dg-bimodule map f∗
+/B ∶ A∗
+/B → B∗
+/B.
+Combining the last two paragraphs, assume that f ∶ B → A is a dg-algebra map, and that
+F/A ∶ A/A → A∗
+/A is a homotopy inner product for A. Then, we get a transferred homotopy
+inner product f(F)/B ∶ B/B → B∗
+/B for B, given by f(F)/B ∶= f∗
+/B ○ F/B ○ f/B,
+B/B
+f/B
+�
+f(F)/B
+�
+A/B
+F/B
+�
+B∗
+/B
+A∗
+/B
+f∗
+/B
+�
+(2.10)
+(f(F)/B)p,q(b1,... ,bp;m;c1,... ,cq)(n)
+= (F/A)p,q(f(b1),... ,f(bp);f(m);f(c1),... ,f(cq))(f(n)),
+where b1,... ,bp,m,c1,... ,cq,n ∈ B.
+Finally, for a dg-algebra map f ∶ B → A, and a dg-bimodule M/A, there is an induced
+cochain map on Hochschild cochains, CH(f,M) ∶ CH
+●(A,M/A) → CH
+●(B,M/B),
+(2.11)
+CH(f;M)(ϕ)(b1,... ,br) ∶= ϕ(f(b1),... ,f(br)),
+∀ϕ ∈ CH
+●(A,M/A),b1,... ,br ∈ B.
+With this we get the following lemma.
+Lemma 2.12. If f ∶ B → A is a dg-algebra map, and F/A ∶ A/A → A∗
+/A is a homotopy inner
+product for A/A, then the following diagram commutes.
+(2.13)
+CH
+●(B,B/B)
+CH(f/B)
+�
+CH(f(F)/B)
+�
+CH
+●(B,A/B)
+CH(F/B)
+�
+CH
+●(A,A/A)
+CH(f,A)
+�
+CH(F/A)
+�
+CH
+●(B,B∗
+/B)
+CH
+●(B,A∗
+/B)
+CH(f∗
+/B)
+�
+CH
+●(A,A∗
+/A)
+CH(f,A∗)
+�
+2.14. Isomorphism on Hochschild cohomology. A particular case of interest is when
+a homotopy inner product F ∶ A → A∗ induces an isomorphism on Hochschild cohomology
+F ∶= HH(F) ∶ HH●(A,A) → HH●(A,A∗). In this case, we can transfer any structure
+between these Hochschild cohomologies, in particular we can transfer the B operator as it
+was done in theorem 1.6.
+Note that in equation (2.13), if the four horizontal maps are quasi-isomorphisms, then,
+obviously, the right vertical map CH(F/A) is a quasi-isomorphism iff the left vertical map
+CH(f(F)/B) is a quasi-isomorphism.
+3. BV algebra on Hochschild cohomology
+We now review how homotopy inner products induce a BV algebra on Hochschild co-
+homology; see theorem 3.7. Our main reference for this is [T2]. We start by defining two
+operators B and ∆F.
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+7
+Definition 3.1. Consider a unital dg-algebra A.
+(1) We define Connes’ B-operator (or more precisely the dual of Connes’ B-operator)
+to be B ∶ CH
+●(A,A∗) → CH
+●(A,A∗) given for ϕ ∈ Hom(A⊗r,A∗) by B(ϕ) ∈
+Hom(A⊗r−1,A∗) with
+(3.2)
+(B(ϕ))(a1,... ,ar−1)(ar)
+∶=
+r
+∑
+j=1
+(−1)(∣aj∣+⋅⋅⋅+∣ar∣)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+∣ar∣ ⋅ ϕ(aj,... ,ar,a1,... ,aj−1)(1),
+A direct but lengthy computation shows that D○B(ϕ) = −B○D(ϕ). By abuse of no-
+tation, we denote the induced map on Hochschild cohomology by B ∶ HH●(A,A∗) →
+HH●(A,A∗) as well.
+(2) Next, assume that we also have a homotopy inner product F = {Fp,q ∶ A⊗p ⊗ A ⊗
+A⊗q → A∗}p,q≥0.
+Then, we define the operator ZF ∶=
+∑
+p,q≥0ZF
+p,q ∶ CH
+●(A,A) →
+CH
+●(A,A∗), where, for ϕ ∈ Hom(A⊗r,A), we set ZF
+p,q(ϕ) ∈ Hom(A⊗p+r+q−1,A∗) to
+be given by
+(ZF
+p,q(ϕ))(a1,... ,ap+r+q−1)(ap+r+q)
+(3.3)
+∶=
+p+q
+∑
+j=p+1
+(−1)∣F∣+(∣ϕ∣+1)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+1
+⋅ Fp,q(a1,... ,ap;1;ap+1,... ,aj−1,ϕ(aj,... ,aj+r−1),aj+r,... ,ap+r+q−1)(ap+r+q)
++
+r
+∑
+j=1
+(−1)(∣aj∣+⋅⋅⋅+∣ap+r+q∣)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+∣ap+r+q∣+∣ϕ∣⋅(∣aj∣+⋅⋅⋅+∣aj+p+q−1∣)
+⋅ Fp,q(aj,... ,aj+p−1;1;aj+p,... ,aj+p+q−1)(ϕ(ap+j+q,... ,ap+r+q,a1,... ,aj−1))
++
+p
+∑
+j=1
+(−1)∣F∣+(∣ϕ∣+1)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+1
+⋅ Fp,q(a1,... ,aj−1,ϕ(aj,... ,aj+r−1),aj+r,... ,ap+r−1;1;ap+r,... ,ap+r+q−1)(ap+r+q)
+One can check again that D ○ ZF(ϕ) = −(−1)∣F∣ ⋅ ZF ○ D(ϕ). By abuse of notation,
+we denote the induced map on Hochschild cohomology by ZF ∶ HH●(A,A) →
+HH●(A,A∗) as well.
+We remark that ZF has appeared in [T2, lemma 17] as the operation associated
+to the symbol
+(−1)µ ⋅ 1
+1
+1
++1
++1
+1
+(3) Let F be a homotopy inner product for A. Assume, moreover, that F ∶ A → A∗
+induces an isomorphism on Hochschild cohomology, F ∶= HH(F) ∶ HH●(A,A) →
+
+8
+K. POIRIER AND T. TRADLER
+HH●(A,A∗). Then denote by ∆F ∶ HH●(A,A) → HH●(A,A) the composition
+∆F ∶= F −1 ○ ZF,
+HH●(A,A)
+ZF
+�→ HH●(A,A∗)
+F−1
+�→ HH●(A,A)
+Lemma 3.4. Let A be a unital dg-algebra.
+(1) On Hochschild cochains HH●(A,A∗), we have that B2 = 0.
+(2) Assume that F ∶ A → A∗ is a homotopy inner product which induces an isomorphism
+on Hochschild cohomology F ∶ HH●(A,A) → HH●(A,A∗). Then the deviation of
+∆F from being a derivation of the cup product is the usual Gerstenhaber bracket.
+Here the cup product on Hochschild cohomology is given for Hochschild cochains
+ϕ ∈ Hom(A⊗r,A) and ρ ∈ Hom(A⊗s,A) to be ϕ ⌣ ρ ∈ Hom(A⊗r+s,A) with
+(3.5)
+(ϕ ⌣ ρ)(a1,... ,ar+s) ∶= (−1)∣ρ∣⋅(∣a1∣+⋅⋅⋅+∣ar∣) ⋅ ϕ(a1,... ,ar) ⋅ ρ(ar+1,... ,ar+s).
+Proof. For (1), note that B2 = 0 follows since normalized Hochschild cochains vanish when
+any input is the unit 1. Part (2) was proved in [T2, section 3.3].
+□
+Since lemma 3.4 (1) and (2) are conditions needed for a BV algebra, i.e., a square zero
+operator (here: B) whose deviation from being a derivation is a Gerstenhaber bracket (here:
+∆F), we make the following definition.
+Definition 3.6. Let A be a dg-algebra, and let F ∶ A → A∗ be a homotopy inner product
+for A. We call F a Poincar´e duality structure for A, if it satisfies if it satisfies the following
+two conditions:
+(1) F induces an isomorphism of graded modules on Hochschild cohomology F =
+HH(F) ∶ HH●(A,A)
+≅
+�→ HH●(A,A∗).
+(2) Transferring B from HH●(A,A∗) to HH●(A,A) via F equals ∆F, i.e.,
+HH●(A,A)
+F
+�
+HH●(A,A∗)
+F−1
+�
+B
+�
+F −1 ○ B ○ F = ∆F = F −1 ○ ZF
+HH●(A,A)
+∆F
+�
+=
+HH●(A,A)
+ZF
+�
+HH●(A,A∗)
+F−1
+�
+With this, we obtain the following theorem, which is [T2, theorem 2].
+Theorem 3.7. Let A be a dg-algebra, and let F ∶ A → A∗ be a Poincar´e duality structure
+for A. Then, HH●(A,A) together with the cup product and ∆F is a BV algebra.
+Examples 3.8. We now give a few examples for how one can check whether a homotopy
+inner product is a Poincar´e duality structure.
+(1) Let F be an inner product for a dg-algebra A (i.e., the only non-zero component is
+F0,0), and assume that F induces an isomorphism on Hochschild cohomology. Then
+F is a Poincar´e duality structure.
+(To see this, we note that this is a special case of the next example (2) below,
+since the condition (D1F)0,1 = 0 gives (using (2.4)) that F(mb)(n) = F(m)(bn),
+and (D1F)1,0 = 0 gives F(am)(n) = (−1)∣a∣⋅(∣m∣+∣n∣)F(m)(na) for all m,n,a,b ∈
+A, and, thus F(m)(n) = F(1 ⋅ m)(n) = F(1)(m ⋅ n) = (−1)∣m∣⋅∣n∣F(n ⋅ 1)(m) =
+(−1)∣m∣⋅∣n∣F(n)(m).)
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+9
+(2) Let F be a homotopy inner product for a dg-algebra A, and assume that F in-
+duces an isomorphism on Hochschild cohomology. Assume further that F is in-
+variant under cyclic rotation of the first p + 1 and last q + 1 inputs, i.e., ∀p,q ≥ 0,
+∀a1,... ,ap,m,b1,... ,bq,n ∈ A:
+Fp,q(a1,... ,ap;m;b1,... ,bq)(n) = (−1)ε ⋅ Fq,p(b1,... ,bq;n;a1,... ,ap)(m)
+a1
+a2
+...
+ap
+m
+b1
+b2
+...
+bq
+n = (−1)ε ⋅
+b1
+b2
+...
+bq
+n
+a1
+a2
+...
+ap
+m
+where (−1)ε = (−1)(∣a1∣+⋅⋅⋅+∣ap∣+∣m∣)⋅(∣b1∣+⋅⋅⋅+∣bq∣+∣n∣). Then F is a Poincar´e duality struc-
+ture. This was proved in [T2, lemma 17].
+(3) In the examples in section 5 of this paper we will check if a given homotopy inner
+product F is a Poincar´e duality structure or not by explicitly computing F, B and
+ZF; cf. the examples given in 5.3, 5.4 and 5.5.
+In particular, it is worth noting that there exist homotopy inner products F for
+which F is an isomorphism, but for which F −1 ○ B ○ F ≠ ∆F. We provide an explicit
+example for such a homotopy inner product in 5.5.
+4. Computations of bimodule maps with higher homotopies
+In this section, we will give explicit bimodule maps and bimodule maps with higher
+homotopies for specific dg-algebras. In particular we will compute the homotopy inner
+product on H●(S2;Z2) coming from a pullback of a local homotopy inner product. In the
+remainder of this paper –with the exception of observation 4.4– the ground ring will always
+be R = Z2.
+Example 4.1 (H●(S2;Z2) inner product without homotopies). Consider the dg-algebra
+A ∶= H●(S2;Z2) ≅ Z2.e ⊕ Z2.s with zero differential and degrees ∣e∣ = 0 and ∣s∣ = −2, and
+where e is the unit and s ⋅ s = 0. For the dg module and dual dg module of A we use
+the notation A ≅ Z2.e ⊕ Z2.s and A∗ ≅ Z2.e∗ ⊕ Z2.s∗, respectively, where e∗ and s∗ are
+the duals of e and s with ∣e∗∣ = 0 and ∣s∗∣ = 2 and the module structure is given by
+e.e∗ = e∗.e = e∗,e.s∗ = s∗.e = s∗,s.e∗ = e∗.s = 0,s.s∗ = s∗.s = e∗ (see 2.1).
+Define the dg bimodule map F ∶ A → A∗ by F(s) = e∗ and F(e) = s∗,
+A
+= ...
+⟨s⟩
+0
+⟨e⟩
+0
+0
+...
+A∗
+= ...
+0
+0
+⟨e∗⟩
+0
+⟨s∗⟩
+...
+F
+F
+F
+Thus, F ∶ A → A∗ is the map F(s)(e) = 1,F(s)(s) = 0,F(e)(e) = 0,F(e)(s) = 1 (in other
+words, F is given by capping with s∗). One can check directly that F is a dg bimodule
+map. Note, that when interpreting F as an inner product <,>= F ∶ A ⊗ A → Z2, the only
+non-vanishing inner products are < s,e >= 1 and < e,s >= 1.
+
+10
+K. POIRIER AND T. TRADLER
+Example 4.2 (H●(S2;Z2) inner product with homotopies). We next consider the same
+algebra A and module structures A and A∗ as in example 4.1, but we define an inner
+product ̃F for A with higher homotopies (in the sense of 2.2). In fact, ̃F has its only
+non-zero components given by ̃F0,0 ∶ A → A∗ and ̃F2,0 ∶ A ⊗ A ⊗ A → A∗ (see 2.2) via
+̃F0,0(s)(e) = 1,
+̃F0,0(e)(s) = 1,
+̃F2,0(s,s;e)(e) = 1.
+(4.3)
+s
+e
+e
+s
+s
+s
+e
+e
+It is again a direct check that ̃F is a homotopy inner product, i.e., it satisfies all equation
+required by 0 = DF = D1F given by (2.4) (since d and thus D0 vanishes). Explicitly, these
+equations are:
+̃F0,0(a ⋅ a1)(̃a) =̃F0,0(a)(a1 ⋅ ̃a),
+̃F0,0(a1 ⋅ a)(̃a) =̃F0,0(a)(̃a ⋅ a1),
+̃F2,0(a1,a2;a ⋅ a3)(̃a) =̃F2,0(a1,a2;a)(a3 ⋅ ̃a),
+0 =̃F2,0(a1,a2;a3 ⋅ a)(̃a) + ̃F2,0(a1,a2 ⋅ a3;a)(̃a)
++ ̃F2,0(a1 ⋅ a2,a3;a)(̃a) + ̃F2,0(a2,a3;a)(̃a ⋅ a1)
+for all a,̃a ∈ A, and a1,a2,a3 ∈ A.
+We next give formulas for calculating homotopy inner products (over Z2) for (triangu-
+lated) 2-dimensional spaces on the cochain level.
+The following observation notes that
+higher homotopies naturally appear for inner products on the cochain level.
+Observation 4.4. Let A be a unital dg-algebra over any commutative ring R, so that
+both A ∶= A and A∗ are dg-modules over A, (see 2.1). Now, let x ∈ A∗ be a fixed closed
+element, d(x) = 0. Define F ∶ A → A∗ for a ∈ A by setting F(a) ∈ A∗ evaluated on some
+̃a ∈ A to be F(a)(̃a) ∶= (x ⌢ a)(̃a) = x(a ⋅ ̃a).
+Claim: F is chain map, and a graded right module map. F is in general not a graded
+left module map. A sufficient condition for F being a graded left module map is that A is
+graded commutative.
+Proof. First, F is chain map, since for a,̃a ∈ A and dx = 0, we have F(da)(̃a) = x(da ⋅ ̃a) =
+x(d(a ⋅ ̃a) − (−1)∣a∣a ⋅ d̃a) = (−1)∣x∣+1dx(a ⋅ ̃a) − (−1)∣a∣ ⋅ F(a)(d̃a) = −(−1)∣a∣ ⋅ (−1)∣F(a)∣+1 ⋅
+d(F(a))(̃a) = (−1)∣F∣⋅d(F(a))(̃a). Next, F is a graded right module map, since for a,̃a ∈ A,
+a1 ∈ A, we have (F(a).a1)(̃a) = F(a)(a1 ⋅ ̃a) = x(a ⋅ a1 ⋅ ̃a) = F(a.a1)(̃a). To check when
+F is a graded left module map, compute F(a1.a)(̃a) = x(a1 ⋅ a ⋅ ̃a) and (a1.F(a))(̃a) =
+(−1)∣a1∣⋅(∣F(a)∣+∣̃a∣) ⋅ F(a)(̃a ⋅ a1) = (−1)∣a1∣⋅(∣F∣+∣a∣+∣̃a∣) ⋅ x(a ⋅ ̃a ⋅ a1). Thus, F is in general not a
+graded left module map. Moreover, if a1 ⋅ a ⋅ ̃a = (−1)∣a1∣⋅(∣a∣+∣̃a∣)a ⋅ ̃a ⋅ a1 for all a,̃a,a1, then
+(a1.F(a))(̃a) = (−1)∣a1∣⋅∣F∣ ⋅ F(a1.a)(̃a), and thus F will be a graded left module map.
+□
+In particular, assume that X is a closed, oriented manifold, and R is a commutative
+ring. In order to calculate the BV algebra on HH●(C●(X;R);(C●(X;R))∗) one might try
+to use capping a cochain with a fundamental cycle x of X as the appropriate dg-bimodule
+map F = x ⌢ − ∶ C●(X;R) → C●(X;R)
+incl
+↪ (C●(X;R))∗. However, the above observation
+shows that capping on the (co-)chain level is in general not a dg-bimodule map, and thus
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+11
+does not induce a chain map on the Hochschild cochains. One way to resolve this issue
+is to provide higher homotopies for the left module structure, i.e., to provide a homotopy
+inner product, which then does give a corresponding chain map on Hochschild cochains.
+This is what is done in this paper for S2 with Z2 coefficients.
+Example 4.5 (The 0-simplex). Let A[0] ∶= Z2.e0, where e0 is the unit. A homotopy inner
+product F[0] ∶ A[0] → (A[0])∗ is given by F[0](e0)(e0) = 1.
+We will also vary the superscript in an obvious way, i.e., the dg-algebra A[1] ∶= Z2.e1 has
+the homotopy inner product F[1] ∶ A[1] → (A[1])∗,F[1](e1)(e1) = 1, etc.
+Example 4.6 (The 1-simplex). Let A[01] ∶= Z2.e0 ⊕ Z2.e1 ⊕ Z2.b01, where ∣e0∣ = ∣e1∣ = 0 and
+∣b01∣ = −1 with differential d(e0) = d(e1) = b01. The product is the usual cup product of
+simplicial cochains, i.e., e0 ⋅ e0 = e0, e1 ⋅ e1 = e1, and e0 ⋅ b01 = b01 ⋅ e1 = b01. Note that the unit
+of A[01] is e0 + e1.
+We want to define an inner product F with lowest component non-vanishing only for
+F(e0)(b01) = F(b01)(e1) = 1. Note that F is not a chain map, but F(d(a))(̃a)+F(a)(d(̃a)) =
+F[0](a)(̃a) + F[1](a)(̃a) for all a,̃a ∈ A[01]. Moreover, F is a graded right module map, i.e.,
+F(a ⋅ a1)(̃a) = F(a)(a1 ⋅ ̃a) for all a,̃a,a1 ∈ A[01], but F is not a graded left module map,
+since F(e0 ⋅ e0)(b01) = 1 ≠ 0 = F(e0)(b01 ⋅ e0).
+There is an inductive procedure (involving choices at each stage) for obtaining higher
+homotopies that provide F with left modules homotopies, making it into a homotopy inner
+product up to F[0] and F[1] from example 4.5 (interpreted as maps A[01] → (A[01])∗) . This
+procedure was described in [TZS, Proposition 3.1.2]. Performing the induction leads to a
+sequence of maps, which was first stated in [TZS, Proposition B.2]:
+∀k ≥ 0 ∶
+F[01]
+k,0 (b01,... ,b01;e0)(b01) = 1,
+F[01]
+k,0 (b01,... ,b01;b01)(e1) = 1,
+(4.7)
+b01
+b01
+...
+b01
+e0
+b01
+b01
+b01
+...
+b01
+b01
+e1
+All other inner products are zero. Note, this resolves the above problem of F[01]
+0,0 not being a
+graded left module map, since, now, F[01]
+0,0 is a graded left module map up to the homotopy
+F[01]
+1,0 ; for example:
+(DF[01])1,0(e0;e0)(b01) = F[01]
+1,0 (d(e0);e0)(b01) + F[01]
+1,0 (e0;d(e0))(b01)
++ F[01]
+0,0 (e0 ⋅ e0)(b01) + F[01]
+0,0 (e0)(b01 ⋅ e0) = 1 + 0 + 1 + 0 = 0.
+From [TZS, Proposition B.2], which we will also prove in appendix A, we have that:
+(4.8)
+DF[01] = F[0] + F[1]
+Again, we will need to vary the superscript in an obvious way, i.e., for the dg-algebra
+A[12] there are maps F[12] which are given by replacing 0 and 1 in the above with 1 and 2,
+respectively, etc.
+Example 4.9 (The 2-simplex). We now describe inner product maps for the 2-simplex,
+which extends the previous two examples of the 0- and 1-simplex. Let A[012] ∶= Z2.e0 ⊕
+Z2.e1 ⊕ Z2.e2 ⊕ Z2.b01 ⊕ Z2.b02 ⊕ Z2.b12 ⊕ Z2.c012 with ∣ej∣ = 0, ∣bij∣ = −1 and ∣c012∣ = −2, and
+
+12
+K. POIRIER AND T. TRADLER
+differential d(e0) = b01 + b02, d(e1) = b01 + b12, d(e2) = b02 + b12 and d(bij) = c012 for all
+0 ≤ i < j ≤ 2. The multiplication is non-zero only for
+∀j ∶
+ej ⋅ ej = ej,
+∀i < j ∶
+ei ⋅ bij = bij,
+bij ⋅ ej = bij,
+(4.10)
+e0 ⋅ c012 = c012,
+c012 ⋅ e2 = c012,
+b01 ⋅ b12 = c012.
+Now, following the procedure (which uses locality) from [TZS, Proposition 3.1.2], we
+define maps F[012]
+k,0
+whose only non-zero maps are given by the following equations (4.11)-
+(4.18):
+∀k ≥ 0 ∶
+F[012]
+k,0 (b02,... ,b02;c012)(e2) = 1,
+(4.11)
+F[012]
+k,0 (b02,... ,b02;e0)(c012) = 1,
+(4.12)
+F[012]
+k,0 (b02,... ,b02;b01)(b12) = 1,
+(4.13)
+b02
+b02
+...
+b02
+e2
+c012
+b02
+b02
+...
+b02
+c012
+e0
+b02
+b02
+...
+b02
+b12
+b01
+∀k ≥ 0 ∶ ∀1 ≤ ℓ ≤ k + 1 ∶
+F[012]
+k+1,0(b01,... ,b01, c012
+�
+ℓth
+,b02,... ,b02;e0)(b01) = 1,
+(4.14)
+F[012]
+k+1,0(b01,... ,b01, c012
+�
+ℓth
+,b02,... ,b02;b01)(e1) = 1,
+(4.15)
+F[012]
+k+1,0(b02,... ,b02, c012
+�
+ℓth
+,b12,... ,b12;e1)(b12) = 1,
+(4.16)
+F[012]
+k+1,0(b02,... ,b02, c012
+�
+ℓth
+,b12,... ,b12;b12)(e2) = 1,
+(4.17)
+b01
+...
+b01
+c012
+b02...b02
+b01
+e0
+b01
+...
+b01
+c012
+b02...b02
+e1
+b01
+b02
+...
+b02
+c012
+b12...b12
+b12
+e1
+b02
+...
+b02
+c012
+b12...b12
+e2
+b12
+(4.18)
+∀k ≥ 0 ∶ ∀1 ≤ ℓ1 < ℓ2 ≤ k + 2 ∶
+F[012]
+k+2,0(b01,... ,b01, c012
+�
+ℓ1th
+,b02,... ,b02, c012
+�
+ℓ2th
+,b12,... ,b12;e1)(e1) = 1.
+b01...
+b01
+c012
+b02
+...
+b02
+c012
+b12...b12
+e1
+e1
+We claim that the following equation holds, which will be proved in appendix A:
+(4.19)
+DF[012] = F[01] + F[02] + F[12]
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+13
+As before, the above will also be applied to obvious variations of the superscript, such
+as, e.g, A[123] with maps F[123] given by replacing 0, 1 and 2 in the above with 1, 2 and 3,
+respectively, etc.
+Example 4.20 (C●(S2;Z2) inner product with homotopies). We now use a tetrahedral
+triangulation of the 2-sphere. More precisely, we set
+A = ⊕
+0≤j≤3
+Z2.ej ⊕
+⊕
+0≤i<j≤3
+Z2.bij ⊕
+⊕
+0≤i<j<l≤3
+Z2.cijl
+e0
+e1
+e2
+e3
+with ∣ej∣ = 0, ∣bij∣ = −1 and ∣cijl∣ = −2 and differential d(ej) = ∑i≠j b{ij}, d(bij) = ∑l≠i,j c{ijl},
+and d(cijl) = 0, where the bracket “{}” denotes indices in ascending order, i.e., b{ij} =
+{bij, for i < j
+bji, for j < i and similarly for c{ijl}. The multiplication is similar to (4.10), where the
+indices “012” are replaced by any 0 ≤ i < j < l ≤ 3:
+∀j ∶
+ej ⋅ ej = ej,
+∀i < j ∶
+ei ⋅ bij = bij,
+bij ⋅ ej = bij,
+(4.21)
+∀i < j < l ∶
+ei ⋅ cijl = cijl,
+cijl ⋅ el = cijl,
+bij ⋅ bjl = cijl,
+and all other multiplications vanish. Note in particular that A has the unit e0 +e1 +e2 +e3.
+We define the homotopy inner product F for A ∶= A to be given by (4.11)-(4.18) with
+“012” in (4.11)-(4.18) replaced by any 0 ≤ i < j < l ≤ 3, i.e.,
+(4.22)
+F ∶= F[012] + F[013] + F[023] + F[123].
+Claim: F is a homotopy inner product, i.e., DF = 0.
+Proof. We compute DF = DF[012] + DF[013] + DF[023] + DF[123]. We claim that we can use
+equation (4.19) to evaluate this expression, which is not completely obvious since D in
+(4.19) is for the dg-algebra A[012], whereas we need to apply D for A.
+To see that (4.19) holds for the dg-algebra A, note that, for example, DF[012](a1,... )(ar)
+applies a differential or a multiplication to the inputs aj ∈ A (as stated in (2.3) and (2.4)).
+Now, if any of the inputs aj are generators of A which has a 3 in the index (such as, e.g.,
+b13 ∈ A), then taking the differential or a multiplication will consist of sums of generators
+with 3 in their indices, and thus DF[012] applied to it would vanish. On the other hand, if
+all inputs aj applied to DF[012] are only given by generators indexed by 0, 1, and 2, then
+DF[012] applied to it coincides with what we would get for A[012], because, the multiplication
+in A equals the one in A[012], and the differential in A equals the differential in A[012] up
+to generators with indices of 3. (For example, applying d(b12) = c012 + c123 in DF[012] will
+vanish on c123 and will coincide with (4.19) on c012 = dA[012](b12) =(differential of b12 in
+A[012]).)
+Thus, DF = DF[012] + DF[013] + DF[023] + DF[123] = (F[01] + F[02] + F[12]) + (F[01] + F[03] +
+F[13]) + (F[02] + F[03] + F[23]) + (F[12] + F[13] + F[23]) = 0.
+□
+
+14
+K. POIRIER AND T. TRADLER
+Proposition 4.23. Let A be the dg-algebra from example 4.20, and let B = H●(S2;Z2) ≅
+Z2.e ⊕ Z2.s be the dg-algebra from examples 4.1 and 4.2.
+Then, the map f ∶ B → A,
+f(e) ∶= e0 + e1 + e2 + e3 and f(s) ∶= c012, is a dg-algebra quasi-isomorphism.
+Transferring the homotopy inner product F/A defined in (4.22) via the map f gives
+f(F)/B = ̃F/B, where ̃F/B is the homotopy inner product defined in equation (4.3) from
+example 4.2.
+Proof. The transfered map of F/A from (4.22) is given by applying f to inputs from B and
+then applying F/A; see (2.10). Since the image of f is spanned by c012 and e0 + e1 + e2 + e3,
+it follows from (4.11)-(4.18) that the only non-zero components (after applying f) are:
+F[012]
+0,0 (c012)(e2) = 1,
+F[012]
+0,0 (e0)(c012) = 1,
+F[012]
+2,0 (c012,c012;e1)(e1) = 1.
+Thus, we get precisely the non-zero components of ̃F from (4.3) for f(F)/B.
+□
+5. Computations of the BV algebras
+We now compute the BV algebras on Hochschild cohomology induced by the homotopy
+inner products considered in examples 4.1 and 4.2 in section 4. In particular, we com-
+plete the computations from theorem 1.6 for the 2-sphere by computing HH●(H●,H●) and
+HH●(H●,H●) for H● = H● = H●(S2;Z2) and H● = H●(S2;Z2), and we will check the for-
+mulas for the maps F and ̃
+F in (1.7) stated in theorem 1.6. In the section the ground ring
+is R = Z2.
+5.1. Hochschild cohomology HH●(H●,H●) and the cup product. Denote by H● =
+H●(S2;Z2) ≅ Z2.e ⊕ Z2.s from example 4.1 with ∣e∣ = 0 and ∣s∣ = −2. The generators of
+normalized Hochschild cochains CH
+●(H●,H●) are φk, ψk for k ≥ 0 given by
+φk(s,... ,s
+��������������������
+k many
+) ∶= e,
+ψk(s,... ,s
+��������������������
+k many
+) ∶= s.
+Note that ∣φk∣ = k, ∣ψk∣ = k − 2, since the shifted generator s is of degree ∣s∣ = −1. Applying
+the Hochschild differential (see 2.1, and using d = 0 and s ⋅ s = 0, s ⋅ e = e ⋅ s = s), we see that
+all φk and ψk are closed (over Z2). Thus,
+HH●(H●,H●) ≅ ⊕
+k≥0
+Z2.φk ⊕ ⊕
+k≥0
+Z2.ψk.
+The cup product (3.5) is readily checked to be φk ⌣ φℓ = φk+ℓ, ψk ⌣ ψℓ = 0, and φk ⌣ ψℓ =
+ψℓ ⌣ φk = ψk+ℓ.
+5.2. Hochschild cohomology HH●(H●,H●) and Connes’ B-operator. Denote by H● =
+H●(S2;Z2) ≅ Z2.e∗ ⊕ Z2.s∗ from example 4.1, with ∣e∗∣ = 0 and ∣s∗∣ = 2 and s.e∗ = e∗.s =
+0,s.s∗ = s∗.s = e∗. The generators of normalized Hochschild cochains CH
+●(H●,H●) are θk,
+χk for k ≥ 0 given by
+θk(s,... ,s
+��������������������
+k many
+) ∶= s∗,
+χk(s,... ,s
+��������������������
+k many
+) ∶= e∗.
+Note that ∣θk∣ = k + 2, ∣χk∣ = k. All θk and χk are closed under the Hochschild cochain
+differential (over Z2), so that
+HH●(H●,H●) ≅ ⊕
+k≥0
+Z2.θk ⊕ ⊕
+k≥0
+Z2.χk.
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+15
+Whereas HH●(H●,H●) has the cup product (3.5), HH●(H●,H●) has Connes’ B-operator
+(3.2). We compute B from (3.2) to be (B(ϕ))( s,... ,s
+��������������������
+(r−1) many
+)(a) ∶= r ⋅ ϕ(s,... ,s,a,s,... ,s)(e),
+which applied to θk and χk yields:
+B(θr) = 0,
+B(χr) = r ⋅ θr−1.
+This confirms the expressions for B stated in theorem 1.6.
+5.3. F induced by example 4.1 and the BV delta ∆F. The dg-bimodule map F ∶ H● →
+H●, F(s) = e∗ and F(e) = s∗, from example 4.1 induced a map on Hochschild cohomologies
+F ∶ HH●(H●,H●) → HH●(H●,H●) via (2.6). Since F only has a lowest component F0,0
+this is simply
+F(φk) = F ○ φk = θk,
+F(ψk) = F ○ ψk = χk.
+F is clearly an isomorphism, and confirms F from theorem 1.6.
+We also compute ∆F and check it is equal to B transferred to HH●(H●,H●), i.e., con-
+dition (2) from definition 3.6 for a Poincar´e duality structure.
+From (3.3), we get for
+ϕ ∈ Hom(H●⊗r,H●) that ZF(ϕ)(s,... ,s)(a) = r ⋅ F(e)(ϕ(s,... ,a,... ,s)), which gives
+ZF(φk) = 0 = B(θk) = B ○ F(φk),
+ZF(ψk) = k ⋅ θk−1 = B(χk) = B ○ F(ψk).
+Therefore, F −1 ○ B ○ F = F −1 ○ ZF = ∆F.
+5.4. ̃
+F induced by example 4.2 and the BV delta ∆̃F. Consider the dg-bimodule
+map ̃F ∶ H● → H● from example 4.2. Since ̃F = ̃F0,0 + ̃F2,0 is ̃F0,0(s) = e∗, ̃F0,0(e) = s∗ with
+one extra homotopy ̃F2,0 which is non-zero only for ̃F2,0(s,s;e) ∶= e∗ (see (4.3)), we get
+̃
+F(φk) = ̃F0,0 ○ φk + ̃F2,0(−,−;φk(−)) = θk + χk+2,
+̃
+F(ψk) = ̃F0,0 ○ ψk = χk.
+This confirms ̃
+F from theorem 1.6. As was noted in theorem 1.6, ̃
+F is an isomorphism with
+inverse ̃
+F −1 given by ̃
+F −1(θk) = φk + ψk+2 and ̃
+F −1(χk) = ψk.
+We also check that in this case ̃
+F −1○B○ ̃
+F = ∆̃F. From (3.3), we get for ϕ ∈ Hom(H●⊗r,H●),
+that
+Z
+̃F(ϕ)(s,... ,s)(a) =r ⋅ ̃F0,0(e)(ϕ(s,... ,a,... ,s)) + r ⋅ ̃F2,0(s,s;e)(ϕ(s,... ,a,... ,s))
++ ̃F2,0(ϕ(s,... ,s),s;e)(a) + ̃F2,0(s,ϕ(s,... ,s);e)(a)
+which gives
+Z
+̃F(φk) = 0 + k ⋅ θk+1 + 0 + 0 = k ⋅ θk+1,
+Z
+̃F(ψk) = k ⋅ θk−1 + 0 + χk+1 + χk+1 = k ⋅ θk−1.
+Since B ○ ̃
+F(φk) = B(θk + χk+2) = k ⋅ θk+1 and B ○ ̃
+F(ψk) = B(χk) = k ⋅ θk−1, this shows that
+Z ̃F = B ○ ̃
+F and, thus, ̃
+F −1 ○ B ○ ̃
+F = ̃
+F −1 ○ Z ̃F = ∆̃F. Explicitly, ∆̃F is given by:
+∆
+̃F(φk) = ̃
+F −1 ○ Z
+̃F(φk) = ̃
+F −1(k ⋅ θk+1) = k ⋅ (φk+1 + ψk+3)
+∆
+̃F(ψk) = ̃
+F −1 ○ Z
+̃F(ψk) = ̃
+F −1(k ⋅ θk−1) = k ⋅ (φk−1 + ψk+1)
+This confirms ̃∆ from (1.9) and thus theorem 1.6.
+
+16
+K. POIRIER AND T. TRADLER
+5.5. Example of a homotopy inner product that is not a Poincar´e duality struc-
+ture. We end this section with an example of a homotopy inner product which is not
+a Poincar´e duality structure due to its failure of condition (2) from definition 3.6 (while
+condition (1) holds). We therefore do not get a BV algebra on Hochschild cohomology
+HH●(A,A) with underlying algebra A, since there is no ∆ operator, but only an operator,
+F −1 ○B ○F, that squares to zero, and a different operator, ∆F, whose deviation from being
+a derivation is a Gerstenhaber bracket; cf. lemma 3.4.
+Let A = Z2.e ⊕ Z2.b ⊕ Z2.c with zero differential, ∣e∣ = 0, ∣b∣ = −1, ∣c∣ = −2, and where
+e is the unit, b ⋅ b = c and all other products vanish.
+We denote A = A and its dual
+A∗ = Z2.e∗ ⊕ Z2.b∗ ⊕ Z2.c∗ where e∗,b∗,c∗ are the duals of e,b,c, respectively, so that
+∣e∗∣ = 0, ∣b∗∣ = 1, ∣c∗∣ = 2. Define the homotopy inner product F whose only non-vanishing
+components are F0,0 and F3,0 given by
+F0,0(c)(e) = F0,0(b)(b) = F0,0(e)(c) = 1,
+F3,0(c,b,c;e)(e) = F3,0(b,b,b;c)(e) = F3,0(b,b,b;e)(c) = F3,0(b,b,b;b)(b) = 1.
+One can check explicitly that the conditions for a homotopy inner product 0 = DF = D1F,
+where D1 is given by (2.4), are satisfied.
+Note, that F induces an isomorphism on Hochschild cochains F ∶ CH
+●(A,A)
+≅
+�→
+CH
+●(A,A∗), since F0,0 ∶ A → A∗ is an isomorphism, and thus F satisfies 3.6 condition (1).
+We next show that 3.6 condition (2) fails, i.e., that B○F ≠ ZF. Consider ϕ ∈ Hom(A⊗0,A),
+ϕ(1) = e. Note that ϕ is a closed element in the Hochschild cochains ϕ ∈ CH
+●(A,A). Plug-
+ging ϕ into (3.3) gives ZF(ϕ) = 0, since each individual summand of (3.3) vanishes. Next,
+to compute F(ϕ), we see the only non-zero evaluation of F(ϕ) are (the ones with “e” in
+the input spot for the module of F):
+F(ϕ)(1) = F0,0(ϕ(1)) = c∗
+F(ϕ)(c,b,c) = F3,0(c,b,c;ϕ(1)) = e∗
+F(ϕ)(b,b,b) = F3,0(b,b,b;ϕ(1)) = c∗
+Applying B from (3.2) (note that the unit in A is in this example written as e) to this is
+only non-zero on the middle term, for which we get the only non-zero evaluations:
+B(F(ϕ))(c,b)(c) = 1,
+B(F(ϕ))(b,c)(c) = 1,
+B(F(ϕ))(c,c)(b) = 1.
+Then, B(F(ϕ)) is non-vanishing, and closed in CH
+●(A,A∗), which follows either because
+ϕ is closed and B ○ F is a chain map or by direct inspection.
+Claim: B(F(ϕ)) is not exact in CH
+●(A,A∗).
+The claim implies that B ○F(ϕ) ≠ 0 = ZF(ϕ) in HH●(A,A∗), and so F is not a Poincar´e
+duality structure as it fails condition (2) from definition 3.6.
+Proof of the claim: To prove the claim, note that d = 0 in A implies that the Hochschild
+differential D = D0 + D1 = D1 from 2.1 must increase the number of tensor factors of A for
+the inputs. Thus any ̃ϕ which makes B(F(ϕ)) exact, i.e., D(̃ϕ) = B(F(ϕ)), must have
+non-vanishing components with exaclty one tensor input. Since the degree ∣B(F(ϕ))∣ = 3,
+it must be that ∣̃ϕ∣ = 4. However, all Hochschild cochains in CH
+●(A,A∗) with one or no
+input are of degree ≤ 3. Thus, B(F(ϕ)) cannot be exact.
+□
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+17
+Appendix A. Proof of equations (4.8) and (4.19)
+In this appendix we prove the identity (4.8), i.e., DF[01] = F[0] + F[1], for the maps F[01]
+given by (4.7) from example 4.6, as well as the identity (4.19), i.e., DF[012] = F[01] + F[02] +
+F[12], for the maps F[012] given by (4.11)-(4.18) from example 4.9. The ground ring in this
+appendix is R = Z2.
+A.1. Preliminaries. Any Fp,q is a map Fp,q ∶ A⊗p ⊗ A ⊗ A⊗q → A∗, which is an element
+of Fp,q ∈ (A∗)⊗p ⊗ A∗ ⊗ (A∗)⊗q ⊗ A∗. Consider A[01] with generators e0,e1,b01, respectively
+A[012] with generators e0,e1,e2,b01,b12,b02,c012). For a multi-index I ∈ {0,1,2,01,02,12,012},
+define a∗
+I to be a∗
+0 ∶= e∗
+0, a∗
+1 ∶= e∗
+1, a∗
+2 ∶= e∗
+2, a∗
+01 ∶= b∗
+01, a∗
+02 ∶= b∗
+02, a∗
+12 ∶= b∗
+12, a∗
+012 ∶= c∗
+012. With
+this we use the following notation, for a given sequence of multi-indices I1,... ,Ip+q+2:
+(A.2)
+[ I1,... ,Ip ∣ Ip+1 ∣ Ip+2,... ,Ip+q+1 ∣ Ip+q+2 ]
+∶= a∗
+I1 ⊗ ... ⊗ a∗
+Ip ⊗ a∗
+Ip+1 ⊗ a∗
+Ip+2 ⊗ ... ⊗ a∗
+Ip+q+1 ⊗ a∗
+Ip+q+2 ∈ (A∗)⊗p ⊗ A∗ ⊗ (A∗)⊗q ⊗ A∗
+Moreover, placing a tilde under the multi-index sums over any number of these indices, i.e.,
+for tildes at positions 1 ≤ k1 < k2 < ⋅⋅⋅ < kr ≤ p, we define
+(A.3)
+[ I1,... ,Ik1
+̃
+,... ,Ik2
+̃
+,... ,Ip ∣ Ip+1 ∣ Ip+2,... ,Ip+q+1 ∣ Ip+q+2 ]
+∶=
+∑
+n1,n2,...,nr≥0
+[ I1,... ,Ik1,... ,Ik1
+���������������������������������������������
+n1 many
+,... ,Ik2,... ,Ik2
+���������������������������������������������
+n2 many
+,... ,Ip ∣ Ip+1 ∣ Ip+2,... ,Ip+q+1 ∣ Ip+q+2 ]
+Since our examples are chosen to be “strict” on the right (see observation 4.4), we only
+need higher homotopies (and thus tildes) on the first p tensor factors, but not on the other
+q tensor factors of A∗.
+We apply D = D0 +D1 from (2.3) and (2.4) to the above. Here, D0 applies the differential
+of A∗, which, in the multi-indiex notation, removes one index, (for example, 01 ↦ 0 + 1,
+012 ↦ 01 + 02 + 12, etc.). D1 applies a multiplication, which on A∗ applies the Alexander-
+Whitney coproduct (for example, 01 ↦ 0 ⊗ 01 + 01 ⊗ 1, 012 ↦ 0 ⊗ 012 + 01 ⊗ 12 + 012 ⊗ 2,
+etc.); see also the calculation below.
+A.4. Proof of (4.8). F[01] from (4.7) is given by F[01] = [ 01
+̃ ∣ 0 ∣ ∣ 01 ] + [ 01
+̃ ∣ 01 ∣ ∣ 1 ].
+We calculate D = D0 + D1 applied to F[01].
+D0([ 01
+̃ ∣ 0 ∣ ∣ 01 ]) =
+[ 01
+̃,0, 01
+̃ ∣ 0 ∣ ∣ 01 ] 01
++[ 01
+̃,1, 01
+̃ ∣ 0 ∣ ∣ 01 ] 02
++[ 01
+̃ ∣ 0 ∣ ∣ 0 ] 03
++[ 01
+̃ ∣ 0 ∣ ∣ 1 ] 04
+D1([ 01
+̃ ∣ 0 ∣ ∣ 01 ]) =
+[ 01
+̃,0,01, 01
+̃ ∣ 0 ∣ ∣ 01 ] 01
++[ 01
+̃,01,1, 01
+̃ ∣ 0 ∣ ∣ 01 ] 02
++[ 01
+̃,0 ∣ 0 ∣ ∣ 01 ] 01
++[ 01
+̃ ∣ 0 ∣ 0 ∣ 01 ] 05
++[ 01
+̃ ∣ 0 ∣ 0 ∣ 01 ] 05
++[ 01
+̃ ∣ 0 ∣ 01 ∣ 1 ] 06
++[ 01, 01
+̃ ∣ 0 ∣ ∣ 0 ] 03
++[ 1, 01
+̃ ∣ 0 ∣ ∣ 01 ] 02
+D0([ 01
+̃ ∣ 01 ∣ ∣ 1 ]) =
+[ 01
+̃,0, 01
+̃ ∣ 01 ∣ ∣ 1 ] 07
++[ 01
+̃,1, 01
+̃ ∣ 01 ∣ ∣ 1 ] 08
++[ 01
+̃ ∣ 0 ∣ ∣ 1 ] 04
++[ 01
+̃ ∣ 1 ∣ ∣ 1 ] 09
+
+18
+K. POIRIER AND T. TRADLER
+D1([ 01
+̃ ∣ 01 ∣ ∣ 1 ]) =
+[ 01
+̃,0,01, 01
+̃ ∣ 01 ∣ ∣ 1 ] 07
++[ 01
+̃,01,1, 01
+̃ ∣ 01 ∣ ∣ 1 ] 08
++[ 01
+̃,0 ∣ 01 ∣ ∣ 1 ] 07
++[ 01
+̃,01 ∣ 1 ∣ ∣ 1 ] 09
++[ 01
+̃ ∣ 0 ∣ 01 ∣ 1 ] 06
++[ 01
+̃ ∣ 01 ∣ 1 ∣ 1 ] 10
++[ 01
+̃ ∣ 01 ∣ 1 ∣ 1 ] 10
++[ 1, 01
+̃ ∣ 01 ∣ ∣ 1 ] 08
+The gray box on the right of each expression indicates which terms can be combined. Note
+that for
+01 ,
+02 ,
+07 , and
+08 , there are always 3 terms that can be combined and cancel. In
+fact, all terms cancel, except for terms labeled
+03 , which has a left over term of [ ∣ 0 ∣ ∣ 0 ],
+and terms labeled
+09 , which has a left over term of [ ∣ 1 ∣ ∣ 1 ]. These two left-over terms
+are F[0] and F[1], which shows that DF[01] = F[0] + F[1], i.e., equation (4.8).
+A.5. Proof of (4.19). F[012] from (4.11)-(4.18) is given by
+F[012] =[ 02
+̃ ∣ 012 ∣ ∣ 2 ] + [ 02
+̃ ∣ 0 ∣ ∣ 012 ] + [ 02
+̃ ∣ 01 ∣ ∣ 12 ]
++ [ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] + [ 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ]
++ [ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] + [ 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ]
++ [ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ]
+We calculate D = D0 + D1 applied to F[012]. Again, the gray box indicates which terms
+combine and cancel. Left-over terms will be collected below.
+D0([ 02
+̃ ∣ 012 ∣ ∣ 2 ]) =
+[ 02
+̃,0, 02
+̃ ∣ 012 ∣ ∣ 2 ] 01
++[ 02
+̃,2, 02
+̃ ∣ 012 ∣ ∣ 2 ] 02
++[ 02
+̃ ∣ 01 ∣ ∣ 2 ] 03
++[ 02
+̃ ∣ 02 ∣ ∣ 2 ] 04
++[ 02
+̃ ∣ 12 ∣ ∣ 2 ] 05
+D1([ 02
+̃ ∣ 012 ∣ ∣ 2 ]) =
+[ 02
+̃,0,02, 02
+̃ ∣ 012 ∣ ∣ 2 ] 01
++[ 02
+̃,02,2, 02
+̃ ∣ 012 ∣ ∣ 2 ] 02
++[ 02
+̃,0 ∣ 012 ∣ ∣ 2 ] 01
++[ 02
+̃,01 ∣ 12 ∣ ∣ 2 ] 06
++[ 02
+̃,012 ∣ 2 ∣ ∣ 2 ] 07
++[ 02
+̃ ∣ 012 ∣ 2 ∣ 2 ] 08
++[ 02
+̃ ∣ 01 ∣ 12 ∣ 2 ] 09
++[ 02
+̃ ∣ 0 ∣ 012 ∣ 2 ] 10
++[ 02
+̃ ∣ 012 ∣ 2 ∣ 2 ] 08
++[ 2, 02
+̃ ∣ 012 ∣ ∣ 2 ] 02
+D0([ 02
+̃ ∣ 0 ∣ ∣ 012 ]) =
+[ 02
+̃,0, 02
+̃ ∣ 0 ∣ ∣ 012 ] 11
++[ 02
+̃,2, 02
+̃ ∣ 0 ∣ ∣ 012 ] 12
++[ 02
+̃ ∣ 0 ∣ ∣ 01 ] 13
++[ 02
+̃ ∣ 0 ∣ ∣ 02 ] 14
++[ 02
+̃ ∣ 0 ∣ ∣ 12 ] 15
+D1([ 02
+̃ ∣ 0 ∣ ∣ 012 ]) =
+[ 02
+̃,0,02, 02
+̃ ∣ 0 ∣ ∣ 012 ] 11
++[ 02
+̃,02,2, 02
+̃ ∣ 0 ∣ ∣ 012 ] 12
+[ 02
+̃,0 ∣ 0 ∣ ∣ 012 ] 11
++[ 02
+̃ ∣ 0 ∣ 0 ∣ 012 ] 16
+[ 02
+̃ ∣ 0 ∣ 0 ∣ 012 ] 16
++[ 02
+̃ ∣ 0 ∣ 01 ∣ 12 ] 17
+[ 02
+̃ ∣ 0 ∣ 012 ∣ 2 ] 10
++[ 2, 02
+̃ ∣ 0 ∣ ∣ 012 ] 12
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+19
+[ 12, 02
+̃ ∣ 0 ∣ ∣ 01 ] 18
++[ 012, 02
+̃ ∣ 0 ∣ ∣ 0 ] 19
+D0([ 02
+̃ ∣ 01 ∣ ∣ 12 ]) =
+[ 02
+̃,0, 02
+̃ ∣ 01 ∣ ∣ 12 ] 20
++[ 02
+̃,2, 02
+̃ ∣ 01 ∣ ∣ 12 ] 21
++[ 02
+̃ ∣ 0 ∣ ∣ 12 ] 15
++[ 02
+̃ ∣ 1 ∣ ∣ 12 ] 22
++[ 02
+̃ ∣ 01 ∣ ∣ 1 ] 23
++[ 02
+̃ ∣ 01 ∣ ∣ 2 ] 03
+D1([ 02
+̃ ∣ 01 ∣ ∣ 12 ]) =
+[ 02
+̃,0,02, 02
+̃ ∣ 01 ∣ ∣ 12 ] 20
++[ 02
+̃,02,2, 02
+̃ ∣ 01 ∣ ∣ 12 ] 21
++[ 02
+̃,0 ∣ 01 ∣ ∣ 12 ] 20
++[ 02
+̃,01 ∣ 1 ∣ ∣ 12 ] 24
++[ 02
+̃ ∣ 01 ∣ 1 ∣ 12 ] 25
++[ 02
+̃ ∣ 0 ∣ 01 ∣ 12 ] 17
++[ 02
+̃ ∣ 01 ∣ 1 ∣ 12 ] 25
++[ 02
+̃ ∣ 01 ∣ 12 ∣ 2 ] 09
++[ 2, 02
+̃ ∣ 01 ∣ ∣ 12 ] 21
++[ 12, 02
+̃ ∣ 01 ∣ ∣ 1 ] 26
+D0([ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ]) =
+[ 01
+̃,0, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 27
++[ 01
+̃,1, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 28
++[ 01
+̃,01, 02
+̃ ∣ 0 ∣ ∣ 01 ] 13
++[ 01
+̃,02, 02
+̃ ∣ 0 ∣ ∣ 01 ] 13
++[ 01
+̃,12, 02
+̃ ∣ 0 ∣ ∣ 01 ] 18
++[ 01
+̃,012, 02
+̃,0, 02
+̃ ∣ 0 ∣ ∣ 01 ] 29
++[ 01
+̃,012, 02
+̃,2, 02
+̃ ∣ 0 ∣ ∣ 01 ] 30
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 0 ] 19
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 1 ] 31
+D1([ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ]) =
+[ 01
+̃,0,01, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 27
++[ 01
+̃,01,1, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 28
++[ 01
+̃,0,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 27
++[ 01
+̃,01,12, 02
+̃ ∣ 0 ∣ ∣ 01 ] 18
++[ 01
+̃,012,2, 02
+̃ ∣ 0 ∣ ∣ 01 ] 30
++[ 01
+̃,012, 02
+̃,0,02, 02
+̃ ∣ 0 ∣ ∣ 01 ] 29
++[ 01
+̃,012, 02
+̃,02,2, 02
+̃ ∣ 0 ∣ ∣ 01 ] 30
++[ 01
+̃,012, 02
+̃,0 ∣ 0 ∣ ∣ 01 ] 29
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ 0 ∣ 01 ] 32
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ 0 ∣ 01 ] 32
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ 01 ∣ 1 ] 33
++[ 1, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 01 ] 28
++[ 01, 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 0 ] 19
+D0([ 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ]) =
+[ 01
+̃,0, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 34
++[ 01
+̃,1, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 35
++[ 01
+̃,01, 02
+̃ ∣ 01 ∣ ∣ 1 ] 23
++[ 01
+̃,02, 02
+̃ ∣ 01 ∣ ∣ 1 ] 23
++[ 01
+̃,12, 02
+̃ ∣ 01 ∣ ∣ 1 ] 26
++[ 01
+̃,012, 02
+̃,0, 02
+̃ ∣ 01 ∣ ∣ 1 ] 36
++[ 01
+̃,012, 02
+̃,2, 02
+̃ ∣ 01 ∣ ∣ 1 ] 37
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ ∣ 1 ] 31
++[ 01
+̃,012, 02
+̃ ∣ 1 ∣ ∣ 1 ] 38
+
+20
+K. POIRIER AND T. TRADLER
+D1([ 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ]) =
+[ 01
+̃,0,01, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 34
++[ 01
+̃,01,1, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 35
++[ 01
+̃,0,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 34
++[ 01
+̃,01,12, 02
+̃ ∣ 01 ∣ ∣ 1 ] 26
++[ 01
+̃,012,2, 02
+̃ ∣ 01 ∣ ∣ 1 ] 37
++[ 01
+̃,012, 02
+̃,0,02, 02
+̃ ∣ 01 ∣ ∣ 1 ] 36
++[ 01
+̃,012, 02
+̃,02,2, 02
+̃ ∣ 01 ∣ ∣ 1 ] 37
++[ 01
+̃,012, 02
+̃,0 ∣ 01 ∣ ∣ 1 ] 36
++[ 01
+̃,012, 02
+̃,01 ∣ 1 ∣ ∣ 1 ] 39
++[ 01
+̃,012, 02
+̃ ∣ 01 ∣ 1 ∣ 1 ] 40
++[ 01
+̃,012, 02
+̃ ∣ 0 ∣ 01 ∣ 1 ] 33
++[ 01
+̃,012, 02
+̃ ∣ 01 ∣ 1 ∣ 1 ] 40
++[ 1, 01
+̃,012, 02
+̃ ∣ 01 ∣ ∣ 1 ] 35
+D0([ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ]) =
+[ 02
+̃,0, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 41
++[ 02
+̃,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 42
++[ 02
+̃,01, 12
+̃ ∣ 1 ∣ ∣ 12 ] 24
++[ 02
+̃,02, 12
+̃ ∣ 1 ∣ ∣ 12 ] 22
++[ 02
+̃,12, 12
+̃ ∣ 1 ∣ ∣ 12 ] 22
++[ 02
+̃,012, 12
+̃,1, 12
+̃ ∣ 1 ∣ ∣ 12 ] 43
++[ 02
+̃,012, 12
+̃,2, 12
+̃ ∣ 1 ∣ ∣ 12 ] 44
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 2 ] 45
+D1([ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ]) =
+[ 02
+̃,0,02, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 41
++[ 02
+̃,02,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 42
++[ 02
+̃,0,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 41
++[ 02
+̃,01,12, 12
+̃ ∣ 1 ∣ ∣ 12 ] 24
++[ 02
+̃,012,2, 12
+̃ ∣ 1 ∣ ∣ 12 ] 44
++[ 02
+̃,012, 12
+̃,1,12, 12
+̃ ∣ 1 ∣ ∣ 12 ] 43
++[ 02
+̃,012, 12
+̃,12,2, 12
+̃ ∣ 1 ∣ ∣ 12 ] 44
++[ 02
+̃,012, 12
+̃,1 ∣ 1 ∣ ∣ 12 ] 43
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ 1 ∣ 12 ] 46
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ 1 ∣ 12 ] 46
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ 12 ∣ 2 ] 47
++[ 2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 12 ] 42
++[ 12, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 48
+D0([ 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ]) =
+[ 02
+̃,0, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 49
++[ 02
+̃,2, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 50
++[ 02
+̃,01, 12
+̃ ∣ 12 ∣ ∣ 2 ] 06
++[ 02
+̃,02, 12
+̃ ∣ 12 ∣ ∣ 2 ] 05
++[ 02
+̃,12, 12
+̃ ∣ 12 ∣ ∣ 2 ] 05
++[ 02
+̃,012, 12
+̃,1, 12
+̃ ∣ 12 ∣ ∣ 2 ] 51
++[ 02
+̃,012, 12
+̃,2, 12
+̃ ∣ 12 ∣ ∣ 2 ] 52
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 2 ] 45
++[ 02
+̃,012, 12
+̃ ∣ 2 ∣ ∣ 2 ] 07
+D1([ 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ]) =
+
+A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS
+21
+[ 02
+̃,0,02, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 49
++[ 02
+̃,02,2, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 50
++[ 02
+̃,0,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 49
++[ 02
+̃,01,12, 12
+̃ ∣ 12 ∣ ∣ 2 ] 06
++[ 02
+̃,012,2, 12
+̃ ∣ 12 ∣ ∣ 2 ] 52
++[ 02
+̃,012, 12
+̃,1,12, 12
+̃ ∣ 12 ∣ ∣ 2 ] 51
++[ 02
+̃,012, 12
+̃,12,2, 12
+̃ ∣ 12 ∣ ∣ 2 ] 52
++[ 02
+̃,012, 12
+̃,1 ∣ 12 ∣ ∣ 2 ] 51
++[ 02
+̃,012, 12
+̃,12 ∣ 2 ∣ ∣ 2 ] 07
++[ 02
+̃,012, 12
+̃ ∣ 12 ∣ 2 ∣ 2 ] 53
++[ 02
+̃,012, 12
+̃ ∣ 1 ∣ 12 ∣ 2 ] 47
++[ 02
+̃,012, 12
+̃ ∣ 12 ∣ 2 ∣ 2 ] 53
++[ 2, 02
+̃,012, 12
+̃ ∣ 12 ∣ ∣ 2 ] 50
+D0([ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ]) =
+[ 01
+̃,0, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 54
++[ 01
+̃,1, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 55
++[ 01
+̃,01, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 01
+̃,02, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 01
+̃,12, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 48
++[ 01
+̃,012, 02
+̃,0, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 56
++[ 01
+̃,012, 02
+̃,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 57
++[ 01
+̃,012, 02
+̃,01, 12
+̃ ∣ 1 ∣ ∣ 1 ] 39
++[ 01
+̃,012, 02
+̃,02, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 01
+̃,012, 02
+̃,12, 12
+̃ ∣ 1 ∣ ∣ 1 ] 38
++[ 01
+̃,012, 02
+̃,012, 12
+̃,1, 12
+̃ ∣ 1 ∣ ∣ 1 ] 58
++[ 01
+̃,012, 02
+̃,012, 12
+̃,2, 12
+̃ ∣ 1 ∣ ∣ 1 ] 59
+D1([ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ]) =
+[ 01
+̃,0,01, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 54
++[ 01
+̃,01,1, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 55
++[ 01
+̃,0,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 54
++[ 01
+̃,01,12, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 48
++[ 01
+̃,012,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 57
++[ 01
+̃,012, 02
+̃,0,02, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 56
++[ 01
+̃,012, 02
+̃,02,2, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 57
++[ 01
+̃,012, 02
+̃,0,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 56
++[ 01
+̃,012, 02
+̃,01,12, 12
+̃ ∣ 1 ∣ ∣ 1 ] 39
++[ 01
+̃,012, 02
+̃,012,2, 12
+̃ ∣ 1 ∣ ∣ 1 ] 59
++[ 01
+̃,012, 02
+̃,012, 12
+̃,1,12, 12
+̃ ∣ 1 ∣ ∣ 1 ] 58
++[ 01
+̃,012, 02
+̃,012, 12
+̃,12,2, 12
+̃ ∣ 1 ∣ ∣ 1 ] 59
++[ 01
+̃,012, 02
+̃,012, 12
+̃,1 ∣ 1 ∣ ∣ 1 ] 58
++[ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ 1 ∣ 1 ] 60
++[ 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ 1 ∣ 1 ] 60
++[ 1, 01
+̃,012, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] 55
+In the above sum, there are several terms which appear twice and which therefore cancel
+(over Z2). The terms that are labeled with the following numbers appear twice and cancel:
+03 , 08 , 09 , 10 , 15 , 16 , 17 , 25 , 31 , 32 , 33 , 40 , 45 , 46 , 47 , 53 , 60 .
+
+22
+K. POIRIER AND T. TRADLER
+Moreover, in some instances the sum of three terms cancel. The following labels appear
+three times and also cancel:
+01 , 02 , 06 , 07 , 11 , 12 , 18 , 19 , 20 , 21 , 24 , 26 , 27 , 28 , 29 , 30 , 34 , 35 ,
+36 , 37 , 39 , 41 , 42 , 43 , 44 , 48 , 49 , 50 , 51 , 52 , 54 , 55 , 56 , 57 , 58 , 59 .
+There are six terms labeled with
+38 , whose sum vanishes:
+38 =([ 01
+̃,012, 02
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,012, 02
+̃,02, 12
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,012, 02
+̃,12, 12
+̃ ∣ 1 ∣ ∣ 1 ])
++ ([ 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,01, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,02, 02
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ])
+=[ 01
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] + [ 01
+̃,012, 12
+̃ ∣ 1 ∣ ∣ 1 ] = 0
+The left-over terms that do not cancel are:
+04 = [ 02
+̃ ∣ 02 ∣ ∣ 2 ]
+05 = [ 02
+̃ ∣ 12 ∣ ∣ 2 ] + [ 02
+̃,02, 12
+̃ ∣ 12 ∣ ∣ 2 ] + [ 02
+̃,12, 12
+̃ ∣ 12 ∣ ∣ 2 ] = [ 12
+̃ ∣ 12 ∣ ∣ 2 ]
+13 = [ 02
+̃ ∣ 0 ∣ ∣ 01 ] + [ 01
+̃,01, 02
+̃ ∣ 0 ∣ ∣ 01 ] + [ 01
+̃,02, 02
+̃ ∣ 0 ∣ ∣ 01 ] = [ 01
+̃ ∣ 0 ∣ ∣ 01 ]
+14 = [ 02
+̃ ∣ 0 ∣ ∣ 02 ]
+22 = [ 02
+̃ ∣ 1 ∣ ∣ 12 ] + [ 02
+̃,02, 12
+̃ ∣ 1 ∣ ∣ 12 ] + [ 02
+̃,12, 12
+̃ ∣ 1 ∣ ∣ 12 ] = [ 12
+̃ ∣ 1 ∣ ∣ 12 ]
+23 = [ 02
+̃ ∣ 01 ∣ ∣ 1 ] + [ 01
+̃,01, 02
+̃ ∣ 01 ∣ ∣ 1 ] + [ 01
+̃,02, 02
+̃ ∣ 01 ∣ ∣ 1 ] = [ 01
+̃ ∣ 01 ∣ ∣ 1 ]
+These terms are precisely F[02], F[12], and F[01], so that DF[012] = D0F[012] + D1F[012] =
+F[01] + F[02] + F[12], which is (4.19).
+References
+[CS]
+Moira Chas, Dennis Sullivan. String Topology. preprint arXiv:math/9911159
+[L]
+Jean-Louis Loday. Cyclic Homology, Grundlehren der mathematischen WIssenschaften 301 (1998),
+Springer.
+[M]
+Luc Menichi. String Topology for Spheres, Comment. Math. Helv. 84 (2009), 135–157.
+[T1]
+Thomas Tradler. Infinity Inner Products on A-Infinity Algebras. J. Homotopy Relat. Struct. 3
+(2008), no. 1, p. 245–271.
+[T2]
+Thomas Tradler. The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products.
+Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, p. 2351–2379.
+[TZS]
+Thomas Tradler, Mahmoud Zeinalian, Dennis Sullivan Infinity structure of Poincar´e duality spaces.
+Algebr. Geom. Topol. 7 (2007), p. 233–260.
+Kate Poirier, Department of Mathematics, New York City College of Technology,
+City University of New York, 300 Jay Street, Brooklyn, NY 11201
+Email address: kpoirier@citytech.cuny.edu
+Thomas Tradler, Department of Mathematics, New York City College of Technology,
+City University of New York, 300 Jay Street, Brooklyn, NY 11201
+Email address: ttradler@citytech.cuny.edu
+
diff --git a/UNE5T4oBgHgl3EQfAw5l/content/tmp_files/load_file.txt b/UNE5T4oBgHgl3EQfAw5l/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1ab129136908e2f7cf441ec4080c1c2e25753559
--- /dev/null
+++ b/UNE5T4oBgHgl3EQfAw5l/content/tmp_files/load_file.txt
@@ -0,0 +1,1740 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf,len=1739
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='05381v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='AT]  13 Jan 2023  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH  Z2 COEFFICIENTS  KATE POIRIER AND THOMAS TRADLER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Luc Menichi showed that the BV algebras on H●(LS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)[−2] coming from  string topology and the one on HH●(H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2),H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)) using Poincar´e duality on  H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  In this note we show how one can obtain the string  topology BV algebra on Hochschild cohomology using a Poincar´e duality structure with  higher homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  This Poincar´e duality (with higher homotopies) on cohomology is  induced by a local Poincar´e duality (with higher homotopies) on the cochain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Main Statements  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' BV algebras for the 2-sphere with Z2 coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In [M, Theorem 24] Luc  Menichi calculated the string topology BV algebra (defined by Moira Chas and Dennis  Sullivan in [CS]) for the 2-sphere M = S2 with Z2 = Z/2Z coefficients to be  H●(LS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) ≅ Λa ⊗ Z2[u] ≅ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='αk ⊕ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='βk,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2)  where αk ⋅ αℓ = αk+ℓ,βk ⋅ βℓ = 0, and αk ⋅ βℓ = βℓ ⋅ αk = βk+ℓ,  and ∆ST(αk) = 0 and ∆ST(βk) = k ⋅ αk−1 + k ⋅ βk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Here, the degrees are given by ∣a∣ = −2, ∣u∣ = 1, and thus, setting αk = 1⊗uk and βk = a⊗uk,  we have ∣αk∣ = k, ∣βk∣ = k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Moreover, in [M, Proposition 20], Menichi calculated the BV algebra for Hochschild  cohomology of the cohomology of S2 with Z2 coefficients to be  HH●(H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)) ≅ Λg ⊗ Z2[f] ≅ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='φk ⊕ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ψk,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3)  where φk ⋅ φℓ = φk+ℓ,ψk ⋅ ψℓ = 0, and φk ⋅ ψℓ = ψℓ ⋅ φk = ψk+ℓ,  and ∆(φk) = 0 and ∆(ψk) = k ⋅ φk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  There degrees are similarly given by ∣g∣ = −2, ∣f∣ = 1, and so, setting φk = 1 ⊗ f k and  ψk = g ⊗ f k, we have ∣φk∣ = k, ∣ψk∣ = k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Note, that these BV algebras differ in their ∆ operators, and, in fact, Menichi obtained  the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' [M, Corollary 30] There is no isomorphism of BV algebras between  (H●(LS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2),⋅,∆ST) and (HH●(H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)),⋅,∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  It is worth noting though, that the induced Gerstenhaber algebras on H●(LS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) and  HH●(H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)) are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' [M, Corollary 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 55P50, 16E40 (primary), 57P10, 08A65 (secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' String topology, Hochschild cohomology, BV algebra, Poincar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  1  2  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The BV algebra coming from local Poincar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' A crucial ingredient for de-  termining the ∆ operator on Hochschild cohomology (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) comes from a choice of Poincar´e  duality structure given as a bimodule isomorphism F ∶ H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)  ≅  �→ H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' For the  (Z2-)cohomology H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) and homology H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2), an obvious choice is to define the  degree +2 isomorphism F as follows:  degree −2  degree −1  degree 0  degree 1  degree 2  H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)  = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Z2  0  Z2  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)  = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  0  0  Z2  0  Z2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  F  F  F  Indeed, this choice of F, which is the map given by capping with the Z2-fundamental class  of S2, leads to the BV algebra in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) as we will confirm in theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6 (together with  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Note that on the cochain level, capping with a fundamental cycle is not a bimodule  map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' see observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Using a generalization of bimodule maps, which, in addition  to F, allows for higher homotopies, we show in this paper that one can obtain the string  topology BV algebra (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2) on Hochschild cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The precise definition of bimodule  maps with higher homotopies was studied in [T1] and will be reviewed in section 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Applying the concept of bimodule maps with higher homotopies, we will construct a spe-  cific example of such a map for the case of a local cochain model of the 2-sphere C●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)  in example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then we pull this map back to obtain a bimodule map with homotopies  ̃F for cohomology H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) (see proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The resulting map ̃F, recorded in  example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2, has precisely one higher homotopy, when compared to F (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Now,  using this ̃F, we obtain the following main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The BV algebra on HH●(H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2)) induced by ̃F (coming  from a local bimodule map with higher homotopies transferred to cohomology) is isomorphic  to the string topology BV algebra (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2) on H●(LS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We compute both BV algebras coming from F and ̃F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Denote by  H● ∶= H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) and H● ∶= H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then the bimodule map F and the bimodule map  ̃F (with higher homotopies) induce respective (graded module) isomorphisms F and ̃  F of  Hochschild cohomologies (see 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7)  F, ̃  F ∶ HH●(H●,H●)  ≅  �→ HH●(H●,H●).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Next, we use the explicit description (as reviewed in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2) stating that  HH●(H●,H●) ≅ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='φk ⊕ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ψk,  where ∣φk∣ = k, ∣ψk∣ = k − 2,  HH●(H●,H●) ≅ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='θk ⊕ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='χk,  where ∣θk∣ = k + 2, ∣χk∣ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  With this notation, F induces in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7) the map F(φk) = θk,F(ψk) = χk (see 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3), while ̃F  induces the map ̃  F(φk) = θk + χk+2, ̃  F(ψk) = χk (computed in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  3  The BV ∆ operator on HH●(H●,H●) is given by transferring Connes’ B-operator from  HH●(H●,H●) to HH●(H●,H●) via the given Poincar´e duality bimodule isomorphism,  where, Connes’ B-operator is the map (computed in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2):  B ∶ HH●(H●,H●) → HH●(H●,H●),  B(θk) = 0,B(χk) = k ⋅ θk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Thus the induced operators ∆ = F −1○B○F and ̃∆ = ̃  F −1○B○ ̃  F from F and ̃F, respectively,  are:  ∆(φk) = 0,  ∆(ψk) = F −1(k ⋅ θk−1) = k ⋅ φk−1,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8)  ̃∆(φk−2) = ̃∆(ψk) = ̃  F −1(k ⋅ θk−1) = k ⋅ (φk−1 + ψk+1),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='9)  where we used that ̃  F −1 is given by ̃  F −1(θk) = φk + ψk+2 and ̃  F −1(χk) = ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Note that the ∆ operator in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8) coincides with ∆ from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' To see that the BV algebra  with ̃∆ is isomorphic to the string topology BV algebra (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2), we use an isomorphism which  was essentially described in [M, last paragraph of Lemma 21]: the map Θ ∶ HH●(H●,H●) →  H●(LS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) with Θ(φk) = αk +k ⋅βk+2 and Θ(ψk) = βk is an algebra isomorphism such that  ∆ST ○ Θ = Θ ○ ̃∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In section 2, we review basics about Hochschild  cochain complexes and morphisms between them coming from bimodule maps up to higher  homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In section 3 we review the induced BV algebra coming from a chosen Poincar´e  duality structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In section 4 we give various computations of bimodule maps (with and  without higher homotopies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In particular, the construction of a bimodule map (up to  certain controlled structures) for the 2-simplex coming from locality (see example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='9) con-  stitutes the main computational aspect of this paper, with its proof being spelled out in  appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In section 5 we compute the BV algebras coming from the Poincar´e duality  structures on cohomology of the 2-sphere from section 4, and with this complete the proof  of theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We would like to thank Mahmoud Zeinalian for discussions about  this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The second author was partially supported by a PSC-CUNY research award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Bimodule maps with higher homotopies  In this section we review bimodule maps, bimodule maps up to higher homotopies,  and homotopy inner products;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' see also [T1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Moreover we review the induced maps on  Hochschild cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Basic setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Denote by R a commutative ring with unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' our main example of interest  is R = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A be a unital dg-algebra over R, where we note that all the differentials in  this paper will always go down, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', d ∶ Aj → Aj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We denote by A the space A shifted up  by one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', Aj ∶= Aj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Let M be a dg-bimodule over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' If we want to emphasize the corresponding algebra A,  then we will also write M/A instead of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' From now on, all modules as well as module  maps will be written in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Note that if M is a dg-bimodule over A, then so is its dual space M∗ with M∗  j ∶= (M−j)∗ =  HomR(M−j,R) with differential d ∶ M∗  j → M∗  j−1,d(n)(m) ∶= (−1)∣n∣+1n(d(m)), and module  maps (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a)(m) ∶= n(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m) and (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='n)(m) ∶= (−1)∣a∣⋅(∣n∣+∣m∣)n(m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a) for n ∈ M∗,m ∈ M,a ∈ A,  4  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  where ∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='∣ denotes the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Moreover, the dg-algebra A is itself a dg-module A ∶= A over  A (with module structure given by the algebra multiplication), and thus A∗ is a dg-module  over A as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Define the Hochschild cochain complex of A with values in M to be given by CH●(A,M) ∶=  ∏r≥0 Hom(A⊗r,M), where the differential D = D0 + D1, with D2 = 0, is defined for ϕ ∈  Hom(A⊗r,M) by setting  D0(ϕ)(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar) ∶= d(ϕ(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar)) +  r  ∑  j=1  (−1)∣ϕ∣+∣a1∣+⋅⋅⋅+∣aj−1∣ ⋅ ϕ(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,daj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar),  D1(ϕ)(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar+1) ∶= (−1)∣ϕ∣⋅∣a1∣ ⋅ a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ϕ(a2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar+1)  +  r  ∑  j=1  (−1)∣ϕ∣+∣a1∣+⋅⋅⋅+∣aj∣ ⋅ ϕ(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj ⋅ aj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar+1) + (−1)∣ϕ∣+1+∣a1∣+⋅⋅⋅+∣ar∣ ⋅ ϕ(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ar+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We will mainly use the normalized Hochschild cochain complex CH  (A,M), which is the  subcomplex of CH●(A,M) consisting of those ϕ ∈ CH●(A,M) which vanish when any of the  inputs is the unit 1 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The inclusion CH  (A,M) ↪ CH●(A,M) is a quasi-isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  see [L, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Inner products and homotopy inner products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let M and N be dg-bimodules  over A, and let F ∶ M → N be a dg-bimodule map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then, there is an induced cochain map  CH(F) ∶ CH  (A,M) → CH  (A,N), ϕ ↦ F ○ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' An inner product on M is a dg-bimodule  map F ∶ M → M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We will need a more general notion of inner product F ∶ M → M∗ which also allows  for higher homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' To this end, consider a sequence of maps F = {Fp,q ∶ A⊗p ⊗ M ⊗  A⊗q → M∗}p,q≥0, and define its differential by setting DF = {(DF)p,q}p,q≥0 to be (DF)p,q =  (D0F)p,q + (D1F)p,q, where ∀a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap,b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq,∈ A and m,n ∈ M:  (D0F)p,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3)  ∶=  p  ∑  j=1  (−1)∣F∣+∣a1∣+⋅⋅⋅+∣aj−1∣ ⋅ Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,daj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  + (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+1 ⋅ Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='dm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  +  q  ∑  j=1  (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bj−1∣ ⋅ Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,dbj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  + (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq∣+1 ⋅ Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(dn)  and, using the notation F−1,q = Fp,−1 = 0:  (D1F)p,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4)  ∶= (−1)∣F∣+∣a1∣⋅(∣a2∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq∣+∣n∣) ⋅ Fp−1,q(a2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a1)  +  p−1  ∑  j=1  (−1)∣F∣+∣a1∣+⋅⋅⋅+∣aj∣ ⋅ Fp−1,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj ⋅ aj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  + (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap−1∣+1 ⋅ Fp−1,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  5  + (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣ ⋅ Fp,q−1(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  +  q−1  ∑  j=1  (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bj∣ ⋅ Fp,q−1(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bj ⋅ bj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  + (−1)∣F∣+∣a1∣+⋅⋅⋅+∣ap∣+∣m∣+∣b1∣+⋅⋅⋅+∣bq−1∣+1 ⋅ Fp,q−1(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq−1)(bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='n)  Then we call F a homotopy inner product for M if DF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' By slight abuse of notation,  we will still use the notation F ∶ M → M∗ for F = {Fp,q}p,q with all its homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We will sometimes depict evaluations of F as follows:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5)  Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n)  a1  a2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  ap  m  b1  b2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  bq  n  Note, that a homotopy inner product F = {Fp,q}p,q with Fp,q = 0 for p + q > 0 is precisely  an inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', a dg-bimodule map F0,0 ∶ M → M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Let F be a homotopy inner product for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We will always assume that F vanishes  when any of the algebra inputs is the unit 1 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then, there is an induced map F ∶=  CH(F) ∶ CH  (A,M) → CH  (A,M∗) by setting F = CH(F) = ∑p,q≥0 CH(F)p,q, where  CH(F)p,q ∶ Hom(A⊗r,M) → Hom(A⊗p+r+q,M∗) is  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6)  (CH(F)p,q(ϕ))(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap+r+q)  ∶= (−1)∣ϕ∣⋅(∣a1∣+⋅⋅⋅+∣ap∣) ⋅ Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ϕ(ap+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap+r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ap+r+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap+r+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  a1  a2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  ap  ap+r+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  ap+r+q  ϕ  ap+1  ap+r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  A direct but lengthy calculation shows that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7)  CH(DF)(ϕ) = D ○ CH(F)(ϕ) − (−1)∣F∣ ⋅ CH(F) ○ D(ϕ),  ∀ϕ ∈ CH●(A,M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' If F is a homotopy inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', DF = 0, then F = CH(F) ∶  CH  (A,M) → CH  (A,M∗) is a cochain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' By abuse of notation, we often write F  for the induced map on Hochschild cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' If F = DF′ for some F′, then the induced  map on Hochschild cohomology vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', HH(F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Pullback under a dg-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A and B be two unital dg-algebras, and  let f ∶ B → A be a dg-algebra map (which is necessarily of degree 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Then any dg-  bimodule M = M/A over A induces a dg-bimodule M/B ∶= M over B via the module structure  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m ∶= f(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b ∶= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='f(b) for any m ∈ M,b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Moreover, any dg-bimodule map  F/A ∶ M/A → N/A also induces a dg-bimodule map F/B ∶ M/B → N/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Moreover, a homotopy  inner product F/A for M/A also induces a homotopy inner product for M/B via  (F/B)p,q(b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='c1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,cq)(n) ∶= (F/A)p,q(f(b1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,f(bp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='f(c1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,f(cq))(n)  6  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  for all b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bp,c1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,cq,∈ B and m,n ∈ M/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Moreover, for a dg-algebra map f ∶ B → A, we have the dg-modules B/B and B∗  /B, and,  from A/A and A∗  /A, we also get A/B and A∗  /B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The dg-algebra map f then induces a dg-  bimodule map f/B ∶ B/B → A/B and, by dualizing, the dg-bimodule map f∗  /B ∶ A∗  /B → B∗  /B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Combining the last two paragraphs, assume that f ∶ B → A is a dg-algebra map, and that  F/A ∶ A/A → A∗  /A is a homotopy inner product for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then, we get a transferred homotopy  inner product f(F)/B ∶ B/B → B∗  /B for B, given by f(F)/B ∶= f∗  /B ○ F/B ○ f/B,  B/B  f/B  �  f(F)/B  �  A/B  F/B  �  B∗  /B  A∗  /B  f∗  /B  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='10)  (f(F)/B)p,q(b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='c1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,cq)(n)  = (F/A)p,q(f(b1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,f(bp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='f(m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='f(c1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,f(cq))(f(n)),  where b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bp,m,c1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,cq,n ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Finally, for a dg-algebra map f ∶ B → A, and a dg-bimodule M/A, there is an induced  cochain map on Hochschild cochains, CH(f,M) ∶ CH  (A,M/A) → CH  (B,M/B),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='11)  CH(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='M)(ϕ)(b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,br) ∶= ϕ(f(b1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,f(br)),  ∀ϕ ∈ CH  (A,M/A),b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,br ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  With this we get the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' If f ∶ B → A is a dg-algebra map, and F/A ∶ A/A → A∗  /A is a homotopy inner  product for A/A, then the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='13)  CH  (B,B/B)  CH(f/B)  �  CH(f(F)/B)  �  CH  (B,A/B)  CH(F/B)  �  CH  (A,A/A)  CH(f,A)  �  CH(F/A)  �  CH  (B,B∗  /B)  CH  (B,A∗  /B)  CH(f∗  /B)  �  CH  (A,A∗  /A)  CH(f,A∗)  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Isomorphism on Hochschild cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' A particular case of interest is when  a homotopy inner product F ∶ A → A∗ induces an isomorphism on Hochschild cohomology  F ∶= HH(F) ∶ HH●(A,A) → HH●(A,A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In this case, we can transfer any structure  between these Hochschild cohomologies, in particular we can transfer the B operator as it  was done in theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Note that in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='13), if the four horizontal maps are quasi-isomorphisms, then,  obviously, the right vertical map CH(F/A) is a quasi-isomorphism iff the left vertical map  CH(f(F)/B) is a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' BV algebra on Hochschild cohomology  We now review how homotopy inner products induce a BV algebra on Hochschild co-  homology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' see theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Our main reference for this is [T2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We start by defining two  operators B and ∆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  7  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Consider a unital dg-algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (1) We define Connes’ B-operator (or more precisely the dual of Connes’ B-operator)  to be B ∶ CH  (A,A∗) → CH  (A,A∗) given for ϕ ∈ Hom(A⊗r,A∗) by B(ϕ) ∈  Hom(A⊗r−1,A∗) with  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2)  (B(ϕ))(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar−1)(ar)  ∶=  r  ∑  j=1  (−1)(∣aj∣+⋅⋅⋅+∣ar∣)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+∣ar∣ ⋅ ϕ(aj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar,a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj−1)(1),  A direct but lengthy computation shows that D○B(ϕ) = −B○D(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' By abuse of no-  tation, we denote the induced map on Hochschild cohomology by B ∶ HH●(A,A∗) →  HH●(A,A∗) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (2) Next, assume that we also have a homotopy inner product F = {Fp,q ∶ A⊗p ⊗ A ⊗  A⊗q → A∗}p,q≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Then, we define the operator ZF ∶=  ∑  p,q≥0ZF  p,q ∶ CH  (A,A) →  CH  (A,A∗), where, for ϕ ∈ Hom(A⊗r,A), we set ZF  p,q(ϕ) ∈ Hom(A⊗p+r+q−1,A∗) to  be given by  (ZF  p,q(ϕ))(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap+r+q−1)(ap+r+q)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3)  ∶=  p+q  ∑  j=p+1  (−1)∣F∣+(∣ϕ∣+1)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+1  ⋅ Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ap+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj−1,ϕ(aj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj+r−1),aj+r,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap+r+q−1)(ap+r+q)  +  r  ∑  j=1  (−1)(∣aj∣+⋅⋅⋅+∣ap+r+q∣)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+∣ap+r+q∣+∣ϕ∣⋅(∣aj∣+⋅⋅⋅+∣aj+p+q−1∣)  ⋅ Fp,q(aj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj+p−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='aj+p,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj+p+q−1)(ϕ(ap+j+q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap+r+q,a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj−1))  +  p  ∑  j=1  (−1)∣F∣+(∣ϕ∣+1)⋅(∣a1∣+⋅⋅⋅+∣aj−1∣)+1  ⋅ Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj−1,ϕ(aj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,aj+r−1),aj+r,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap+r−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ap+r,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap+r+q−1)(ap+r+q)  One can check again that D ○ ZF(ϕ) = −(−1)∣F∣ ⋅ ZF ○ D(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' By abuse of notation,  we denote the induced map on Hochschild cohomology by ZF ∶ HH●(A,A) →  HH●(A,A∗) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We remark that ZF has appeared in [T2, lemma 17] as the operation associated  to the symbol  (−1)µ ⋅ 1  1  1  +1  +1  1  (3) Let F be a homotopy inner product for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Assume, moreover, that F ∶ A → A∗  induces an isomorphism on Hochschild cohomology, F ∶= HH(F) ∶ HH●(A,A) →  8  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  HH●(A,A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then denote by ∆F ∶ HH●(A,A) → HH●(A,A) the composition  ∆F ∶= F −1 ○ ZF,  HH●(A,A)  ZF  �→ HH●(A,A∗)  F−1  �→ HH●(A,A)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A be a unital dg-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (1) On Hochschild cochains HH●(A,A∗), we have that B2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (2) Assume that F ∶ A → A∗ is a homotopy inner product which induces an isomorphism  on Hochschild cohomology F ∶ HH●(A,A) → HH●(A,A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then the deviation of  ∆F from being a derivation of the cup product is the usual Gerstenhaber bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Here the cup product on Hochschild cohomology is given for Hochschild cochains  ϕ ∈ Hom(A⊗r,A) and ρ ∈ Hom(A⊗s,A) to be ϕ ⌣ ρ ∈ Hom(A⊗r+s,A) with  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5)  (ϕ ⌣ ρ)(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar+s) ∶= (−1)∣ρ∣⋅(∣a1∣+⋅⋅⋅+∣ar∣) ⋅ ϕ(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar) ⋅ ρ(ar+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ar+s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' For (1), note that B2 = 0 follows since normalized Hochschild cochains vanish when  any input is the unit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Part (2) was proved in [T2, section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  □  Since lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4 (1) and (2) are conditions needed for a BV algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', a square zero  operator (here: B) whose deviation from being a derivation is a Gerstenhaber bracket (here:  ∆F), we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A be a dg-algebra, and let F ∶ A → A∗ be a homotopy inner product  for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We call F a Poincar´e duality structure for A, if it satisfies if it satisfies the following  two conditions:  (1) F induces an isomorphism of graded modules on Hochschild cohomology F =  HH(F) ∶ HH●(A,A)  ≅  �→ HH●(A,A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (2) Transferring B from HH●(A,A∗) to HH●(A,A) via F equals ∆F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=',  HH●(A,A)  F  �  HH●(A,A∗)  F−1  �  B  �  F −1 ○ B ○ F = ∆F = F −1 ○ ZF  HH●(A,A)  ∆F  �  =  HH●(A,A)  ZF  �  HH●(A,A∗)  F−1  �  With this, we obtain the following theorem, which is [T2, theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A be a dg-algebra, and let F ∶ A → A∗ be a Poincar´e duality structure  for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then, HH●(A,A) together with the cup product and ∆F is a BV algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We now give a few examples for how one can check whether a homotopy  inner product is a Poincar´e duality structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (1) Let F be an inner product for a dg-algebra A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', the only non-zero component is  F0,0), and assume that F induces an isomorphism on Hochschild cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then  F is a Poincar´e duality structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (To see this, we note that this is a special case of the next example (2) below,  since the condition (D1F)0,1 = 0 gives (using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4)) that F(mb)(n) = F(m)(bn),  and (D1F)1,0 = 0 gives F(am)(n) = (−1)∣a∣⋅(∣m∣+∣n∣)F(m)(na) for all m,n,a,b ∈  A, and, thus F(m)(n) = F(1 ⋅ m)(n) = F(1)(m ⋅ n) = (−1)∣m∣⋅∣n∣F(n ⋅ 1)(m) =  (−1)∣m∣⋅∣n∣F(n)(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=')  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  9  (2) Let F be a homotopy inner product for a dg-algebra A, and assume that F in-  duces an isomorphism on Hochschild cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Assume further that F is in-  variant under cyclic rotation of the first p + 1 and last q + 1 inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', ∀p,q ≥ 0,  ∀a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap,m,b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq,n ∈ A:  Fp,q(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq)(n) = (−1)ε ⋅ Fq,p(b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,bq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,ap)(m)  a1  a2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  ap  m  b1  b2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  bq  n = (−1)ε ⋅  b1  b2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  bq  n  a1  a2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  ap  m  where (−1)ε = (−1)(∣a1∣+⋅⋅⋅+∣ap∣+∣m∣)⋅(∣b1∣+⋅⋅⋅+∣bq∣+∣n∣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Then F is a Poincar´e duality struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' This was proved in [T2, lemma 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (3) In the examples in section 5 of this paper we will check if a given homotopy inner  product F is a Poincar´e duality structure or not by explicitly computing F, B and  ZF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' the examples given in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  In particular, it is worth noting that there exist homotopy inner products F for  which F is an isomorphism, but for which F −1 ○ B ○ F ≠ ∆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We provide an explicit  example for such a homotopy inner product in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Computations of bimodule maps with higher homotopies  In this section, we will give explicit bimodule maps and bimodule maps with higher  homotopies for specific dg-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In particular we will compute the homotopy inner  product on H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) coming from a pullback of a local homotopy inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In the  remainder of this paper –with the exception of observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4– the ground ring will always  be R = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 (H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) inner product without homotopies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Consider the dg-algebra  A ∶= H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) ≅ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s with zero differential and degrees ∣e∣ = 0 and ∣s∣ = −2, and  where e is the unit and s ⋅ s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' For the dg module and dual dg module of A we use  the notation A ≅ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s and A∗ ≅ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e∗ ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s∗, respectively, where e∗ and s∗ are  the duals of e and s with ∣e∗∣ = 0 and ∣s∗∣ = 2 and the module structure is given by  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e∗ = e∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e = e∗,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s∗ = s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e = s∗,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e∗ = e∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s = 0,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s∗ = s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s = e∗ (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Define the dg bimodule map F ∶ A → A∗ by F(s) = e∗ and F(e) = s∗,  A  = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  ⟨s⟩  0  ⟨e⟩  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  A∗  = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  0  0  ⟨e∗⟩  0  ⟨s∗⟩  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  F  F  F  Thus, F ∶ A → A∗ is the map F(s)(e) = 1,F(s)(s) = 0,F(e)(e) = 0,F(e)(s) = 1 (in other  words, F is given by capping with s∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' One can check directly that F is a dg bimodule  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Note, that when interpreting F as an inner product <,>= F ∶ A ⊗ A → Z2, the only  non-vanishing inner products are < s,e >= 1 and < e,s >= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  10  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2 (H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) inner product with homotopies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We next consider the same  algebra A and module structures A and A∗ as in example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1, but we define an inner  product ̃F for A with higher homotopies (in the sense of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In fact, ̃F has its only  non-zero components given by ̃F0,0 ∶ A → A∗ and ̃F2,0 ∶ A ⊗ A ⊗ A → A∗ (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2) via  ̃F0,0(s)(e) = 1,  ̃F0,0(e)(s) = 1,  ̃F2,0(s,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e)(e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3)  s  e  e  s  s  s  e  e  It is again a direct check that ̃F is a homotopy inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', it satisfies all equation  required by 0 = DF = D1F given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4) (since d and thus D0 vanishes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Explicitly, these  equations are:  ̃F0,0(a ⋅ a1)(̃a) =̃F0,0(a)(a1 ⋅ ̃a),  ̃F0,0(a1 ⋅ a)(̃a) =̃F0,0(a)(̃a ⋅ a1),  ̃F2,0(a1,a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a ⋅ a3)(̃a) =̃F2,0(a1,a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a)(a3 ⋅ ̃a),  0 =̃F2,0(a1,a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a3 ⋅ a)(̃a) + ̃F2,0(a1,a2 ⋅ a3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a)(̃a)  + ̃F2,0(a1 ⋅ a2,a3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a)(̃a) + ̃F2,0(a2,a3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a)(̃a ⋅ a1)  for all a,̃a ∈ A, and a1,a2,a3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We next give formulas for calculating homotopy inner products (over Z2) for (triangu-  lated) 2-dimensional spaces on the cochain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  The following observation notes that  higher homotopies naturally appear for inner products on the cochain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A be a unital dg-algebra over any commutative ring R, so that  both A ∶= A and A∗ are dg-modules over A, (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Now, let x ∈ A∗ be a fixed closed  element, d(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Define F ∶ A → A∗ for a ∈ A by setting F(a) ∈ A∗ evaluated on some  ̃a ∈ A to be F(a)(̃a) ∶= (x ⌢ a)(̃a) = x(a ⋅ ̃a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Claim: F is chain map, and a graded right module map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' F is in general not a graded  left module map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' A sufficient condition for F being a graded left module map is that A is  graded commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' First, F is chain map, since for a,̃a ∈ A and dx = 0, we have F(da)(̃a) = x(da ⋅ ̃a) =  x(d(a ⋅ ̃a) − (−1)∣a∣a ⋅ d̃a) = (−1)∣x∣+1dx(a ⋅ ̃a) − (−1)∣a∣ ⋅ F(a)(d̃a) = −(−1)∣a∣ ⋅ (−1)∣F(a)∣+1 ⋅  d(F(a))(̃a) = (−1)∣F∣⋅d(F(a))(̃a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Next, F is a graded right module map, since for a,̃a ∈ A,  a1 ∈ A, we have (F(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a1)(̃a) = F(a)(a1 ⋅ ̃a) = x(a ⋅ a1 ⋅ ̃a) = F(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a1)(̃a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' To check when  F is a graded left module map, compute F(a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a)(̃a) = x(a1 ⋅ a ⋅ ̃a) and (a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='F(a))(̃a) =  (−1)∣a1∣⋅(∣F(a)∣+∣̃a∣) ⋅ F(a)(̃a ⋅ a1) = (−1)∣a1∣⋅(∣F∣+∣a∣+∣̃a∣) ⋅ x(a ⋅ ̃a ⋅ a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Thus, F is in general not a  graded left module map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Moreover, if a1 ⋅ a ⋅ ̃a = (−1)∣a1∣⋅(∣a∣+∣̃a∣)a ⋅ ̃a ⋅ a1 for all a,̃a,a1, then  (a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='F(a))(̃a) = (−1)∣a1∣⋅∣F∣ ⋅ F(a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='a)(̃a), and thus F will be a graded left module map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  □  In particular, assume that X is a closed, oriented manifold, and R is a commutative  ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In order to calculate the BV algebra on HH●(C●(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='(C●(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='R))∗) one might try  to use capping a cochain with a fundamental cycle x of X as the appropriate dg-bimodule  map F = x ⌢ − ∶ C●(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='R) → C●(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='R)  incl  ↪ (C●(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='R))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' However, the above observation  shows that capping on the (co-)chain level is in general not a dg-bimodule map, and thus  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  11  does not induce a chain map on the Hochschild cochains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' One way to resolve this issue  is to provide higher homotopies for the left module structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', to provide a homotopy  inner product, which then does give a corresponding chain map on Hochschild cochains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  This is what is done in this paper for S2 with Z2 coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5 (The 0-simplex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A[0] ∶= Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e0, where e0 is the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' A homotopy inner  product F[0] ∶ A[0] → (A[0])∗ is given by F[0](e0)(e0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We will also vary the superscript in an obvious way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', the dg-algebra A[1] ∶= Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e1 has  the homotopy inner product F[1] ∶ A[1] → (A[1])∗,F[1](e1)(e1) = 1, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6 (The 1-simplex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A[01] ∶= Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e0 ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e1 ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b01, where ∣e0∣ = ∣e1∣ = 0 and  ∣b01∣ = −1 with differential d(e0) = d(e1) = b01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The product is the usual cup product of  simplicial cochains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', e0 ⋅ e0 = e0, e1 ⋅ e1 = e1, and e0 ⋅ b01 = b01 ⋅ e1 = b01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Note that the unit  of A[01] is e0 + e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We want to define an inner product F with lowest component non-vanishing only for  F(e0)(b01) = F(b01)(e1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Note that F is not a chain map, but F(d(a))(̃a)+F(a)(d(̃a)) =  F[0](a)(̃a) + F[1](a)(̃a) for all a,̃a ∈ A[01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Moreover, F is a graded right module map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=',  F(a ⋅ a1)(̃a) = F(a)(a1 ⋅ ̃a) for all a,̃a,a1 ∈ A[01], but F is not a graded left module map,  since F(e0 ⋅ e0)(b01) = 1 ≠ 0 = F(e0)(b01 ⋅ e0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  There is an inductive procedure (involving choices at each stage) for obtaining higher  homotopies that provide F with left modules homotopies, making it into a homotopy inner  product up to F[0] and F[1] from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5 (interpreted as maps A[01] → (A[01])∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' This  procedure was described in [TZS, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Performing the induction leads to a  sequence of maps, which was first stated in [TZS, Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2]:  ∀k ≥ 0 ∶  F[01]  k,0 (b01,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e0)(b01) = 1,  F[01]  k,0 (b01,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b01)(e1) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7)  b01  b01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b01  e0  b01  b01  b01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b01  b01  e1  All other inner products are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Note, this resolves the above problem of F[01]  0,0 not being a  graded left module map, since, now, F[01]  0,0 is a graded left module map up to the homotopy  F[01]  1,0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' for example:  (DF[01])1,0(e0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e0)(b01) = F[01]  1,0 (d(e0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e0)(b01) + F[01]  1,0 (e0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='d(e0))(b01)  + F[01]  0,0 (e0 ⋅ e0)(b01) + F[01]  0,0 (e0)(b01 ⋅ e0) = 1 + 0 + 1 + 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  From [TZS, Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2], which we will also prove in appendix A, we have that:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8)  DF[01] = F[0] + F[1]  Again, we will need to vary the superscript in an obvious way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', for the dg-algebra  A[12] there are maps F[12] which are given by replacing 0 and 1 in the above with 1 and 2,  respectively, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='9 (The 2-simplex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We now describe inner product maps for the 2-simplex,  which extends the previous two examples of the 0- and 1-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A[012] ∶= Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e0 ⊕  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e1 ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e2 ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b01 ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b02 ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b12 ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='c012 with ∣ej∣ = 0, ∣bij∣ = −1 and ∣c012∣ = −2, and  12  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  differential d(e0) = b01 + b02, d(e1) = b01 + b12, d(e2) = b02 + b12 and d(bij) = c012 for all  0 ≤ i < j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The multiplication is non-zero only for  ∀j ∶  ej ⋅ ej = ej,  ∀i < j ∶  ei ⋅ bij = bij,  bij ⋅ ej = bij,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='10)  e0 ⋅ c012 = c012,  c012 ⋅ e2 = c012,  b01 ⋅ b12 = c012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Now, following the procedure (which uses locality) from [TZS, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2], we  define maps F[012]  k,0  whose only non-zero maps are given by the following equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='11)-  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='18):  ∀k ≥ 0 ∶  F[012]  k,0 (b02,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='c012)(e2) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='11)  F[012]  k,0 (b02,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e0)(c012) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='12)  F[012]  k,0 (b02,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b01)(b12) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='13)  b02  b02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b02  e2  c012  b02  b02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b02  c012  e0  b02  b02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b02  b12  b01  ∀k ≥ 0 ∶ ∀1 ≤ ℓ ≤ k + 1 ∶  F[012]  k+1,0(b01,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b01, c012  �  ℓth  ,b02,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e0)(b01) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='14)  F[012]  k+1,0(b01,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b01, c012  �  ℓth  ,b02,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b01)(e1) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='15)  F[012]  k+1,0(b02,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b02, c012  �  ℓth  ,b12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e1)(b12) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='16)  F[012]  k+1,0(b02,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b02, c012  �  ℓth  ,b12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b12)(e2) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='17)  b01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b01  c012  b02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b02  b01  e0  b01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b01  c012  b02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b02  e1  b01  b02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b02  c012  b12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b12  b12  e1  b02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b02  c012  b12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b12  e2  b12  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='18)  ∀k ≥ 0 ∶ ∀1 ≤ ℓ1 < ℓ2 ≤ k + 2 ∶  F[012]  k+2,0(b01,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b01, c012  �  ℓ1th  ,b02,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b02, c012  �  ℓ2th  ,b12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,b12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e1)(e1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b01  c012  b02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  b02  c012  b12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b12  e1  e1  We claim that the following equation holds, which will be proved in appendix A:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19)  DF[012] = F[01] + F[02] + F[12]  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  13  As before, the above will also be applied to obvious variations of the superscript, such  as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='g, A[123] with maps F[123] given by replacing 0, 1 and 2 in the above with 1, 2 and 3,  respectively, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='20 (C●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) inner product with homotopies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We now use a tetrahedral  triangulation of the 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' More precisely, we set  A = ⊕  0≤j≤3  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ej ⊕  ⊕  0≤i<j≤3  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='bij ⊕  ⊕  0≤i<j<l≤3  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='cijl  e0  e1  e2  e3  with ∣ej∣ = 0, ∣bij∣ = −1 and ∣cijl∣ = −2 and differential d(ej) = ∑i≠j b{ij}, d(bij) = ∑l≠i,j c{ijl},  and d(cijl) = 0, where the bracket “{}” denotes indices in ascending order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', b{ij} =  {bij, for i < j  bji, for j < i and similarly for c{ijl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The multiplication is similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='10), where the  indices “012” are replaced by any 0 ≤ i < j < l ≤ 3:  ∀j ∶  ej ⋅ ej = ej,  ∀i < j ∶  ei ⋅ bij = bij,  bij ⋅ ej = bij,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='21)  ∀i < j < l ∶  ei ⋅ cijl = cijl,  cijl ⋅ el = cijl,  bij ⋅ bjl = cijl,  and all other multiplications vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Note in particular that A has the unit e0 +e1 +e2 +e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We define the homotopy inner product F for A ∶= A to be given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='11)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='18) with  “012” in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='11)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='18) replaced by any 0 ≤ i < j < l ≤ 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=',  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='22)  F ∶= F[012] + F[013] + F[023] + F[123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Claim: F is a homotopy inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', DF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We compute DF = DF[012] + DF[013] + DF[023] + DF[123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We claim that we can use  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19) to evaluate this expression, which is not completely obvious since D in  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19) is for the dg-algebra A[012], whereas we need to apply D for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  To see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19) holds for the dg-algebra A, note that, for example, DF[012](a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' )(ar)  applies a differential or a multiplication to the inputs aj ∈ A (as stated in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Now, if any of the inputs aj are generators of A which has a 3 in the index (such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=',  b13 ∈ A), then taking the differential or a multiplication will consist of sums of generators  with 3 in their indices, and thus DF[012] applied to it would vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' On the other hand, if  all inputs aj applied to DF[012] are only given by generators indexed by 0, 1, and 2, then  DF[012] applied to it coincides with what we would get for A[012], because, the multiplication  in A equals the one in A[012], and the differential in A equals the differential in A[012] up  to generators with indices of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' (For example, applying d(b12) = c012 + c123 in DF[012] will  vanish on c123 and will coincide with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19) on c012 = dA[012](b12) =(differential of b12 in  A[012]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=')  Thus, DF = DF[012] + DF[013] + DF[023] + DF[123] = (F[01] + F[02] + F[12]) + (F[01] + F[03] +  F[13]) + (F[02] + F[03] + F[23]) + (F[12] + F[13] + F[23]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  □  14  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Let A be the dg-algebra from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='20, and let B = H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) ≅  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s be the dg-algebra from examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Then, the map f ∶ B → A,  f(e) ∶= e0 + e1 + e2 + e3 and f(s) ∶= c012, is a dg-algebra quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Transferring the homotopy inner product F/A defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='22) via the map f gives  f(F)/B = ̃F/B, where ̃F/B is the homotopy inner product defined in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) from  example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The transfered map of F/A from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='22) is given by applying f to inputs from B and  then applying F/A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Since the image of f is spanned by c012 and e0 + e1 + e2 + e3,  it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='11)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='18) that the only non-zero components (after applying f) are:  F[012]  0,0 (c012)(e2) = 1,  F[012]  0,0 (e0)(c012) = 1,  F[012]  2,0 (c012,c012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e1)(e1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Thus, we get precisely the non-zero components of ̃F from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) for f(F)/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Computations of the BV algebras  We now compute the BV algebras on Hochschild cohomology induced by the homotopy  inner products considered in examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2 in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In particular, we com-  plete the computations from theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6 for the 2-sphere by computing HH●(H●,H●) and  HH●(H●,H●) for H● = H● = H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) and H● = H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2), and we will check the for-  mulas for the maps F and ̃  F in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7) stated in theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In the section the ground ring  is R = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Hochschild cohomology HH●(H●,H●) and the cup product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Denote by H● =  H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) ≅ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 with ∣e∣ = 0 and ∣s∣ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The generators of  normalized Hochschild cochains CH  (H●,H●) are φk, ψk for k ≥ 0 given by  φk(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s  ��������������������  k many  ) ∶= e,  ψk(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s  ��������������������  k many  ) ∶= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Note that ∣φk∣ = k, ∣ψk∣ = k − 2, since the shifted generator s is of degree ∣s∣ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Applying  the Hochschild differential (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1, and using d = 0 and s ⋅ s = 0, s ⋅ e = e ⋅ s = s), we see that  all φk and ψk are closed (over Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Thus,  HH●(H●,H●) ≅ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='φk ⊕ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  The cup product (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5) is readily checked to be φk ⌣ φℓ = φk+ℓ, ψk ⌣ ψℓ = 0, and φk ⌣ ψℓ =  ψℓ ⌣ φk = ψk+ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Hochschild cohomology HH●(H●,H●) and Connes’ B-operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Denote by H● =  H●(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='Z2) ≅ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e∗ ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s∗ from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1, with ∣e∗∣ = 0 and ∣s∗∣ = 2 and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e∗ = e∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s =  0,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s∗ = s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='s = e∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The generators of normalized Hochschild cochains CH  (H●,H●) are θk,  χk for k ≥ 0 given by  θk(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s  ��������������������  k many  ) ∶= s∗,  χk(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s  ��������������������  k many  ) ∶= e∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Note that ∣θk∣ = k + 2, ∣χk∣ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' All θk and χk are closed under the Hochschild cochain  differential (over Z2), so that  HH●(H●,H●) ≅ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='θk ⊕ ⊕  k≥0  Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='χk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  15  Whereas HH●(H●,H●) has the cup product (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5), HH●(H●,H●) has Connes’ B-operator  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We compute B from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2) to be (B(ϕ))( s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s  ��������������������  (r−1) many  )(a) ∶= r ⋅ ϕ(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s,a,s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s)(e),  which applied to θk and χk yields:  B(θr) = 0,  B(χr) = r ⋅ θr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  This confirms the expressions for B stated in theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' F induced by example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 and the BV delta ∆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The dg-bimodule map F ∶ H● →  H●, F(s) = e∗ and F(e) = s∗, from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 induced a map on Hochschild cohomologies  F ∶ HH●(H●,H●) → HH●(H●,H●) via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Since F only has a lowest component F0,0  this is simply  F(φk) = F ○ φk = θk,  F(ψk) = F ○ ψk = χk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  F is clearly an isomorphism, and confirms F from theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We also compute ∆F and check it is equal to B transferred to HH●(H●,H●), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', con-  dition (2) from definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6 for a Poincar´e duality structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3), we get for  ϕ ∈ Hom(H●⊗r,H●) that ZF(ϕ)(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s)(a) = r ⋅ F(e)(ϕ(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,a,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s)), which gives  ZF(φk) = 0 = B(θk) = B ○ F(φk),  ZF(ψk) = k ⋅ θk−1 = B(χk) = B ○ F(ψk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Therefore, F −1 ○ B ○ F = F −1 ○ ZF = ∆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ̃  F induced by example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2 and the BV delta ∆̃F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Consider the dg-bimodule  map ̃F ∶ H● → H● from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Since ̃F = ̃F0,0 + ̃F2,0 is ̃F0,0(s) = e∗, ̃F0,0(e) = s∗ with  one extra homotopy ̃F2,0 which is non-zero only for ̃F2,0(s,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e) ∶= e∗ (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3)), we get  ̃  F(φk) = ̃F0,0 ○ φk + ̃F2,0(−,−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='φk(−)) = θk + χk+2,  ̃  F(ψk) = ̃F0,0 ○ ψk = χk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  This confirms ̃  F from theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' As was noted in theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6, ̃  F is an isomorphism with  inverse ̃  F −1 given by ̃  F −1(θk) = φk + ψk+2 and ̃  F −1(χk) = ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We also check that in this case ̃  F −1○B○ ̃  F = ∆̃F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3), we get for ϕ ∈ Hom(H●⊗r,H●),  that  Z  ̃F(ϕ)(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s)(a) =r ⋅ ̃F0,0(e)(ϕ(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,a,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s)) + r ⋅ ̃F2,0(s,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e)(ϕ(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,a,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s))  + ̃F2,0(ϕ(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s),s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e)(a) + ̃F2,0(s,ϕ(s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e)(a)  which gives  Z  ̃F(φk) = 0 + k ⋅ θk+1 + 0 + 0 = k ⋅ θk+1,  Z  ̃F(ψk) = k ⋅ θk−1 + 0 + χk+1 + χk+1 = k ⋅ θk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Since B ○ ̃  F(φk) = B(θk + χk+2) = k ⋅ θk+1 and B ○ ̃  F(ψk) = B(χk) = k ⋅ θk−1, this shows that  Z ̃F = B ○ ̃  F and, thus, ̃  F −1 ○ B ○ ̃  F = ̃  F −1 ○ Z ̃F = ∆̃F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Explicitly, ∆̃F is given by:  ∆  ̃F(φk) = ̃  F −1 ○ Z  ̃F(φk) = ̃  F −1(k ⋅ θk+1) = k ⋅ (φk+1 + ψk+3)  ∆  ̃F(ψk) = ̃  F −1 ○ Z  ̃F(ψk) = ̃  F −1(k ⋅ θk−1) = k ⋅ (φk−1 + ψk+1)  This confirms ̃∆ from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='9) and thus theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  16  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Example of a homotopy inner product that is not a Poincar´e duality struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We end this section with an example of a homotopy inner product which is not  a Poincar´e duality structure due to its failure of condition (2) from definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6 (while  condition (1) holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' We therefore do not get a BV algebra on Hochschild cohomology  HH●(A,A) with underlying algebra A, since there is no ∆ operator, but only an operator,  F −1 ○B ○F, that squares to zero, and a different operator, ∆F, whose deviation from being  a derivation is a Gerstenhaber bracket;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Let A = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='c with zero differential, ∣e∣ = 0, ∣b∣ = −1, ∣c∣ = −2, and where  e is the unit, b ⋅ b = c and all other products vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We denote A = A and its dual  A∗ = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e∗ ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b∗ ⊕ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='c∗ where e∗,b∗,c∗ are the duals of e,b,c, respectively, so that  ∣e∗∣ = 0, ∣b∗∣ = 1, ∣c∗∣ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Define the homotopy inner product F whose only non-vanishing  components are F0,0 and F3,0 given by  F0,0(c)(e) = F0,0(b)(b) = F0,0(e)(c) = 1,  F3,0(c,b,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e)(e) = F3,0(b,b,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='c)(e) = F3,0(b,b,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e)(c) = F3,0(b,b,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='b)(b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  One can check explicitly that the conditions for a homotopy inner product 0 = DF = D1F,  where D1 is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4), are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Note, that F induces an isomorphism on Hochschild cochains F ∶ CH  (A,A)  ≅  �→  CH  (A,A∗), since F0,0 ∶ A → A∗ is an isomorphism, and thus F satisfies 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6 condition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We next show that 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6 condition (2) fails, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', that B○F ≠ ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Consider ϕ ∈ Hom(A⊗0,A),  ϕ(1) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Note that ϕ is a closed element in the Hochschild cochains ϕ ∈ CH  (A,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Plug-  ging ϕ into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) gives ZF(ϕ) = 0, since each individual summand of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Next,  to compute F(ϕ), we see the only non-zero evaluation of F(ϕ) are (the ones with “e” in  the input spot for the module of F):  F(ϕ)(1) = F0,0(ϕ(1)) = c∗  F(ϕ)(c,b,c) = F3,0(c,b,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ϕ(1)) = e∗  F(ϕ)(b,b,b) = F3,0(b,b,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='ϕ(1)) = c∗  Applying B from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2) (note that the unit in A is in this example written as e) to this is  only non-zero on the middle term, for which we get the only non-zero evaluations:  B(F(ϕ))(c,b)(c) = 1,  B(F(ϕ))(b,c)(c) = 1,  B(F(ϕ))(c,c)(b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Then, B(F(ϕ)) is non-vanishing, and closed in CH  (A,A∗), which follows either because  ϕ is closed and B ○ F is a chain map or by direct inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Claim: B(F(ϕ)) is not exact in CH  (A,A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  The claim implies that B ○F(ϕ) ≠ 0 = ZF(ϕ) in HH●(A,A∗), and so F is not a Poincar´e  duality structure as it fails condition (2) from definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Proof of the claim: To prove the claim, note that d = 0 in A implies that the Hochschild  differential D = D0 + D1 = D1 from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 must increase the number of tensor factors of A for  the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Thus any ̃ϕ which makes B(F(ϕ)) exact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', D(̃ϕ) = B(F(ϕ)), must have  non-vanishing components with exaclty one tensor input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Since the degree ∣B(F(ϕ))∣ = 3,  it must be that ∣̃ϕ∣ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' However, all Hochschild cochains in CH  (A,A∗) with one or no  input are of degree ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Thus, B(F(ϕ)) cannot be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  □  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  17  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Proof of equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19)  In this appendix we prove the identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', DF[01] = F[0] + F[1], for the maps F[01]  given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7) from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='6, as well as the identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', DF[012] = F[01] + F[02] +  F[12], for the maps F[012] given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='11)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='18) from example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The ground ring in this  appendix is R = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Any Fp,q is a map Fp,q ∶ A⊗p ⊗ A ⊗ A⊗q → A∗, which is an element  of Fp,q ∈ (A∗)⊗p ⊗ A∗ ⊗ (A∗)⊗q ⊗ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Consider A[01] with generators e0,e1,b01, respectively  A[012] with generators e0,e1,e2,b01,b12,b02,c012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' For a multi-index I ∈ {0,1,2,01,02,12,012},  define a∗  I to be a∗  0 ∶= e∗  0, a∗  1 ∶= e∗  1, a∗  2 ∶= e∗  2, a∗  01 ∶= b∗  01, a∗  02 ∶= b∗  02, a∗  12 ∶= b∗  12, a∗  012 ∶= c∗  012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' With  this we use the following notation, for a given sequence of multi-indices I1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ip+q+2:  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2)  [ I1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ip ∣ Ip+1 ∣ Ip+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ip+q+1 ∣ Ip+q+2 ]  ∶= a∗  I1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ⊗ a∗  Ip ⊗ a∗  Ip+1 ⊗ a∗  Ip+2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ⊗ a∗  Ip+q+1 ⊗ a∗  Ip+q+2 ∈ (A∗)⊗p ⊗ A∗ ⊗ (A∗)⊗q ⊗ A∗  Moreover, placing a tilde under the multi-index sums over any number of these indices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=',  for tildes at positions 1 ≤ k1 < k2 < ⋅⋅⋅ < kr ≤ p, we define  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3)  [ I1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ik1  ̃  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ik2  ̃  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ip ∣ Ip+1 ∣ Ip+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ip+q+1 ∣ Ip+q+2 ]  ∶=  ∑  n1,n2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=',nr≥0  [ I1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ik1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ik1  ���������������������������������������������  n1 many  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ik2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ik2  ���������������������������������������������  n2 many  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ip ∣ Ip+1 ∣ Ip+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' ,Ip+q+1 ∣ Ip+q+2 ]  Since our examples are chosen to be “strict” on the right (see observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4), we only  need higher homotopies (and thus tildes) on the first p tensor factors, but not on the other  q tensor factors of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We apply D = D0 +D1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4) to the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Here, D0 applies the differential  of A∗, which, in the multi-indiex notation, removes one index, (for example, 01 ↦ 0 + 1,  012 ↦ 01 + 02 + 12, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' D1 applies a multiplication, which on A∗ applies the Alexander-  Whitney coproduct (for example, 01 ↦ 0 ⊗ 01 + 01 ⊗ 1, 012 ↦ 0 ⊗ 012 + 01 ⊗ 12 + 012 ⊗ 2,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' see also the calculation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' F[01] from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='7) is given by F[01] = [ 01  ̃ ∣ 0 ∣ ∣ 01 ] + [ 01  ̃ ∣ 01 ∣ ∣ 1 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  We calculate D = D0 + D1 applied to F[01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  D0([ 01  ̃ ∣ 0 ∣ ∣ 01 ]) =  [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃ ∣ 0 ∣ ∣ 01 ] 01  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃ ∣ 0 ∣ ∣ 01 ] 02  +[ 01  ̃ ∣ 0 ∣ ∣ 0 ] 03  +[ 01  ̃ ∣ 0 ∣ ∣ 1 ] 04  D1([ 01  ̃ ∣ 0 ∣ ∣ 01 ]) =  [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃ ∣ 0 ∣ ∣ 01 ] 01  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃ ∣ 0 ∣ ∣ 01 ] 02  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0 ∣ 0 ∣ ∣ 01 ] 01  +[ 01  ̃ ∣ 0 ∣ 0 ∣ 01 ] 05  +[ 01  ̃ ∣ 0 ∣ 0 ∣ 01 ] 05  +[ 01  ̃ ∣ 0 ∣ 01 ∣ 1 ] 06  +[ 01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃ ∣ 0 ∣ ∣ 0 ] 03  +[ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃ ∣ 0 ∣ ∣ 01 ] 02  D0([ 01  ̃ ∣ 01 ∣ ∣ 1 ]) =  [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃ ∣ 01 ∣ ∣ 1 ] 07  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃ ∣ 01 ∣ ∣ 1 ] 08  +[ 01  ̃ ∣ 0 ∣ ∣ 1 ] 04  +[ 01  ̃ ∣ 1 ∣ ∣ 1 ] 09  18  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  D1([ 01  ̃ ∣ 01 ∣ ∣ 1 ]) =  [ 01  ̃,0,01, 01  ̃ ∣ 01 ∣ ∣ 1 ] 07  +[ 01  ̃,01,1, 01  ̃ ∣ 01 ∣ ∣ 1 ] 08  +[ 01  ̃,0 ∣ 01 ∣ ∣ 1 ] 07  +[ 01  ̃,01 ∣ 1 ∣ ∣ 1 ] 09  +[ 01  ̃ ∣ 0 ∣ 01 ∣ 1 ] 06  +[ 01  ̃ ∣ 01 ∣ 1 ∣ 1 ] 10  +[ 01  ̃ ∣ 01 ∣ 1 ∣ 1 ] 10  +[ 1, 01  ̃ ∣ 01 ∣ ∣ 1 ] 08  The gray box on the right of each expression indicates which terms can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Note  that for  01 ,  02 ,  07 , and  08 , there are always 3 terms that can be combined and cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' In  fact, all terms cancel, except for terms labeled  03 , which has a left over term of [ ∣ 0 ∣ ∣ 0 ],  and terms labeled  09 , which has a left over term of [ ∣ 1 ∣ ∣ 1 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' These two left-over terms  are F[0] and F[1], which shows that DF[01] = F[0] + F[1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=', equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' F[012] from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='11)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='18) is given by  F[012] =[ 02  ̃ ∣ 012 ∣ ∣ 2 ] + [ 02  ̃ ∣ 0 ∣ ∣ 012 ] + [ 02  ̃ ∣ 01 ∣ ∣ 12 ]  + [ 01  ̃,012, 02  ̃ ∣ 0 ∣ ∣ 01 ] + [ 01  ̃,012, 02  ̃ ∣ 01 ∣ ∣ 1 ]  + [ 02  ̃,012, 12  ̃ ∣ 1 ∣ ∣ 12 ] + [ 02  ̃,012, 12  ̃ ∣ 12 ∣ ∣ 2 ]  + [ 01  ̃,012, 02  ̃,012, 12  ̃ ∣ 1 ∣ ∣ 1 ]  We calculate D = D0 + D1 applied to F[012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Again, the gray box indicates which terms  combine and cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Left-over terms will be collected below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  D0([ 02  ̃ ∣ 012 ∣ ∣ 2 ]) =  [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 012 ∣ ∣ 2 ] 01  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 012 ∣ ∣ 2 ] 02  +[ 02  ̃ ∣ 01 ∣ ∣ 2 ] 03  +[ 02  ̃ ∣ 02 ∣ ∣ 2 ] 04  +[ 02  ̃ ∣ 12 ∣ ∣ 2 ] 05  D1([ 02  ̃ ∣ 012 ∣ ∣ 2 ]) =  [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 012 ∣ ∣ 2 ] 01  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 012 ∣ ∣ 2 ] 02  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0 ∣ 012 ∣ ∣ 2 ] 01  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='01 ∣ 12 ∣ ∣ 2 ] 06  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012 ∣ 2 ∣ ∣ 2 ] 07  +[ 02  ̃ ∣ 012 ∣ 2 ∣ 2 ] 08  +[ 02  ̃ ∣ 01 ∣ 12 ∣ 2 ] 09  +[ 02  ̃ ∣ 0 ∣ 012 ∣ 2 ] 10  +[ 02  ̃ ∣ 012 ∣ 2 ∣ 2 ] 08  +[ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 012 ∣ ∣ 2 ] 02  D0([ 02  ̃ ∣ 0 ∣ ∣ 012 ]) =  [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 012 ] 11  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 012 ] 12  +[ 02  ̃ ∣ 0 ∣ ∣ 01 ] 13  +[ 02  ̃ ∣ 0 ∣ ∣ 02 ] 14  +[ 02  ̃ ∣ 0 ∣ ∣ 12 ] 15  D1([ 02  ̃ ∣ 0 ∣ ∣ 012 ]) =  [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 012 ] 11  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 012 ] 12  [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0 ∣ 0 ∣ ∣ 012 ] 11  +[ 02  ̃ ∣ 0 ∣ 0 ∣ 012 ] 16  [ 02  ̃ ∣ 0 ∣ 0 ∣ 012 ] 16  +[ 02  ̃ ∣ 0 ∣ 01 ∣ 12 ] 17  [ 02  ̃ ∣ 0 ∣ 012 ∣ 2 ] 10  +[ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 012 ] 12  A NOTE ON THE STRING TOPOLOGY BV-ALGEBRA FOR S2 WITH Z2 COEFFICIENTS  19  [ 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 01 ] 18  +[ 012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 0 ] 19  D0([ 02  ̃ ∣ 01 ∣ ∣ 12 ]) =  [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 01 ∣ ∣ 12 ] 20  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 01 ∣ ∣ 12 ] 21  +[ 02  ̃ ∣ 0 ∣ ∣ 12 ] 15  +[ 02  ̃ ∣ 1 ∣ ∣ 12 ] 22  +[ 02  ̃ ∣ 01 ∣ ∣ 1 ] 23  +[ 02  ̃ ∣ 01 ∣ ∣ 2 ] 03  D1([ 02  ̃ ∣ 01 ∣ ∣ 12 ]) =  [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+page_content=' 02  ̃ ∣ 01 ∣ ∣ 12 ] 20  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 01 ∣ ∣ 12 ] 21  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0 ∣ 01 ∣ ∣ 12 ] 20  +[ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='01 ∣ 1 ∣ ∣ 12 ] 24  +[ 02  ̃ ∣ 01 ∣ 1 ∣ 12 ] 25  +[ 02  ̃ ∣ 0 ∣ 01 ∣ 12 ] 17  +[ 02  ̃ ∣ 01 ∣ 1 ∣ 12 ] 25  +[ 02  ̃ ∣ 01 ∣ 12 ∣ 2 ] 09  +[ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 01 ∣ ∣ 12 ] 21  +[ 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 01 ∣ ∣ 1 ] 26  D0([ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 01 ]) =  [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+page_content=' 02  ̃ ∣ 0 ∣ ∣ 01 ] 27  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+page_content=' 02  ̃ ∣ 0 ∣ ∣ 01 ]) =  [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] 56  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+page_content='01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] 39  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] 59  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] 58  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] 59  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='1 ∣ 1 ∣ ∣ 1 ] 58  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ 1 ∣ 1 ] 60  +[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ 1 ∣ 1 ] 60  +[ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] 55  In the above sum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' there are several terms which appear twice and which therefore cancel  (over Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The terms that are labeled with the following numbers appear twice and cancel:  03 , 08 , 09 , 10 , 15 , 16 , 17 , 25 , 31 , 32 , 33 , 40 , 45 , 46 , 47 , 53 , 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  22  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' POIRIER AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' TRADLER  Moreover, in some instances the sum of three terms cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The following labels appear  three times and also cancel:  01 , 02 , 06 , 07 , 11 , 12 , 18 , 19 , 20 , 21 , 24 , 26 , 27 , 28 , 29 , 30 , 34 , 35 ,  36 , 37 , 39 , 41 , 42 , 43 , 44 , 48 , 49 , 50 , 51 , 52 , 54 , 55 , 56 , 57 , 58 , 59 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  There are six terms labeled with  38 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' whose sum vanishes:  38 =([ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 1 ∣ ∣ 1 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ])  + ([ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ])  =[ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 1 ] = 0  The left-over terms that do not cancel are:  04 = [ 02  ̃ ∣ 02 ∣ ∣ 2 ]  05 = [ 02  ̃ ∣ 12 ∣ ∣ 2 ] + [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 12 ∣ ∣ 2 ] + [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 12 ∣ ∣ 2 ] = [ 12  ̃ ∣ 12 ∣ ∣ 2 ]  13 = [ 02  ̃ ∣ 0 ∣ ∣ 01 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 01 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 0 ∣ ∣ 01 ] = [ 01  ̃ ∣ 0 ∣ ∣ 01 ]  14 = [ 02  ̃ ∣ 0 ∣ ∣ 02 ]  22 = [ 02  ̃ ∣ 1 ∣ ∣ 12 ] + [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 12 ] + [ 02  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 12  ̃ ∣ 1 ∣ ∣ 12 ] = [ 12  ̃ ∣ 1 ∣ ∣ 12 ]  23 = [ 02  ̃ ∣ 01 ∣ ∣ 1 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 01 ∣ ∣ 1 ] + [ 01  ̃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 02  ̃ ∣ 01 ∣ ∣ 1 ] = [ 01  ̃ ∣ 01 ∣ ∣ 1 ]  These terms are precisely F[02],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' F[12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' and F[01],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' so that DF[012] = D0F[012] + D1F[012] =  F[01] + F[02] + F[12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' which is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  References  [CS]  Moira Chas, Dennis Sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' String Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' preprint arXiv:math/9911159  [L]  Jean-Louis Loday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Cyclic Homology, Grundlehren der mathematischen WIssenschaften 301 (1998),  Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  [M]  Luc Menichi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' String Topology for Spheres, Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 84 (2009), 135–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  [T1]  Thomas Tradler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Infinity Inner Products on A-Infinity Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Homotopy Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 3  (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 245–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  [T2]  Thomas Tradler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Fourier (Grenoble) 58 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 2351–2379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  [TZS]  Thomas Tradler, Mahmoud Zeinalian, Dennis Sullivan Infinity structure of Poincar´e duality spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 7 (2007), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content=' 233–260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='  Kate Poirier, Department of Mathematics, New York City College of Technology,  City University of New York, 300 Jay Street, Brooklyn, NY 11201  Email address: kpoirier@citytech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='cuny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='edu  Thomas Tradler, Department of Mathematics, New York City College of Technology,  City University of New York, 300 Jay Street, Brooklyn, NY 11201  Email address: ttradler@citytech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='cuny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE5T4oBgHgl3EQfAw5l/content/2301.05381v1.pdf'}
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+MATHEMATICA MONTISNIGRI 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+  
+ 
+2010 Mathematics Subject Classification:  30B10, 11M05, 11M41, 33B15.  
+Key words and Phrases: Riemann’s zeta function, Lerch transcendent, polylogarithm, digamma function, 
+Euler’s constant. 
+SEVERAL CLASSICAL IDENTITIES VIA MELLIN’S TRANSFORM 
+KHRISTO N. BOYADZHIEV 
+Department of mathematics, Ohio Northern University 
+Ada, Ohio, 45810, USA 
+ E-mail: k-boyadzhiev@onu.edu 
+DOI: 
+Summary. We present a summation rule using Mellin’s transform to give short proofs of 
+some important classical relations between special functions and Bernoulli and Euler 
+polynomials. For example, the values of the Hurwitz zeta function at the negative integers are 
+expressed in terms of Bernoulli polynomials. We also show identities involving exponential 
+and Hermite polynomials. 
+1. INTRODUCTION 
+Throughout we use the notation ( )
+,
+a
+a
+it t
+=
++
+∈  for the vertical line with abscissa
+0
+1
+a
+<
+< , oriented from minus to plus infinity. First we recall the formulas for the Mellin 
+transform 
+1
+0
+( )
+( )
+s
+G s
+x
+g x dx
+∞
+−
+= ∫
+ 
+and its inverse  
+ 
+( )
+1
+( )
+( )
+,
+0
+2
+s
+a
+g x
+x G s ds
+x
+i
+π
+−
+=
+>
+∫
+. 
+(1) 
+Definition. For 
+0
+n >
+ consider integrals of the form 
+( )
+( )
+,
+0
+s
+L n x G s ds
+x
+−
+>
+∫
+ 
+where ( )
+L n  consists of the line segment [
+,
+]
+ni ni
+−
+ together with the semicircle 
+( )
+R n  in the left 
+half plane for which the line segment is the diagonal. If the integrals 
+( )
+( )
+s
+R n x G s ds
+−
+∫
+ 
+approach zero when n → ∞, we say that the line of integration in (1) can be closed to the left. 
+In a similar manner we define integrals where the line of integration can be closed to the right. 
+Proposition. Suppose that the function 
+( )
+G s  is meromorphic on the half plane 
+( )
+Re s
+a
+ε
+<
++
+for some small 
+0
+ε >
+, and has only simple poles at  
+0, 1, 2,...
+s =
+− −
+, with residues 
+0
+1
+2
+,
+,
+,...
+c c c
+. 
+If the line of integration in (1) can be closed to the left, the residue theorem provides the 
+representation 
+
+Khristo N. Boyadzhiev 
+ 
+0
+( )
+n
+n
+n
+g x
+c x
+∞
+=
+= ∑
+ 
+(2) 
+for the function 
+( )
+g x  from (1), i.e. this function is a power series. If now 
+( )
+f s  is an 
+appropriate holomorphic function on 
+( )
+Re s
+a
+ε
+<
++
+ without poles, we can write  
+ 
+( )
+0
+1
+( ) ( )
+(
+)
+2
+s
+n
+n
+a
+n
+x
+f s G s ds
+c f
+n x
+i
+π
+∞
+−
+=
+=
+−
+∑
+∫
+, 
+(3) 
+when the power series on the right side converges.  
+Formulas of this type are used for summation of series or interpolation, and are present in 
+many publications (see [2, 4, 9, 10] and the references there).  
+In this note we focus on a special area of applications for the proposition - obtaining some 
+classical identities by using Mellin inversion and comparing coefficients. In order to keep the 
+paper short we omit details and do not discuss convergence of some integrals and series. The 
+validity of such formulas is considered in [4, 9, 10]. 
+The illustration of the method is given in the following examples.  
+2 EXAMPLES 
+Remind that the residues of the gamma function at zero and the negative integers are given 
+by 
+( 1)
+Res( ,
+)
+!
+n
+n
+n
+−
+Γ −
+=
+ for 
+0,1,2,...
+n =
+. Also ( )s
+Γ
+ has rapid decay on vertical lines. We have 
+the estimate ([15, (20)] 
+(
+)
+it
+a +
+Γ
+ ~ 
+1
+2
+2
+2
+| |
+,
+a
+t
+t
+e
+a
+π
+π
+−
+−
+∈  
+when | |t → ∞ . The estimate helps for the convergence of our integrals.  
+The above facts will be used in the following examples. We note that in these examples the 
+lines of integration can be closed to the left (proofs are standard, using the growth estimate for 
+( )s
+Γ
+). Note also that when we replace 
+( )
+G s  by ( )s
+Γ
+ in (3) we have 
+ 
+( )
+0
+1
+( 1)
+( ) ( )
+(
+)
+2
+!
+n
+s
+n
+a
+n
+x
+f s
+s ds
+f
+n x
+i
+n
+π
+∞
+−
+=
+−
+Γ
+=
+−
+∑
+∫
+. 
+(4) 
+Example 1. Ramanujan’s Master Theorem. As Hardy writes in [8], Ramanujan was very fond 
+of his integral formula [8, p.186] 
+ 
+1
+0
+0
+( )(
+)
+(
+) ( )
+!
+n
+s
+n
+f n
+x
+x
+dx
+f
+s
+s
+n
+∞
+∞
+−
+=
+
+
+−
+=
+−
+Γ
+
+
+
+
+∑
+∫
+ 
+ 
+(5) 
+and used it for many applications. Berndt rightly calls it Ramanujan’s Master Theorem [2, 
+Entry 11, p.105]. Details, comments, and applications of (5) are given in these two books and 
+also in [1, 4, 6]. Clearly, after replacing 
+( )
+f s  by 
+(
+)
+f
+s
+−
+ equation (5) turns into (4) after 
+Mellin inversion. Ramanujan did not use the residue theorem for his proof but only standard 
+calculus (see [2, p.106]).  
+
+Khristo N. Boyadzhiev 
+Example 2. The Hurwitz zeta function is defined by 
+0
+1
+( , )
+(Re( )
+0, Re( )
+1)
+(
+)s
+n
+s z
+z
+s
+n
+z
+ζ
+∞
+=
+=
+>
+>
++
+∑
+ 
+with integral representation 
+(1
+)
+1
+0
+( , ) ( )
+(Re( )
+1)
+1
+t
+z
+s
+t
+e
+s z
+s
+t
+dt
+s
+e
+ζ
+∞
+−
+−
+Γ
+=
+>
+−
+∫
+ 
+When 
+1
+z = , 
+( ,1)
+( )
+s
+s
+ζ
+ζ
+=
+ is the Riemann zeta function [5]. 
+We have also the modified integral representation (argument is the same as on pp. 61-62 in 
+[13]) 
+(1
+)
+1
+0
+1
+( , ) ( )
+(0
+Re( )
+1)
+1
+t
+z
+s
+t
+e
+s z
+s
+t
+dt
+s
+e
+t
+ζ
+∞
+−
+− 
+
+Γ
+=
+<
+<
+−
+
+
+−
+
+
+∫
+ 
+which is a Mellin transform formula. By Mellin inversion 
+(1
+)
+( )
+0
+1
+1
+( 1)
+( , ) ( )
+(
+, )
+1
+2
+!
+t
+z
+n
+s
+t
+a
+n
+e
+t
+s z
+s ds
+n z
+e
+t
+i
+n
+ζ
+ζ
+π
+−
+∞
+−
+=
+−
+−
+=
+Γ
+=
+−
+−
+∑
+∫
+ 
+At the same time, the Bernoulli polynomials 
+( )
+n
+B z  have the generating function 
+0
+( )
+1
+!
+xz
+k
+k
+x
+k
+B z
+xe
+x
+e
+k
+∞
+=
+=
+−
+∑
+ 
+and from this 
+(1
+)
+1
+1
+1
+0
+(1
+)
+(1
+)
+1
+( , )
+1
+!
+(
+1)!
+x
+z
+k
+n
+k
+n
+x
+k
+n
+B
+z
+B
+z
+e
+g x z
+x
+x
+e
+x
+k
+n
+−
+∞
+∞
+−
++
+=
+=
+−
+−
+≡
+−
+=
+=
+−
++
+∑
+∑
+. 
+Therefore, by comparing coefficients for 
+0, 1,...
+n =
+we find the classical formula 
+1
+1
+( 1)
+(1
+)
+( )
+(
+, )
+1
+1
+n
+n
+n
+B
+z
+B
+z
+n z
+n
+n
+ζ
++
++
+−
+−
+−
+=
+= −
++
++
+ 
+(using the property ( 1)
+(1
+)
+( )
+n
+n
+n
+B
+z
+B z
+−
+−
+=
+, so that 
+1
+1
+( 1)
+(1
+)
+( )
+n
+n
+n
+B
+z
+B
+z
++
++
+−
+−
+= −
+).  
+In particular,  
+1
+1
+(0, )
+( )
+2
+z
+B z
+z
+ζ
+= −
+=
+− . 
+For the Bernoulli numbers 
+(0)
+( 1)
+(1)
+n
+n
+n
+n
+B
+B
+B
+=
+= −
+ we have 
+1
+1
+( 1)
+(0)
+( 1)
+(
+)
+(
+,1)
+1
+1
+n
+n
+n
+n
+B
+B
+n
+n
+n
+n
+ζ
+ζ
++
++
+−
+−
+−
+=
+−
+=
+=
++
++
+. 
+The odd Bernoulli numbers are zeros except 
+1
+1/ 2
+B = −
+. Thus 
+(0)
+1/ 2
+ζ
+= −
+. 
+
+Khristo N. Boyadzhiev 
+Next we give a new proof of a result of Kenneth Williams and Zhang Nan-Yue [14]. 
+Example 3. Consider now the alternating Hurwitz zeta function 
+0
+( 1)
+( , )
+(Re( )
+0, Re( )
+0)
+(
+)
+n
+s
+n
+s z
+z
+s
+n
+z
+η
+∞
+=
+−
+=
+>
+>
++
+∑
+ 
+which extends to the entire complex plane as analytic in the variable s  and has the integral 
+representation 
+(1
+)
+1
+0
+( , ) ( )
+(Re( )
+0)
+1
+t
+z
+s
+t
+e
+s z
+s
+t
+dt
+s
+e
+η
+∞
+−
+−
+Γ
+=
+>
++
+∫
+. 
+By inversion 
+(1
+)
+( )
+0
+1
+( 1)
+( , ) ( )
+(
+, )
+1
+2
+!
+t
+z
+n
+s
+n
+t
+a
+n
+e
+t
+s z
+s ds
+n z x
+e
+i
+n
+η
+η
+π
+−
+∞
+−
+=
+−
+=
+Γ
+=
+−
++
+∑
+∫
+. 
+Euler’s polynomials 
+( )
+n
+E
+z  are defined by the generating function 
+0
+2
+( )
+(| |
+)
+1
+!
+t x
+n
+n
+t
+n
+e
+t
+E
+x
+t
+e
+n
+π
+∞
+=
+=
+<
++
+∑
+. 
+Comparing coefficients gives 
+( 1)
+1
+(
+, )
+(1
+)
+( )
+2
+2
+n
+n
+n
+n z
+E
+z
+E
+z
+η
+−
+−
+=
+−
+=
+ 
+by the property 
+(1
+)
+( 1)
+( )
+n
+n
+n
+E
+z
+E
+z
+−
+= −
+. 
+Example 4. Euler worked with the function 
+0
+( 1)
+( )
+(Re
+0)
+(2
+1)
+n
+s
+n
+L s
+s
+n
+∞
+=
+−
+=
+>
++
+∑
+ 
+which we call here Euler’s L -function (sometimes it is called Dirichlet’s L -function). This 
+function has the integral representation 
+ 
+1
+0
+2 ( ) ( )
+cosh
+sx
+s L s
+dx
+x
+∞
+−
+Γ
+= ∫
+. 
+It also has analytic extension on the complex plane.  
+By Mellin inversion and using equation (4) 
+( )
+0
+1
+1
+( 1)
+( ) ( )
+(
+)
+2cosh( )
+2
+!
+n
+s
+n
+a
+n
+x L s
+s ds
+L
+n x
+x
+i
+n
+π
+∞
+−
+=
+−
+=
+Γ
+=
+−
+∑
+∫
+ 
+Euler’s numbers 
+n
+E  are defined by the generating function 
+0
+1
+cosh
+!
+n
+n
+n
+E x
+x
+n
+∞
+=
+= ∑
+. 
+
+Khristo N. Boyadzhiev 
+This function is even, so the Euler numbers with odd indices are zeros. By comparing 
+coefficients we find 
+2
+1
+( 2 )
+(
+0,1, 2,...)
+2
+n
+L
+n
+E
+n
+−
+=
+=
+. 
+Example 5.  The exponential polynomials 
+n
+ϕ  are defined by the generating function 
+ 
+(
+1)
+0
+( )
+!
+x
+n
+z e
+n
+n
+x
+e
+z n
+ϕ
+∞
+−
+=
+= ∑
+ 
+(see [3, 12]). We will use the function   
+ 
+(
+1)
+0
+( , )
+( 1)
+( )
+!
+x
+n
+z e
+n
+n
+n
+x
+x z
+e
+z n
+ψ
+ϕ
+−
+∞
+−
+=
+=
+=
+−
+∑
+   
+ which has Mellin transform 
+ 
+1
+1
+0
+0
+( , )
+( , )
+s
+z
+s
+x
+ze
+s z
+x
+x z dx
+e
+x
+e
+dx
+ψ
+∞
+∞
+−
+−
+−
+−
+Ψ
+=
+=
+∫
+∫
+ 
+ 
+1
+0
+0
+0
+( )
+( ) ( , )
+!
+!
+n
+n
+z
+s
+z
+s
+n
+n
+nx
+z
+z
+e
+x
+e
+dx
+e
+s
+s f s z
+n
+n n
+∞
+∞
+∞
+−
+−
+−
+−
+=
+=
+
+
+=
+=
+Γ
+= Γ
+
+
+
+
+∑
+∑
+∫
+ 
+with  
+0
+( , )
+!
+n
+z
+s
+n
+z
+f s z
+e
+n n
+∞
+−
+=
+=
+∑
+. 
+Now we have from equation (4) 
+( )
+( )
+0
+1
+1
+( 1)
+( , )
+( , )
+( , ) ( )
+(
+, )
+2
+2
+!
+n
+s
+s
+n
+a
+a
+n
+x z
+x
+x s ds
+x
+f s z
+s ds
+f
+n z x
+i
+i
+n
+ψ
+π
+π
+∞
+−
+−
+=
+−
+=
+Ψ
+=
+Γ
+=
+−
+∑
+∫
+∫
+ 
+Thus for 
+0
+n ≥
+(with the agreent 
+00
+1
+= ) 
+ 
+0
+( )
+(
+, )
+!
+n
+z
+k
+n
+k
+k
+z
+f
+n z
+e
+z
+k
+ϕ
+∞
+−
+=
+=
+−
+=
+∑
+ 
+(6) 
+which is one of the fundamental properties of the exponential polynomials. 
+Note that identity (6) can be used for a meaningful extension of 
+( )
+n z
+ϕ
+ to 
+( )z
+λ
+ϕ
+ with non-
+integer index λ . For instance, we have (with 0
+0
+λ =
+) 
+ 
+1
+( )
+!
+z
+k
+k
+k
+z
+e
+z
+k
+λ
+λ
+ϕ
+∞
+−
+=
+=
+∑
+. 
+Example 6.  The generating function for the Hermite polynomials is 
+
+Khristo N. Boyadzhiev 
+ 
+2
+2
+0
+( , )
+( )
+!
+n
+xz
+x
+n
+n
+x
+x z
+e
+H
+z n
+ψ
+∞
+−
+=
+=
+= ∑
+ 
+with Mellin transform [11, p. 27] 
+ 
+ 
+2
+/2
+2
+( , )
+2
+( 2 ) ( )
+z
+s
+s
+s z
+e
+D
+z
+s
+−
+−
+Ψ
+=
+Γ
+. 
+where 
+p
+D  are the parabolic cylinder functions [7, pp. 1065-1067]. Let now  
+2
+/2
+2
+( , )
+2
+( 2 )
+z
+s
+s
+f s z
+e
+D
+z
+−
+−
+=
+. 
+Then 
+( )
+( 1)
+(
+, )
+n
+n
+H
+z
+f
+n z
+= −
+−
+ and since  
+( )
+( 1)
+(
+)
+n
+n
+n
+H
+z
+H
+z
+= −
+−
+ one finds the classical result 
+ 
+2
+2
+2
+( )
+2
+( 2 )
+n
+z
+n
+n
+H
+z
+e D
+z
+=
+ 
+([7, entry 9.253, p.1067]). 
+6.  CONCLUSIONS 
+In this note we presented a rule how Mellin’s transform can be used to give short proofs of 
+several classical identities connecting, in particular, the Bernoulli and Euler polynomials to 
+the values of the Hurwitz and alternating Hurwitz functions at the negative integers. We also 
+proved identities for the exponential and Hermite polynomials. 
+REFERENCES 
+[1] 
+T. Amdeberhan, I. Gonzalez, M. Harrison, V. H. Moll, A. Straub, “Ramanujan's Master 
+Theorem”, The Ramanujan Journal, 29, 103–120 (2012). 
+[2] 
+B. C. Berndt, Ramanujan’s Notebooks, Parts I and II, Springer-Verlag, New York (1985, 
+1989). 
+[3] 
+K. N. Boyadzhiev, “Exponential polynomials, Stirling numbers, and evaluation of some 
+Gamma integrals”, Abstract and Applied Analysis, Article ID 168672 (2009).  
+[4] 
+G. Dahlquist, “On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-
+Christoffel rules, I”, BIT Numer. Math., 37(2), 256-295 (1997); “On summation formulas 
+due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules, II”, BIT Numer. Math., 
+37(4), 804-832 (1997); “On Summation Formulas Due to Plana, Lindelöf and Abel, and 
+Related Gauss-Christoffel Rules, III”, BIT Numer. Math. 39, 51–78 (1999).  
+[5] 
+H. M. Edwards, Riemann’s Zeta Function, Academic Press, Boston, (1974). 
+[6] 
+I. González, V. H. Moll, I. Schmidt, “A generalized Ramanujan Master Theorem applied to 
+the evaluation of Feynman diagrams”, Adv. Appl. Math., 63, 214-230 (2015). 
+[7] 
+I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic 
+Press, (1980). 
+[8] 
+G. H. Hardy, Ramanujan, Cambridge University Press, (1940). 
+[9] 
+E. Lindelöf, Le Calcul des Résidues et ses Applications à la Theorie des Fonctions, 
+Gauthier-Villars, Paris, (1905). 
+[10] D.S. Mitrinovic and J. D. Keckic, The Cauchy Method of Residues, D. Reidel Publ. Co., 
+Dordrecht/Boston, (1984). 
+[11] F. Oberhettinger, Tables of Mellin Transforms, Springer-Verlag, New York,1974. 
+ 
+
+Khristo N. Boyadzhiev 
+[12] Gian-Carlo Rota, Finite Operator Calculus, Academic Press, new York, (1975). 
+[13] D.V. Widder, An Introduction to Transform Theory, Academic Press, New York, (1971). 
+[14] K. S. Williams and Z. Nan-Yue, “Special values of the Lerch zeta function and the 
+evaluation of certain integrals”, Proc. Amer. Math. Soc., 119, 35-49 (1993). 
+[15] W. Heap, Notes on the gamma function and the Riemann zeta function (online publication). 
+https://wiki.math.ntnu.no/_media/ma3001/2014v/analytisktallteori/the_riemann_zeta_functi
+on_notes.pdf 
+ 
+ 
+ 
+
diff --git a/UtAzT4oBgHgl3EQf0_7l/content/tmp_files/load_file.txt b/UtAzT4oBgHgl3EQf0_7l/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9f0e778bc5676eb0ffc191781d4b1cb4c31c527c
--- /dev/null
+++ b/UtAzT4oBgHgl3EQf0_7l/content/tmp_files/load_file.txt
@@ -0,0 +1,194 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf,len=193
+page_content='MATHEMATICA MONTISNIGRI  2010 Mathematics Subject Classification:  30B10, 11M05, 11M41, 33B15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Key words and Phrases: Riemann’s zeta function, Lerch transcendent, polylogarithm, digamma function,  Euler’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  SEVERAL CLASSICAL IDENTITIES VIA MELLIN’S TRANSFORM  KHRISTO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' BOYADZHIEV  Department of mathematics, Ohio Northern University  Ada, Ohio, 45810, USA  E-mail: k-boyadzhiev@onu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='edu  DOI:  Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We present a summation rule using Mellin’s transform to give short proofs of  some important classical relations between special functions and Bernoulli and Euler  polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' For example, the values of the Hurwitz zeta function at the negative integers are  expressed in terms of Bernoulli polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We also show identities involving exponential  and Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' INTRODUCTION  Throughout we use the notation ( )  ,  a  a  it t  =  +  ∈\uf0a1  for the vertical line with abscissa  0  1  a  <  < , oriented from minus to plus infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' First we recall the formulas for the Mellin  transform  1  0  ( )  ( )  s  G s  x  g x dx  ∞  −  = ∫  and its inverse  ( )  1  ( )  ( )  ,  0  2  s  a  g x  x G s ds  x  i  π  −  =  >  ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  (1)  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' For  0  n >  consider integrals of the form  ( )  ( )  ,  0  s  L n x G s ds  x  −  >  ∫  where ( )  L n  consists of the line segment [  ,  ]  ni ni  −  together with the semicircle  ( )  R n  in the left  half plane for which the line segment is the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' If the integrals  ( )  ( )  s  R n x G s ds  −  ∫  approach zero when n → ∞, we say that the line of integration in (1) can be closed to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  In a similar manner we define integrals where the line of integration can be closed to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Suppose that the function  ( )  G s  is meromorphic on the half plane  ( )  Re s  a  ε  <  +  for some small  0  ε >  , and has only simple poles at  0, 1, 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  s =  − −  , with residues  0  1  2  ,  ,  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  c c c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  If the line of integration in (1) can be closed to the left, the residue theorem provides the  representation  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  0  ( )  n  n  n  g x  c x  ∞  =  = ∑  (2)  for the function  ( )  g x  from (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' this function is a power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' If now  ( )  f s  is an  appropriate holomorphic function on  ( )  Re s  a  ε  <  +  without poles, we can write  ( )  0  1  ( ) ( )  (  )  2  s  n  n  a  n  x  f s G s ds  c f  n x  i  π  ∞  −  =  =  −  ∑  ∫  ,  (3)  when the power series on the right side converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Formulas of this type are used for summation of series or interpolation, and are present in  many publications (see [2, 4, 9, 10] and the references there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  In this note we focus on a special area of applications for the proposition - obtaining some  classical identities by using Mellin inversion and comparing coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' In order to keep the  paper short we omit details and do not discuss convergence of some integrals and series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The  validity of such formulas is considered in [4, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  The illustration of the method is given in the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  2 EXAMPLES  Remind that the residues of the gamma function at zero and the negative integers are given  by  ( 1)  Res( ,  )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  n  n  −  Γ −  =  for  0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Also ( )s  Γ  has rapid decay on vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We have  the estimate ([15, (20)]  (  )  it  a +  Γ  ~  1  2  2  2  | |  ,  a  t  t  e  a  π  π  −  −  ∈\uf0a1  when | |t → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The estimate helps for the convergence of our integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  The above facts will be used in the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We note that in these examples the  lines of integration can be closed to the left (proofs are standard, using the growth estimate for  ( )s  Γ  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Note also that when we replace  ( )  G s  by ( )s  Γ  in (3) we have  ( )  0  1  ( 1)  ( ) ( )  (  )  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  s  n  a  n  x  f s  s ds  f  n x  i  n  π  ∞  −  =  −  Γ  =  −  ∑  ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  (4)  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Ramanujan’s Master Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' As Hardy writes in [8], Ramanujan was very fond  of his integral formula [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='186]  1  0  0  ( )(  )  (  ) ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  s  n  f n  x  x  dx  f  s  s  n  ∞  ∞  −  =  \uf8f1  \uf8fc  −  =  −  Γ  \uf8f2  \uf8fd  \uf8f3  \uf8fe  ∑  ∫  (5)  and used it for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Berndt rightly calls it Ramanujan’s Master Theorem [2,  Entry 11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Details, comments, and applications of (5) are given in these two books and  also in [1, 4, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Clearly, after replacing  ( )  f s  by  (  )  f  s  −  equation (5) turns into (4) after  Mellin inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Ramanujan did not use the residue theorem for his proof but only standard  calculus (see [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='106]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The Hurwitz zeta function is defined by  0  1  ( , )  (Re( )  0, Re( )  1)  (  )s  n  s z  z  s  n  z  ζ  ∞  =  =  >  >  +  ∑  with integral representation  (1  )  1  0  ( , ) ( )  (Re( )  1)  1  t  z  s  t  e  s z  s  t  dt  s  e  ζ  ∞  −  −  Γ  =  >  −  ∫  When  1  z = ,  ( ,1)  ( )  s  s  ζ  ζ  =  is the Riemann zeta function [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  We have also the modified integral representation (argument is the same as on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' 61-62 in  [13])  (1  )  1  0  1  ( , ) ( )  (0  Re( )  1)  1  t  z  s  t  e  s z  s  t  dt  s  e  t  ζ  ∞  −  − \uf8eb  \uf8f6  Γ  =  <  <  −  \uf8ec  \uf8f7  −  \uf8ed  \uf8f8  ∫  which is a Mellin transform formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' By Mellin inversion  (1  )  ( )  0  1  1  ( 1)  ( , ) ( )  (  , )  1  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  t  z  n  s  t  a  n  e  t  s z  s ds  n z  e  t  i  n  ζ  ζ  π  −  ∞  −  =  −  −  =  Γ  =  −  −  ∑  ∫  At the same time, the Bernoulli polynomials  ( )  n  B z  have the generating function  0  ( )  1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  xz  k  k  x  k  B z  xe  x  e  k  ∞  =  =  −  ∑  and from this  (1  )  1  1  1  0  (1  )  (1  )  1  ( , )  1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  (  1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  x  z  k  n  k  n  x  k  n  B  z  B  z  e  g x z  x  x  e  x  k  n  −  ∞  ∞  −  +  =  =  −  −  ≡  −  =  =  −  +  ∑  ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Therefore, by comparing coefficients for  0, 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n =  we find the classical formula  1  1  ( 1)  (1  )  ( )  (  , )  1  1  n  n  n  B  z  B  z  n z  n  n  ζ  +  +  −  −  −  =  = −  +  +  (using the property ( 1)  (1  )  ( )  n  n  n  B  z  B z  −  −  =  , so that  1  1  ( 1)  (1  )  ( )  n  n  n  B  z  B  z  +  +  −  −  = −  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  In particular,  1  1  (0, )  ( )  2  z  B z  z  ζ  = −  =  − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  For the Bernoulli numbers  (0)  ( 1)  (1)  n  n  n  n  B  B  B  =  = −  we have  1  1  ( 1)  (0)  ( 1)  (  )  (  ,1)  1  1  n  n  n  n  B  B  n  n  n  n  ζ  ζ  +  +  −  −  −  =  −  =  =  +  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  The odd Bernoulli numbers are zeros except  1  1/ 2  B = −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Thus  (0)  1/ 2  ζ  = −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  Next we give a new proof of a result of Kenneth Williams and Zhang Nan-Yue [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Consider now the alternating Hurwitz zeta function  0  ( 1)  ( , )  (Re( )  0, Re( )  0)  (  )  n  s  n  s z  z  s  n  z  η  ∞  =  −  =  >  >  +  ∑  which extends to the entire complex plane as analytic in the variable s  and has the integral  representation  (1  )  1  0  ( , ) ( )  (Re( )  0)  1  t  z  s  t  e  s z  s  t  dt  s  e  η  ∞  −  −  Γ  =  >  +  ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  By inversion  (1  )  ( )  0  1  ( 1)  ( , ) ( )  (  , )  1  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  t  z  n  s  n  t  a  n  e  t  s z  s ds  n z x  e  i  n  η  η  π  −  ∞  −  =  −  =  Γ  =  −  +  ∑  ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Euler’s polynomials  ( )  n  E  z  are defined by the generating function  0  2  ( )  (| |  )  1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  t x  n  n  t  n  e  t  E  x  t  e  n  π  ∞  =  =  <  +  ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Comparing coefficients gives  ( 1)  1  (  , )  (1  )  ( )  2  2  n  n  n  n z  E  z  E  z  η  −  −  =  −  =  by the property  (1  )  ( 1)  ( )  n  n  n  E  z  E  z  −  = −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Euler worked with the function  0  ( 1)  ( )  (Re  0)  (2  1)  n  s  n  L s  s  n  ∞  =  −  =  >  +  ∑  which we call here Euler’s L -function (sometimes it is called Dirichlet’s L -function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' This  function has the integral representation  1  0  2 ( ) ( )  cosh  sx  s L s  dx  x  ∞  −  Γ  = ∫  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  It also has analytic extension on the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  By Mellin inversion and using equation (4)  ( )  0  1  1  ( 1)  ( ) ( )  (  )  2cosh( )  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  s  n  a  n  x L s  s ds  L  n x  x  i  n  π  ∞  −  =  −  =  Γ  =  −  ∑  ∫  Euler’s numbers  n  E  are defined by the generating function  0  1  cosh  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  n  n  E x  x  n  ∞  =  = ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  This function is even, so the Euler numbers with odd indices are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' By comparing  coefficients we find  2  1  ( 2 )  (  0,1, 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=')  2  n  L  n  E  n  −  =  =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The exponential polynomials  n  ϕ  are defined by the generating function  (  1)  0  ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  x  n  z e  n  n  x  e  z n  ϕ  ∞  −  =  = ∑  (see [3, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We will use the function  (  1)  0  ( , )  ( 1)  ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  x  n  z e  n  n  n  x  x z  e  z n  ψ  ϕ  −  ∞  −  =  =  =  −  ∑  which has Mellin transform  1  1  0  0  ( , )  ( , )  s  z  s  x  ze  s z  x  x z dx  e  x  e  dx  ψ  ∞  ∞  −  −  −  −  Ψ  =  =  ∫  ∫  1  0  0  0  ( )  ( ) ( , )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  n  z  s  z  s  n  n  nx  z  z  e  x  e  dx  e  s  s f s z  n  n n  ∞  ∞  ∞  −  −  −  −  =  =  \uf8f1  \uf8fc  =  =  Γ  = Γ  \uf8f2  \uf8fd  \uf8f3  \uf8fe  ∑  ∑  ∫  with  0  ( , )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  z  s  n  z  f s z  e  n n  ∞  −  =  =  ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Now we have from equation (4)  ( )  ( )  0  1  1  ( 1)  ( , )  ( , )  ( , ) ( )  (  , )  2  2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  s  s  n  a  a  n  x z  x  x s ds  x  f s z  s ds  f  n z x  i  i  n  ψ  π  π  ∞  −  −  =  −  =  Ψ  =  Γ  =  −  ∑  ∫  ∫  Thus for  0  n ≥  (with the agreent  00  1  = )  0  ( )  (  , )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  z  k  n  k  k  z  f  n z  e  z  k  ϕ  ∞  −  =  =  −  =  ∑  (6)  which is one of the fundamental properties of the exponential polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Note that identity (6) can be used for a meaningful extension of  ( )  n z  ϕ  to  ( )z  λ  ϕ  with non-  integer index λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' For instance, we have (with 0  0  λ =  )  1  ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  z  k  k  k  z  e  z  k  λ  λ  ϕ  ∞  −  =  =  ∑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' The generating function for the Hermite polynomials is  Khristo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev  2  2  0  ( , )  ( )  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  n  xz  x  n  n  x  x z  e  H  z n  ψ  ∞  −  =  =  = ∑  with Mellin transform [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' 27]  2  /2  2  ( , )  2  ( 2 ) ( )  z  s  s  s z  e  D  z  s  −  −  Ψ  =  Γ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  where  p  D  are the parabolic cylinder functions [7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' 1065-1067].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Let now  2  /2  2  ( , )  2  ( 2 )  z  s  s  f s z  e  D  z  −  −  =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  Then  ( )  ( 1)  (  , )  n  n  H  z  f  n z  = −  −  and since  ( )  ( 1)  (  )  n  n  n  H  z  H  z  = −  −  one finds the classical result  2  2  2  ( )  2  ( 2 )  n  z  n  n  H  z  e D  z  =  ([7, entry 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='253, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='1067]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' CONCLUSIONS  In this note we presented a rule how Mellin’s transform can be used to give short proofs of  several classical identities connecting, in particular, the Bernoulli and Euler polynomials to  the values of the Hurwitz and alternating Hurwitz functions at the negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' We also  proved identities for the exponential and Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  REFERENCES  [1]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Amdeberhan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Gonzalez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Harrison, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Moll, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=" Straub, “Ramanujan's Master  Theorem”, The Ramanujan Journal, 29, 103–120 (2012)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  [2]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Berndt, Ramanujan’s Notebooks, Parts I and II, Springer-Verlag, New York (1985,  1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  [3]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Boyadzhiev, “Exponential polynomials, Stirling numbers, and evaluation of some  Gamma integrals”, Abstract and Applied Analysis, Article ID 168672 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  [4]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Dahlquist, “On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-  Christoffel rules, I”, BIT Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
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+page_content=' “On summation formulas  due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules, II”, BIT Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=',  37(4), 804-832 (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' “On Summation Formulas Due to Plana, Lindelöf and Abel, and  Related Gauss-Christoffel Rules, III”, BIT Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' 39, 51–78 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  [5]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Edwards, Riemann’s Zeta Function, Academic Press, Boston, (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  [6]  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' González, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Moll, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Schmidt, “A generalized Ramanujan Master Theorem applied to  the evaluation of Feynman diagrams”, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=', 63, 214-230 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='  [7]  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
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+page_content='  https://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
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+page_content='no/_media/ma3001/2014v/analytisktallteori/the_riemann_zeta_functi  on_notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
+page_content='pdf' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQf0_7l/content/2301.01794v1.pdf'}
diff --git a/W9E1T4oBgHgl3EQfvgXD/content/tmp_files/2301.03401v1.pdf.txt b/W9E1T4oBgHgl3EQfvgXD/content/tmp_files/2301.03401v1.pdf.txt
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+A reappraisal of the principle of equivalent time based on
+physicochemical methods
+M. Rufinoa, A.L. Lixandrão-Filhoa and S. Guedesa,∗
+aDepartamento de Raios Cósmicos e Cronologia, Grupo de Cronologia, Instituto de Física “Gleb Wataghin", Universidade Estadual de
+Campinas, R. Sérgio Buarque de Holanda, 777 - Cidade Universitária, Campinas - SP, 13083-859, Brazil
+A R T I C L E I N F O
+Keywords:
+Fission-track thermochronology
+Equivalent time
+Effective rate constant
+Physicochemical techniques
+A B S T R A C T
+The main feature of the Fission-Track Thermochronology is its ability to infer the thermal histo-
+ries of mineral samples in regions of interest for geological studies. The ingredients that make
+the thermal history inference possible are the annealing models, which capture the annealing ki-
+netics of fission tracks for isothermal heating experiments, and the Principle of Equivalent Time
+(PET), which allows the application of the annealing models to variable temperatures. It turns
+out that the PET only applies to specific types of annealing models describing single activation
+energy annealing mechanisms (parallel models). However, the PET has been extensively applied
+to models related to multiple activation energy mechanisms (fanning models). This procedure is
+an approximation that has been overlooked due to the lack of a suitable alternative. To deal with
+this difficult, a formalism, based on physicochemical techniques, that allows to quantify the ef-
+fects of annealing on the fission tracks for variable temperatures, is developed. It is independent
+of the annealing mechanism and, therefore, is applicable to any annealing model. In the cases
+in which the PET is valid, parallel models, the proposed method and the PET predict the same
+degrees of annealing. However, deviations appear when the methods are applied to the fanning
+models, with the PET underestimating annealing effects. The consequences for the inference of
+thermal histories are discussed.
+1. Introduction
+The Principle of Equivalent Time (PET) is one of the basic ingredients for the inference of thermal histories in
+Fission-Track Thermochronology (FTT). The PET states that the rate of track shortening due to temperature is inde-
+pendent of its previous thermal history. Thus, any thermal history may be replaced with a constant temperature heating
+for an equivalent time resulting in the current fission-track length. Since the models that constrain the annealing ki-
+netics are only applicable to constant temperature heating events, it is the PET that allows for the inference of variable
+thermal histories from the fission-track age and from the distribution of fission-track lengths measured in the sample.
+The PET was first proposed by (Goswami et al., 1984). Later on, Duddy et al. (1988) established a practical method
+of finding thermal histories, applying the PET, that has been mostly unchanged since then. They also demonstrated
+that the PET is only valid for the single activation energy Arrhenius annealing equation. Such equation is represented
+as parallel straight isoretention (same fission-track length) curves on the pseudo-Arrhenius space (logarithm of time as
+a function of inverse temperature, Fig. 1a) and is called Parallel Arrhenius equation. However, they applied the PET to
+the Fanning Arrhenius (FA) model (Laslett et al., 1987), in which the isoretention curves diverge from a single point
+with different slopes, implying different activation energies. They recognized that this procedure is an approximation,
+since the PET only applies to parallel models, but argued that their fanning model deviated only slightly from a parallel
+one.
+Annealing models continued to evolve. Carlson (1990) presented a modified version of the parallel model (Fig. 1a).
+Crowley et al. (1991) proposed versions of the Parallel and Fanning equations that are curved in the pseudo-Arrhenius
+space (Fig. 1b), implying activation energies that vary with the temperature of annealing. The Fanning Curvilinear
+(FC) model has been shown to produce better geological extrapolations than the other models for the apatite (Ketcham
+et al., 2007) and zircon (Guedes et al., 2013) fission-track systems. It is currently the model of choice in most geological
+studies using the FTT (Ketcham, 2019) in thermal history codes relying on the PET to apply the annealing equations for
+∗Principal corresponding author
+rufino@ifi.unicamp.br (M. Rufino); allfilho@ifi.unicamp.br (A.L. Lixandrão-Filho); sguedes@ifi.unicamp.br (S.
+Guedes)
+ORCID(s): 0000-0003-4871-5120 (M. Rufino); 0000-0002-8343-8942 (A.L. Lixandrão-Filho); 0000-0002-7753-8584 (S. Guedes)
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 1 of 15
+arXiv:2301.03401v1  [physics.geo-ph]  6 Jan 2023
+
+athe inference of thermal histories (Ketcham, 2005; Gallagher, 2012). The joint application of the PET and FC equation
+is an approximation that has been overlooked due to the lack of an alternative to deal with the variable temperature
+thermal histories.
+Recently, Rufino and Guedes (2022) applied a physicochemical technique to the Fission-Tack Arrhenius equations
+and were able to formulate the annealing kinetics in terms of the reaction rate constant, which is the fundamental
+quantity related to the activation energy (Arrhenius, 1889; Cohen et al., 2007). The reaction for the annealing process
+is the recombination of displaced atoms and vacant sites that form the track. Once the rate constant is determined for
+the annealing equations, they can be represented in the Arrhenius space (logarithm of the rate constant as a function
+of the inverse temperature) and their trends can be used to retrieve the general mechanisms underlying the Arrhenius
+models. The rate constant encodes the most fundamental features of annealing. Once it is determined, the shortening
+of the fission tracks may in principle be quantified not only for constant temperatures but also for varying ones, being
+an alternative to the PET, without the single activation energy restriction.
+The Arrhenius annealing equations can be derived from the rate constant for the case of constant temperature an-
+nealing (Rufino and Guedes, 2022). The next step is to apply the physicochemical approach to the variable temperature
+annealing of the fission tracks and compare the results to the ones obtained with the PET. The length shortening of
+fission tracks is calculated for cooling temperature-time (T-t) paths with different slopes using the parallel, fanning and
+Carlson models as they are the representations of different activation energy mechanisms. The temperature indexes
+Closure and Total Annealing temperatures, calculated using the PET and the rate constant techniques, are presented
+and compared to illustrate the differences between both approaches for the geological extrapolation.
+2. Method
+2.1. The physicochemical perspective of fission track annealing
+The kinetics of chemical reactions can be described by the Arrhenius equation (Arrhenius, 1889), which relates
+the temperature derivative of the reaction rate 푘, the universal gas constant 푅 and a constant 푞, related to a change in
+the standard internal energy (Laidler, 1984, p.494):
+d ln 푘(푇 )
+d푇
+=
+푞
+푅푇 2 .
+(1)
+Eq. (1) can be solved for the reaction rate as a function of temperature 푘(푇 ) using a pre-exponential factor 퐴 and
+the Arrhenius activation energy 퐸푎:
+푘(푇 ) = 퐴 exp (−퐸푎∕푅푇 ) .
+(2)
+Chemical processes that obey Eq. (2) result in straight lines with slope −퐸푎∕푅 in Arrhenius plots (ln 푘 × 1∕푇 ).
+Among the fission-track annealing models, the Parallel Arrhenius is the only one that actually fits this formulation
+of a single constant activation energy. Deviations from Eq. (2) are quite common. To enable a more complete study
+of chemical reactions, the International Union of Pure and Applied Chemistry (IUPAC) has defined the Arrhenius
+activation energy (Cohen et al., 2007):
+퐸푎 = −푅 d ln(푘)
+d(1∕푇 ).
+(3)
+The Arrhenius activation energy, as defined by Eq. (3), is an empirical quantity aimed to be a kinetic parameter that
+can vary with the temperature of the reaction medium. Its determination depends on the previous knowledge of the rate
+constant, the quantity that encodes the reaction kinetics. Thus, for application to the fission-track system, the reaction
+rate constant associated with the annealing mechanisms must be found, which can be done using the formalism of
+studies in solid state processes (Vyazovkin, 2015).
+The annealing kinetics of fission tracks is described by empirical (Laslett et al., 1987; Crowley et al., 1991; Laslett
+and Galbraith, 1996; Rana et al., 2021) or semi-empirical (Carlson, 1990; Guedes et al., 2006, 2013) equations relating
+the reduced track length, 푟 = 퐿∕퐿0 (where 퐿 is the length of the fission track after heating and 퐿0 is the unannealed
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 2 of 15
+
+fission-track length), with the duration, 푡, of the constant temperature (푇 ) heating. The general form of the annealing
+equations is:
+푔(푟) = 푓(푡, 푇 ),
+(4)
+in which 푔(푟) is a transformation of 푟 and 푓(푡, 푇 ) defines the geometrical characteristics of the isoretention curves
+in the pseudo-Arrhenius space (ln 푡 × 1∕푇 , Fig. 1). The Parallel Arrhenius (PA, Eq. (PA1)) and Fanning Arrhenius
+(FA, Eq. (FA1)) equations (Laslett et al., 1987), the Parallel Curvilinear (PC, Eq. (PC1)) and Fanning Curvilinear
+(FC, Eq. (FC1)) models (Crowley et al., 1991) as well as the Carlson Model (CM, Eq. (CM1)) that mixes the Parallel
+Arrhenius and Parallel Curvilinear models in the same equation (Carlson, 1990), are used in this analysis. The trans-
+formation function 푔(푟) = ln(1 − 푟) was chosen because it carries no fitting parameters and was shown to produce
+good fits to annealing data (Guedes et al., 2022). In addition, it arises naturally from the physicochemical formulation
+of the fission track annealing (Rufino and Guedes, 2022), as will be shown below. More comprehensive descriptions
+of the annealing models can be found elsewhere (Carlson, 1990; Ketcham, 2019; Guedes et al., 2022).
+(a)
+(b)
+Figure 1: Representation of the Arrhenius fission-track annealing models in the pseudo-Arrhenius plot.
+(a) Fanning
+Arrhenius, Parallel Arrhenius, and Carlson models. (b) Fanning Curvilinear and Parallel Curvilinear models. Laboratory
+annealing data are c-axis projected reduced fission-track lengths from Durango apatite (Carlson et al., 1999). Data from
+the geological benchmark KTB (Wauschkuhn et al., 2015) are included only for reference. The models are represented as
+isoretention curves. Points on these curves are the temperature and time of constant temperature heating resulting in the
+same reduced length.
+The annealing data set on the c-axis projected fission tracks for Durango apatite (Carlson et al., 1999) was used for
+model fitting. Durango apatite annealing data was chosen because Durango is a well-known standard sample often used
+in methodological studies (Green et al., 1986; Carlson, 1990; Ketcham et al., 1999, 2007; Rana et al., 2021; Guedes
+et al., 2022; Rufino and Guedes, 2022). The fitting parameters for PA, PC, FA and FC models are the same presented in
+Rufino and Guedes (2022). They were numerically determined using the function nlsLM of the package minpack.lm
+(Elzhov et al., 2016) written in R language, which applies the Levenberg– Marquardt algorithm to minimize the residual
+sum of square (RSS), using the squared inverse of 푟 uncertainties as weights. With the same method, fitting parameters
+were also obtained for the CM model. The fitting parameters are presented in the last column of Table 1.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 3 of 15
+
+40
+Model
+r = 0.55
+=0.9
+ParallelArrhenius
+In(time), time in hours
+Carlson Model
+20
+ Fanning Arrhenius
+88880
+0
+Length reduction exp. data
+-20
+r> 0.9
+A
+0.8 < r <0.9
+0.7 < r < 0.8
+-40
+口
+0.6
+5 < r<0.7
+0
+1
+2
+3
+4
+1000/T, K-140
+r = 0.55
+Model
+r = 0.9
+Parallel Curvilinear
+In(time), time in hours
+AM
+Fanning Curvilinear
+20
+88880
+0
+Length reduction exp. data
+-20
+r> 0.9
+A
+0.8 < r < 0.9
+0.7 < r < 0.8
+-40
+口
+0.6
+5 < r<0.7
+0
+1
+2
+3
+4
+1000/T, K-1Table 1
+Effective reaction rate constant, 푟 reduction and equivalent time equations associated with
+the fission-track annealing models
+FT models
+Equations
+Parameters (standard error)
+Parallel Arrhenius
+(PA)
+푓푃퐴(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐2
+푅푇
+(PA1)
+푘푒푓(푇 )푃퐴 = 푐1푒푐0∕푐1 exp
+(푐2∕푐1
+푅푇
+)
+(PA2)
+퐸푃퐴
+푎
+= −푐2
+푐1
+(PA3)
+푐0 = 5.631 (0.220)
+푐1 = 0.1865 (0.0066)
+푐2 = -10.46 (0.31) kcal/mol
+휒2
+휈 = 2.65
+Parallel curvilinear
+(PC)
+푓푃퐶(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐2 ln
+( 1
+푅푇
+)
+(PC1)
+푘푒푓(푇 )푃퐶 = 푐1푒푐0∕푐1 (푅푇 )−푐2∕푐1
+(PC2)
+퐸푃퐶
+푎 (푇 ) = −푐2
+푐1
+푅푇
+(PC3)
+푐0 = -4.910 (0.096)
+푐1 = 0.1944 (0.0060)
+푐2 = -9.610 (0.244)
+휒2
+휈 = 2.12
+Carlson Model
+(CM)
+푓퐶푀(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐1 ln(푅푇 ) + 푐2
+푅푇
+(CM1)
+푘푒푓(푇 )퐶푀 = 푐1푒푐0∕푐1 exp
+(푐2∕푐1
+푅푇
+)
+푅푇
+(CM2)
+퐸퐶푀
+푎
+(푇 ) = −푐2
+푐1
++ 푅푇
+(CM3)
+푐0 = 5.426 (0.2155)
+푐1 = 0.1867 (0.0066)
+푐2 = -10.25 (0.2994) kcal/mol
+휒2
+휈 = 2.63
+Fanning Arrhenius
+(FA)
+푓퐹퐴(푡, 푇 ) = 푐0 + 푐1
+ln(푡) − 푐2
+1
+푅푇 − 푐3
+(FA1)
+푘푒푓(푡, 푇 )퐹퐴 =
+푐1 exp[(1 − 푛)푓퐹퐴(푡, 푇 )]
+푡(1 − 푐3푅푇 )
+(FA2)
+퐸퐹퐴
+푎 (푡, 푇 ) =
+(푅푇 )2 [푐1(푛 − 1)(푐2 − ln 푡) − 푐3 + 1∕푅푇 ]
+(푐3푅푇 − 1)2
+(FA3)
+푐0 = -8.518 (1.072)
+푐1 = 0.1266 (0.0191) mol/kcal
+푐2 = -20.99 (5.81)
+푐3 = 0.2985 (0.1026) mol/kcal
+휒2
+휈 = 1.66
+0.5 ≤ 푛 < 1
+Fanning curvilinear
+(FC)
+푓퐹퐶(푡, 푇 ) = 푐0 + 푐1
+ln(푡) − 푐2
+ln
+(
+1
+푅푇
+)
+− 푐3
+(FC1)
+푘푒푓(푡, 푇 )퐹퐶 =
+푐1 exp[(1 − 푛)푓퐹퐶(푡, 푇 )]
+푡
+(
+ln
+(
+1
+푅푇
+)
+− 푐3
+)
+(FC2)
+퐸퐹퐶
+푎 (푡, 푇 ) =
+푅푇
+(
+푐1푐2푛 − 푐1푐2 + (푐1 − 푐1푛) ln(푡) − 푐3 + ln
+(
+1
+푅푇
+))
+(
+푐3 − ln
+(
+1
+푅푇
+))2
+(FC3)
+푐0 = -9.449 (1.480)
+푐1 = 0.1627 (0.0298)
+푐2 = -24.58 (7.75)
+푐3 = -0.8626 (0.1549)
+휒2
+휈 = 1.88
+0.5 ≤ 푛 < 1
+Notes: 1. For each fission-track annealing model (Eqs. (PA1), (PC1), (CM1) (FA1), (FC1)), the reaction rate constants,
+푘푒푓 and Arrhenius Activation energies were obtained using 푔(푟) = ln(1 − 푟) and 푓푟(푟) = (1 − 푟)푛. 2. Rate constants (Eqs.
+(PA2), (PC2), (CM2) (FA2), (FC2)) were calculated after Eq. (15). 3. Arrhenius activation energies (Eqs. (PA3), (PC3),
+(CM3) (FA3), (FC3)) were obtained by the application of Eq. (3), and are average values for constant heating
+experiments.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 4 of 15
+
+Fission tracks are formed by displaced atoms and vacant sites, in concentrations high enough to change the structure
+of the mineral in a volume of about 2-10 nm in diameter and around 20 휇m in length. The annealing process is the
+recombination of defects and vacancies, which also changes the neighbor structure and consequently the recombination
+rate. This kind of solid-state reaction is described by the conversion rate equation (Vyazovkin, 2015):
+d훼
+d푡 = 푘(푇 )푓훼(훼).
+(5)
+The Eq. (5) relates the rate of conversion of the reactant 훼 with the constant rate and with the reaction function
+푓훼(훼). For the fission tracks, 훼 is the concentration of recombined atoms, and 푓훼(훼) describes how the recombination
+process changes the surrounding structure. The track length can be used as a proxy for the concentration of displaced
+atoms (Rufino and Guedes, 2022) and:
+훼 = 퐿0 − 퐿
+퐿0
+= 1 − 푟
+(6)
+and with this change of variable:
+d푟
+d푡 = −푘푒푓(푡, 푇 )푓푟(푟).
+(7)
+The rate constant has been replaced with an effective rate constant, 푘푒푓(푡, 푇 ), which may depend on time and
+temperature and is suitable to describe more complex reactions (Vyazovkin, 2016). For the reaction function, the
+reaction-order function has already been shown to produce consistent results mainly for the single activation energy
+mechanisms of annealing (Green et al., 1988; Rufino and Guedes, 2022):
+푓푟 = (1 − 푟)푛
+(8)
+in which 푛 is the reaction order.
+Eq. (7) is a differential equation that can be solved by the separation of variables. To define the limits of the integral,
+consider that at the beginning of the thermal history (푡 = 0), the track is unannealed (푟 = 1). After a heating duration
+푡, the track length has been shortened to 푟. Then:
+∫
+1
+푟
+d푟
+푓푟(푟) = ∫
+푡
+0
+−푘푒푓(푡, 푇 )d푡.
+(9)
+Eq. (9) is the basic equation from which the annealing kinetics can be studied from a physicochemical perspective.
+Once the reaction function and the rate constant are chosen, the dependence of the reduced fission-track length can be
+calculated over any T-t path. Let’s start with the known case of constant temperature heating, from which the annealing
+equations should be obtained. For the single activation energy models, PA, PC, and CM, the rate constants are given
+by:
+푘푒푓(푇 )푃 퐴 = 퐴1 exp
+(−푄1
+푅푇
+)
+,
+(10a)
+푘푒푓(푇 )푃 퐶 = 퐴2(푅푇 )푚,
+(10b)
+푘푒푓(푇 )퐶푀 = 퐴3(푅푇 ) exp
+(
+− 푄3
+푅푇
+)
+,
+(10c)
+where 퐴푖, 푄푖, and 푚 are constants. Eq. (10a) is the original Arrhenius equation from which the PA equation is derived.
+푄1 can be directly identified with the activation energy only in this case. Eq. (10b) generates the PC equation with
+a temperature-dependent activation energy (Table 1, Eq. (PC3)). Eq. (10c) generates the Carlson Model, also with a
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 5 of 15
+
+temperature-dependent activation energy (Table 1, Eq. (CM3)). It is the product of Eqs. (10a) and (10b), with 푚 = 1,
+and has been proposed soon after the original Arrhenius equation to deal with reactions that deviate from the expected
+Arrhenius behavior (Kooij, 1893). Note that although the activation energies in the Eqs. (10b) and (10c) depend on
+temperature, they still fall into the category of single activation energy processes, meaning that all recombination events
+at a given temperature have the same activation energy.
+Annealing experiments are isothermal heating procedures. Then, substituting the effective rate constants (Eqs. (10))
+into the integral equation (Eq. 9) together with the reaction-order function defined in Eq. (8) and solving it considering
+the temperature as a constant results in:
+ln(1 − 푟) =
+ln [퐴1(1 − 푛)]
+1 − 푛
++
+1
+1 − 푛 ln(푡) −
+푄1
+1 − 푛
+1
+푅푇 ,
+(11a)
+ln(1 − 푟) = ln[퐴2(1 − 푛)]
+1 − 푛
++
+1
+1 − 푛 ln(푡) −
+푚
+1 − 푛 ln
+( 1
+푅푇
+)
+,
+(11b)
+ln(1 − 푟) =
+ln [퐴3(1 − 푛)]
+1 − 푛
++
+1
+1 − 푛 ln(푡) −
+푄3
+1 − 푛
+1
+푅푇 −
+1
+1 − 푛 ln
+( 1
+푅푇
+)
+,
+(11c)
+which are the equations for the PA (11a), PC (11b), CM (11c) models with 푔(푟) = ln(1 − 푟). For the chosen reaction
+function, the integral only has a real solution if 푛 < 1. Comparing the right sides of these equations respectively with
+Eqs. (PA1), (PC1), and (CM1), one can find out that the rate constant parameters are related to the fitting parameters
+of the annealing equations as
+PA ∶ 푛 = 푐1 − 1
+푐1
+푄1 = −푐2
+푐1
+퐴1 = 푐1 exp(푐0∕푐1
+)
+(12)
+PC ∶ 푛 = 푐1 − 1
+푐1
+푚 = −푐2
+푐1
+퐴2 = 푐1 exp(푐0∕푐1
+)
+(13)
+CM ∶ 푛 = 푐1 − 1
+푐1
+푄3 = −푐2
+푐1
+퐴3 = 푐1 exp(푐0∕푐1
+)
+(14)
+In this way, the rate constants can be expressed in terms of the fitting parameters of the annealing models as shown
+in Eqs. (PA2), (PC2), and (CM2) of Table 1. The values for the reaction order 푛 for the three models are 푛 ≈ −4, in
+agreement with a similar analysis carried out by Green et al. (1988) for the PA model. Therefore, the parallel models
+are not compatible with first-order annealing kinetics, meaning that the neighbor structure has a strong influence on
+the rate of defect recombination during annealing.
+There are no obvious expressions for the rate constant for the fanning models. A physicochemical analysis of their
+trends indicates that multiple concurring processes with different activation energies are occurring during the annealing
+of the fission tracks (Rufino and Guedes, 2022), in agreement with previous suggestions (Green et al., 1988; Tamer
+and Ketcham, 2020). Rufino and Guedes (2022) derived an expression from Eq. (7) to find the effective rate constant
+from the annealing model:
+푘푒푓(푡, 푇 ) = −
+1
+푓푟(푟)
+[휕푔(푟)
+휕푟
+]−1 휕푓(푡, 푇 )
+휕푡
+|||||푇
+(15)
+Eq. (15) provides a direct way to calculate this effective reaction rate constant from the model functions that fit the
+experimental annealing data, 푓(푡, 푇 ) and 푔(푟), and from the reaction function 푓푟(푟). The partial derivative in relation to
+time is taken because the annealing models were designed to describe constant temperature experiments. As a check,
+before applying Eq. (15) to the fanning models, one can show that Eqs. (PA2), (PC2), and (CM2) are found by the
+application of Eq. (15) respectively to Eqs. (PA1), (PC1), and (CM1), with 푔(푟) = ln(1 − 푟) and 푓푟(푟) = (1 − 푟)푛.
+The same procedure can be applied to Eq. (FA1) and (FC1) to find the effective reaction rates respectively for the
+Fanning Arrhenius (Eq. (FA2)) and Fanning Curvilinear (Eq. (FC2)) models. An alternative way to infer the effective
+rate constants for the FA and FC models is departing from the hypothesis that the Arrhenius activation energies and,
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 6 of 15
+
+therefore, the rate constants are dependent on the fission-track reduced length. Then, integration of Eq. (9), on the
+isothermal condition, results in
+− ∫
+푟
+1
+d푟
+푓푟(푟)푘푒푓(푟) = ∫
+푡
+0
+d푡 = 푡.
+(16)
+It can be shown that the primitive functions that make Eq. (16) true for the FA and FC models, with 푓푟(푟) = (1−푛)푛
+and 푔(푟) = ln(1 − 푟) are the ones with the effective rate constants given by Eqs. (FA2) and (FC2). This approach also
+illustrate how the incorporation of the time in the rate constant and, therefore, in the activation energies for the fanning
+models are implied from the dependence of the activation energies on the values of 푟.
+To obtain the reaction order 푛 for the FA and FC models, the effective reaction rate constants given by Eqs. (FA2)
+and (FC2) are integrated with Eq. (9) considering constant temperatures (isothermal experiments):
+∫
+푟
+1
+d푟
+(1 − 푟)푛 = − ∫
+푡
+0
+푐1
+1
+푅푇 − 푐3
+1
+푡 exp
+⎡
+⎢
+⎢⎣
+−(푛 − 1)
+⎛
+⎜
+⎜⎝
+푐0 + 푐1
+ln 푡 − 푐2
+1
+푅푇 − 푐3
+⎞
+⎟
+⎟⎠
+⎤
+⎥
+⎥⎦
+d푡
+(17a)
+∫
+푟
+1
+d푟
+(1 − 푟)푛 = − ∫
+푡
+0
+푐1
+ln
+(
+1
+푅푇
+)
+− 푐3
+1
+푡 exp
+⎡
+⎢
+⎢
+⎢⎣
+−(푛 − 1)
+⎛
+⎜
+⎜
+⎜⎝
+푐0 + 푐1
+ln 푡 − 푐2
+ln
+(
+1
+푅푇
+)
+− 푐3
+⎞
+⎟
+⎟
+⎟⎠
+⎤
+⎥
+⎥
+⎥⎦
+d푡
+(17b)
+With the necessary condition of 푛 < 1, the solution of the integral equation (17a) is
+(1 − 푟) = (−1)−1∕(푛−1) exp
+⎡
+⎢
+⎢⎣
+푐0 + 푐1
+ln(푡) − 푐2
+1
+푅푇 − 푐3
+⎤
+⎥
+⎥⎦
+(18)
+As the solution of this equation is to represent the shortening of the fission tracks, (1 − 푟) must be a real value
+between 0 and 1, which is true only if −1∕(푛 − 1) is an even and positive integer value 2푗. Then, the values of 푛 are
+restricted to
+푛 = 2푗 − 1
+2푗
+,
+(19)
+where 푗 = 1, 2, 3, .... With this condition and 푔(푟) = ln(1 − 푟), the FA model (Table 1, Eq.(FA1)) is recovered. The
+solutions for Eqs. (17a) and (17b) are similar, differing only on the logarithm of 1∕푅푇 for the FC model instead
+of the 1∕푅푇 for the FA model, which are both constants in this case. The previous analysis holds also for the FC
+model. The values of 푛 will be fractional for FA and FC (푛 = 1∕2, 3∕4, 5∕6, 7∕8, ...), according to Eq. (19). Fractional
+reaction orders are characteristics of multiple-step reactions or some more complex kinetic mechanism, as it has been
+explained for the decomposition of acetaldehyde (Laidler et al., 1965), a well know example of fractional reaction order
+in chemistry. However, for the fission tracks, where the displaced atoms and vacant sites take the role of reactants and
+the deformed track structure is the reaction medium, explanations of the kinetics of a single reactant via a mean-
+field approximation (MFA) may not be appropriate (Córdoba-Torres et al., 2003). Thus, for the effective reaction
+rate constant 푘푒푓 of the fanning annealing models, mechanistic modeling considering the intermediate steps, i.e.,
+recognizing the reaction order of each mechanism involved in annealing, would be desirable to elucidate the meaning
+of the fractional reaction order found (Koga et al., 1992). The rate constants for FA and FC are to be viewed as effective
+equations constraining the general behavior of annealing but that does not allow the description of the specifics of the
+annealing kinetics.
+In this physicochemical framework of the fission-track annealing, the effective reaction rate constant, 푘푒푓(푡, 푇 ),
+and the reaction function, 푓푟(푟), are the fundamental building blocks from which the fission-track annealing kinetics
+can be studied. The application to the constant temperature annealing made it possible to determine the rate constant
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 7 of 15
+
+parameters from the empirically determined parameters of the annealing equations. The calculation of the Arrhenius
+activation energies (퐸푎) for different models becomes possible through Eq. (3). The Arrhenius activation energies of
+the parallel annealing models (Eq. PA3, PC3 and CM3) will be constants with respect to the variable 푡. As for the
+fanning equations, 퐸푎 will vary with time and temperature (Eq. FA3 and FC3). However, the main advantage of this
+approach is the possibility of calculating the fission-track length reduction over any T-t path using Eq. (9), without
+recurring to the interactive application of the Principle of Equivalent Time.
+2.2. Fission track annealing under variable temperature thermal histories
+Fission-track thermal history inference is based on the Principle of Equivalent Time (PET) (Goswami et al., 1984;
+Duddy et al., 1988), which is an interactive method that allows the application of isothermal annealing models to
+variable temperature T-t paths. It is detailed in Appendix A. In general, a given variable temperature thermal history
+is divided into finite time intervals Δ푡푖, centered at times 푡푖 and temperatures 푇푖. At the time interval in which the
+population was born, a first reduced length is calculated by applying the annealing equation, using the temperature of
+the T-t path and the duration of the interval (푇푖, Δ푡푖). In the next interval, at a different temperature on T-t path, the
+annealing model is used to find an equivalent time capable of producing the same length shortening of the previous
+interval but at the new temperature. A new length shortening is then calculated by applying the annealing model to
+the period of time that is the sum of the equivalent time and the length of the time interval. This procedure is repeated
+and at any given temperature, 푇푖 on the T-t path, an equivalent time, 휏푖, which reproduces the length shortening at the
+previous interval, 푟푖−1, is determined, so that the new length shortening can be calculated as if the track had been at
+the same constant temperature from the beginning. The reduced length is updated (푟푖) by calculating it as a result of
+heating at 푇푖 for the duration 휏푖 + Δ푡푖. The hypothesis that the annealing kinetics does not depend on the previous
+thermal history of the track, but only on its current length so that any previous T-t path can be replaced with a constant
+temperature heating resulting in this length, is the basis of this procedure and defines the Principle of Equivalent Time.
+This means, in practice, that the track will have no memory of the material conditions of time and temperature of its
+previous shortenings. The equations for the application of the PET with the PA, PC, CM, FA, and FC models can be
+found in Table A1 in Appendix A.
+The physicochemical tool presented in the previous section provides an alternative way to access variable temper-
+ature annealing kinetics by solving the integral in the right side of Eq. (9) over a T-t path. Eq. (9) is solved as a line
+integral. A suitable parameterization is:
+푠 =
+{
+푇 = 푇 (푢)
+푡 = 푢
+,
+d푠 =
+√
+1 +
+(d푇
+d푢
+)2
+d푢.
+(20)
+Implementing the parameterized variables on the right side of the integral equation (Eq. 9),
+퐼 = ∫
+푡
+0
+푘푒푓 (푡(푢), 푇 (푢))
+d푠
+√
+1 +
+(
+d푇
+d푢
+)2
+⟹ 퐼 = ∫
+푡
+0
+푘푒푓 (푡(푢), 푇 (푢)) d푢.
+(21)
+Solving the left side of Eq. (9) for the 푓푟(푟) function given by Eq. (8) and the parameterized integral for the rate
+constant (Eq. (21)), the reduced length, after the track has experienced the thermal history given by the T-t path, is
+푟 = 1 −
+(
+(1 − 푛) ∫
+푡
+0
+푘푒푓 (푡(푢), 푇 (푢)) d푢
+)1∕1−푛
+,
+(22)
+in which 푛 < 1 as usual. At a first glance, the advantage of the Rate Constant path Integral (RCI, Eq. (22)) is that it is a
+one-shot calculation of the reduced track length. In addition, there was no need to restrict the form of the rate constant
+function and therefore the annealing mechanism it is related to.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 8 of 15
+
+3. Results and Discussion
+The RCI Eq. (22) can be applied to calculate the shortening in the reduced length of a single fission-track population
+submitted to any T-t path. The same calculation can be carried out using the interactive technique based on the Principle
+of Equivalent Time (PET). To compare the outcomes of the two methods, both calculations will be performed for the
+parallel (including CM) and fanning models. The case will be made for the linear cooling, 푇 (푡) = 푇0 − ̇푇 푡, where ̇푇
+is the cooling rate and 푇0 is the temperature at the time the track was generated. The temperature at the end of the T-t
+path (present time) was fixed to be 20 ◦C (293.15 K) for this analysis.
+3.1. Parallel models
+To solve the RCI, the effective rate constant functions for the parallel models (Table 1, Eqs. (PA2), (PC2), and
+(CM2)) are inserted in Eq. (22) with the variable 푇 replaced with 푇 (푡) = 푇0 − ̇푇 푡 wherever it appears. The analytical
+solutions for the reduced length shortening calculated for the PA, PC, and CM are
+푟푃 퐴 = 1 −
+⎛
+⎜
+⎜
+⎜
+⎜⎝
+푒푐0∕푐1
+(
+푐2Ei
+(
+푐2
+푐1푅(푇0− ̇푇 푡)
+)
+− 푐2Ei
+(
+푐2
+푐1푅푇0
+)
++ 푐1푅
+(
+( ̇푇 푡 − 푇0)푒
+푐2
+푐1푅(푇0− ̇푇 푡) + 푇0푒
+푐2
+푐1푅푇0
+))
+푐1 ̇푇 푅
+⎞
+⎟
+⎟
+⎟
+⎟⎠
+푐1
+,
+(23a)
+푟푃 퐶 = 1 −
+⎛
+⎜
+⎜
+⎜
+⎜⎝
+푐1푒푐0∕푐1
+(
+( ̇푇 푡 − 푇0)(푅(푇0 − ̇푇 푡))
+− 푐2
+푐1 + 푇0(푅푇0)
+− 푐2
+푐1
+)
+푐1 ̇푇 − 푐2 ̇푇
+⎞
+⎟
+⎟
+⎟
+⎟⎠
+푐1
+,
+(23b)
+푟퐶푀 = 1 − 2−푐1푒푐0 푐−2푐1
+1
+( ̇푇 푅)푐1
+[
+푐1푅
+(
+푇0푒
+푐2
+푐1푅푇0 (푐1푅푇0 + 푐2) − (푇0 − ̇푇 푡)푒
+푐2
+푐1푅푇0−푐1 ̇푇 푅푡 (푐1푅(푇0 − ̇푇 푡) + 푐2)
+)
++푐2
+2
+(
+Ei
+(
+푐2
+푐1푅푇0 − 푐1푅푡 ̇푇
+)
+− Ei
+(
+푐2
+푐1푅푇0
+))]푐1
+,
+(23c)
+where Ei is the exponential integral function. Eqs. (23a) - (23c) give the resulting reduced length 푟 for the parallel
+models, as functions of the three variables that characterize the thermal history: the duration of the T-t path (푡), the
+cooling rate ( ̇푇 ), and the temperature at the time when the track was born (푇0). The parameters 푐푖 are given in the last
+column of Table 1. The values of 푟 for the cooling path with the cooling rate ̇푇 = 1.0◦C/Ma calculated with the three
+parallel models are presented in Fig. 2. For each point, the value of 푟 is the length reduction after a linear cooling
+duration 푡 and measured in the present. Values of 푟 = 0 mean that the tracks have been erased before the present.
+Values obtained by the RCI solutions (Eqs. (23a) - (23c)) are represented as red curves marked with red circles and the
+values calculated using the PET are represented as blue curves marked with blue squares. RCI and PET calculations
+produce very close values of 푟 for the three parallel models (Figs. 2a, 2b, 2c).
+The temperature indexes Closure Temperature (푇퐶) and Total Annealing Temperature (푇퐴) were also calculated
+for the three parallel models, applying both methods of calculation, for cooling T-t paths with cooling rates of 1, 10,
+and 100 ◦C/Ma. 푇퐶 is, for a monotonic cooling thermal history, the temperature at the apparent sample age (Dodson,
+1973). 푇퐴 is the age of the oldest track that has not been erased and can be counted in the sample (Issler, 1996).
+Details for the method of calculation of the index temperatures can be found in Guedes et al. (2013). 푇퐶 and 푇퐴 are
+meaningful quantities that allow quantifying the impact of using the RCI instead of the interactive PET calculation.
+The uncertainties in 푇퐶 and in 푇퐴 were estimated by simple error propagation of apparent (푇퐶) or retention (푇퐴) ages
+and present time temperature. Results are shown in Table 2. Setting the PET results as the reference values, given
+that PET is the method established in the literature, a relative error analysis can be carried out to verify the internal
+consistency between PET and RCI calculations. The relative error between PA PET and PA RCI is on average 0.69%
+for 푇퐶 and 1.19% 푇퐴. The same trend is found for calculations of 푇퐶 and 푇퐴 with PC and CM: 0.66% and 0.58% (PC)
+and 0.7% and 1.19% (CM). All errors are well inside the estimated uncertainties for temperature index values and are
+most probably artifacts of the PET numerical calculation.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 9 of 15
+
+,
+(a)
+(b)
+(c)
+Figure 2: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.0◦C/Ma) starting at time 푡 and ending at
+the present time at a fixed temperature of 20◦C, calculated using the Parallel models: (a) Arrhenius, (b) Curvilinear, and
+(c) Carlson). The points that form the curve in red (circle marks) were calculated by applying the RCI (Eq. 23a, 23b, and
+23c). The values calculated using the PET are in blue (square marks).
+Table 2
+Comparison of thermal indexes for the models presented in this work, calculated using the PET and the RCI
+Principle of Equivalent Time (PET) thermal indexes
+Fit
+푇퐶1
+푇퐶10
+푇퐶100
+푇퐴1
+푇퐴10
+푇퐴100
+PA
+130(7)
+144(7)
+158(8)
+153(8)
+168(8)
+184(9)
+PC
+105(6)
+121(6)
+139(7)
+132(8)
+151(8)
+171(9)
+CM
+130(7)
+143(7)
+157(8)
+153(8)
+168(8)
+184(9)
+FA
+134(7)
+146(7)
+160(8)
+163(8)
+176(8)
+191(10)
+FC
+111(6)
+126(6)
+143(7)
+143(7)
+160(8)
+179(9)
+Rate constant integral on a path (RCI) thermal indexes
+Fit
+푇퐶1
+푇퐶10
+푇퐶100
+푇퐴1
+푇퐴10
+푇퐴100
+PA
+131(7)
+145(7)
+159(8)
+155(8)
+170(8)
+186(9)
+PC
+106(6)
+122(6)
+140(7)
+133(8)
+152(7)
+172(9)
+CM
+131(7)
+144(7)
+158(8)
+155(8)
+170(8)
+186(9)
+FA
+125(7)
+137(7)
+150(7)
+153(9)
+166(8)
+181(10)
+FC
+100(6)
+115(6)
+130(6)
+130(8)
+148(8)
+165(8)
+Notes: 1. Temperatures in ◦C. 2. The standard errors are given in parentheses. 3. 푇퐶: Closure Temperature. 4. 푇퐴:
+Total Annealing Temperature. 5. The numbers to the right of 푇퐶 and 푇퐴 are the cooling rates in ◦C/Ma.
+The PET was formulated under the hypothesis that the annealing of fission tracks is a single activation energy pro-
+cess Duddy et al. (1988). The internal consistency between PET and RCI values of 푟, 푇퐶, and 푇퐴 calculated with the
+parallel models is a check for the robustness of the physicochemical approach to deal with variable temperature thermal
+histories. It is to be noted that not only the PA model, in which the activation energy is temperature-independent (Ta-
+ble 1, Eq. (PA3)), but also the PC model, in which the activation energy is temperature-dependent (Table 1, Eqs. (FC3)),
+show such internal consistency. The same agreement is observed for CM. The CM activation energy may vary with
+temperature but, with the parameters shown in Table 1, its Arrhenius activation energy is approximately constant since
+the value of 푐2∕푐1 (54.9 kcal/mol) is much higher than typical values of 푅푇 (< 1.0 kcal/mol). Although the activation
+energies may vary with temperature, these models imply that at any given temperature, the recombination events are
+taking place with the same activation energy. This is a sufficient condition for the applicability of the PET.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 10 of 15
+
+1.0
+PA
+S0.8
+Reduction
+0.6
+Method
+Length F
+RCI
+0.4
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)1.0
+PC
+S0.8
+Reduction
+0.6
+Method
+Length F
+RCI
+0.4
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)1.0
+CM
+C0.8
+Reduction
+0.6
+Method
+Length F
+RCI
+0.4
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)3.2. Fanning models
+The values of 푟, 푇퐶, and 푇퐴 for the cooling T-t path were also calculated for the fanning models, using both the
+interactive PET and RCI methods. However, the RCI (Eq. (22)) could not be solved analytically for the FA and FC
+rate constants (Table 1, Eqs. (FA2) and (FC2)). The integrals were then solved numerically with the Wolfram Mathe-
+matica software (Wolfram-Research-Inc., 2021). For validation, the integrals for the parallel models were also solved
+numerically resulting in exactly the same values obtained with the analytical solutions (Eqs. (23a)-(23c)). Another
+feature to be considered is that there are certain fractional values allowed for the reaction order 푛, given by Eq. (19).
+The analysis will be limited to 푛 = 0.5, 푛 = 0.75 and 푛 = 0.9. The numerical method breaks down when 푛 > 0.95
+although its mathematical upper bound is 푛 < 1. The reduced length calculation results are shown in Fig. 3. Values
+calculated with the PET are shown in blue, with triangle marks, while values found by solving the RCI are shown,
+in red (푛 = 0.5), purple (푛 = 0.75), and light purple (푛 = 0.9), respectively with circle, square, and diamond marks.
+RCI 푟 curves are very close to each other but depart from the 푟 values calculated with the PET. Significant differences
+between RCI and PET 푟 values are observed for the FC (Fig. 3b) and for the FA (Fig. 3a) models.
+(a)
+(b)
+Figure 3: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.0◦C/Ma) starting at time 푡 and ending at
+the present time at a fixed temperature of 20◦C, calculated using the Fanning models: (a) Arrhenius and (b) Curvilinear.
+The points that form the curves in red (circle marks), purple (square marks), and light purple (diamond marks) were
+calculated by applying the RCI, respectively with 푛 = 0.5, 푛 = 0.75, and 푛 = 0.9. The values calculated using the PET are
+in blue.
+The values of 푇퐶 and 푇퐴 for the fanning models, calculated using the PET and the RCI (푛 = 0.5) are in Table 2. For
+the FA model, the mean relative errors between the PET and the RCI 푇퐶 and 푇퐴 calculations are respectively 6.25%
+and 5.68%. For the FC model, the same comparisons result in still more significant differences: 9.57% (푇퐶) and 8.48%
+(푇퐴). The deviations between the values calculated using the PET and the RCI are much more significant than the ones
+found for the parallel model calculations. One major issue is that the fanning models do not fulfill the single activation
+energy hypothesis on which the PET is founded. The fanning models emerge from multiple concurring processes with
+different activation energies (Tamer and Ketcham, 2020; Rufino and Guedes, 2022). The effective Arrhenius activation
+energies incorporate a time dependence (Table 1, Eqs. (FA3) and (FC3)) that is the consequence of their dependence
+on the current reduced fission-track length (different slopes of the isoretention curves in the pseudo-Arrhenius plot).
+On the other hand, the rate constant integral (Eq. (22)) was obtained in a physicochemical framework developed to
+deal with chemical reactions that did not fit the single activation energy Arrhenius law (Vyazovkin, 2015). It is by
+design suitable for complex activation energy systems like the ones pictured by the fanning models. Note also that
+the presented figures are particular of the fitting parameters in Table 1. A different set of parameters would result in
+different values without changing the conclusion that RCI and PET predictions deviate from each other.
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 11 of 15
+
+1.0
+FA
+0.8
+Reduction
+0.6
+Method
+RCI (0.5)
+Length F
+RCI (0.75)
+0.4
+RCI (0.9)
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)1.0
+FC
+0.8
+Reduction
+0.6
+Method
+RCI (0.5)
+Length F
+RCI (0.75)
+0.4
+RCI (0.9)
+PET
+0.2
+0.0
+50
+100
+150
+200
+Thermal history duration (t)3.3. Implications for the thermal history modeling
+The fanning models, especially the Fanning Curvilinear, have been shown to produce better fits to laboratory data
+and better geological extrapolation of annealing effects (Ketcham et al., 2007; Guedes et al., 2013). However, the
+application of the FC along with the PET is an approximation. Compared with the RCI formulation, it underestimates
+the annealing effect in about 10%, i.e., it predicts that higher temperatures are necessary for the same length shortening
+as calculated with the RCI for the tested cooling histories. In the context of the inverse problem of inferring T-t paths
+from the FT age and track length distribution of a mineral sample, it implies the requirement of a longer residence
+time in the partial annealing zone. For instance, compare, in Table 2, the FC 푇퐴 calculated with RCI for a cooling rate
+of 10◦C/Ma (148◦C) with the FC 푇퐴 calculated with PET for a cooling rate of 1◦C/Ma (143◦C). The same analysis
+applies to the FA model with less significant relative error figures (about 6%).
+The Parallel models (PA, PC, and CM), which can be safely applied along with the PET, have long been ruled out for
+FTT studies (Laslett et al., 1987; Guedes et al., 2013; Ketcham et al., 1999, 2007; Ketcham, 2019). Duddy et al. (1988)
+had argued that the FA deviated only slightly from the PA model and applied it along with the PET. The isoretention
+curves for the two models follow approximately the same trends (Fig. 1a). The same behavior is observed for the
+curvilinear models (Fig. 1b). FC and PC isoretention curves bend together towards lower temperatures. Their argument
+can be better appreciated in Fig. 4. All the PET and RCI predictions for reduced lengths after the track underwent the
+cooling history are gathered in the same plot. Note that the linear models (PA and FA) and the approximately linear CM
+form a cluster, while the curvilinear models (PC and FC) form a separate set. The predictions with the fanning models
+and PET are closer to the predictions of the parallel models for track populations born when the sample passed through
+intermediate temperatures (partial annealing zone), which results in closer 푇퐶 values (compare values in Table 2). For
+populations born at higher temperatures, the fanning-PET predictions depart from the parallel model ones, resulting in
+a more significant difference between calculated 푇퐴 values. Calculations with RCI approximate fanning and parallel
+model predictions for populations born at higher temperatures. Within this approximation, it could be possible to
+engineer fanning model parameters to make the model even closer to a parallel model.
+Figure 4: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.0◦C/Ma) starting at time 푡 and ending at
+the present time at a fixed temperature of 20◦C, calculated for the parallel and fanning models. Calculations using RCI
+are shown as solid geometric forms, while calculations using PET are represented by empty geometric forms.
+4. Concluding remarks
+Departing from Eq. (9), a physicochemical framework was built to deal with the effects of annealing in variable tem-
+perature T-t paths. The basic building blocks are the reaction function 푓푟(푟) and the effective rate constant, 푘푒푓(푡, 푇 ).
+The parallel models (PA, PC, and CM) were shown to be consistent with the single activation energy rate constants
+given by Eqs. (10a)-(10c) and with the reaction-order function (Eq. (8)). The fanning models (FA and FC) are the
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 12 of 15
+
+1.0
+0.8
+Reduction
+0.6
+RCI Method
+PET Method
+Length F
+0.4
+PA
+0
+PA
+PC
+FA
+CM
+CM
+FA
+口
+PC
+0.2
+FC
+FC
+0.0
+50
+100
+150
+200
+Thermal history duration (t)representation of multiple concurrent recombination processes with different activation energies (Rufino and Guedes,
+2022). The 푘푒푓(푡, 푇 ) functions were built (Eq. (15)) to be consistent with the reaction-order function (Eq.(8)). Obtain-
+ing FA and FC rate constants from first principle, i.e., a composition of rate constants for individual processes, and
+validating them experimentally is still an open issue that has to be dealt with. The Eq. (22) is the line integral related
+to Eq. (9) from which length shortening due to cooling T-t paths can be directly calculated, independently of whether
+the rate constant represents a single or multiple activation energy mechanism.
+The Principle of Equivalent Time, on the other hand, is only valid for single activation energy equations, which, for
+the fission-track system, are the parallel models. In these cases, the RCI-based calculations are in agreement with the
+PET ones (Fig. 2), indicating the robustness of RCI formulation. For the fanning models, the use of the PET has long
+been recognized as an approximation (Duddy et al., 1988). Deviations have indeed been observed between RCI and
+PET-based calculations (Fig. 3). Compared to the application of RCI, the PET calculation underestimates annealing
+effects in variable temperature T-t paths (Table 2).
+The PET along with FA or FC models is the calculation method used to infer most published thermal histories.
+This procedure introduces a systematic deviation that should be considered in the geological interpretation of the
+thermal history modeling. Alternatively, the rate constant integral (Eq. (22)) could be considered to substitute the
+PET in inversion thermal history codes (Ketcham, 2005; Gallagher, 2012). Computationally, solving an integral,
+even numerically, is a routine faster than the interactive steps necessary to apply the PET. More importantly, if the
+rate constants are representative of the track annealing kinetics, this framework results, in principle, in more accurate
+predictions of the annealing effects in samples submitted to variable temperature thermal histories.
+Appendix
+A. Equations for the application of the Principle of Equivalent Time
+The proposed method to deal with annealing in thermal histories with variable temperatures is also based on the
+Arrhenius equation and was first proposed by Goswami et al. (1984). The Principle of Equivalent Time (PET), on
+which this method is founded, states that the annealing rate of a track does not depend on its previous thermal history,
+but only on its current length.
+The procedure to solve this problem is to infer the magnitude of annealing recursively, dividing the thermal history
+into appropriate intervals Δ푡푖 and starting from a given value of 푟. For each step, annealing is carried out at constant
+temperature. In the first step, centered at (푡1, 푇1). The procedure will be shown for the Parallel Arrhenius model. The
+reduced fission-track length is calculated using Eq. (PA1), along with 푔(푟) = ln(1 − 푟):
+푟1 = 1 − (Δ푡1)푐1 exp
+(
+푐0 + 푐2
+푅푇1
+)
+(24)
+For the next step, the calculation of 푟2 incorporates the principle in the form that this new shortening is postulated
+to be independent of the previous one. Thus, one can find a length of time, 휏푟1, that will yield the length 푟1 but for
+heating at the temperature of the second interval, 푇2:
+휏푟1 = (1 − 푟1)−1∕푐1 exp
+(
+− 1
+푅푇2
+푐2
+푐1
++ 푐0
+푐1
+)
+.
+(25)
+The value of 푟2 is then found by the application of the annealing model to the interval 휏푟1 + Δ푡2:
+푟2 = 1 − (휏푟1 + Δ푡2)푐1 exp
+(
+푐0 + 푐2
+푅푇2
+)
+.
+(26)
+This procedure is interactively repeated for the entire T-t path. The last value of 푟 will be the reduced length of
+the population born in the first interval after experiencing the entire thermal history. For each interval 푗, the formulas
+above are:
+푟푗
+= 1 − (휏푟푗−1 + Δ푡푗)푐1 exp
+(
+푐0 +
+푐2
+푅푇푗
+)
+(27a)
+M. Rufino, A.L. Lixandrão-Filho, S. Guedes
+Page 13 of 15
+
+Table A1
+Equations for the application of the Principle of Equivalent Time
+FT models
+Equations
+Parameters (standard error)
+Parallel Arrhenius
+(PA)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓푃 퐴
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(PA4)
+ln
+(
+휏푟푗−1
+)
+=
+ln (1 − 푟푗−1
+)
+푐1
+−
+푐2
+푐1푅푇푗
+− 푐0
+푐1
+(PA5)
+푐0 = 5.631 (0.220)
+푐1 = 0.1865 (0.0066)
+푐2 = -10.46 (0.31) kcal/mol
+휒2
+휈 = 2.65
+Parallel curvilinear
+(PC)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓푃 퐶
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(PC4)
+ln
+(
+휏푟푗−1
+)
+=
+ln (1 − 푟푗−1
+)
+푐1
+− 푐2
+푐1
+ln
+(
+1
+푅푇푗
+)
+− 푐0
+푐1
+(PC5)
+푐0 = -4.910 (0.096)
+푐1 = 0.1944 (0.0060)
+푐2 = -9.610 (0.244)
+휒2
+휈 = 2.12
+Carlson Model
+(CM)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓퐶푀
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(CM4)
+ln
+(
+휏푟푗−1
+)
+=
+ln (1 − 푟푗−1
+)
+푐1
+−
+푐2
+푐1푅푇푗
+− 푐0
+푐1
+− ln (푅푇푗
+)
+(CM5)
+푐0 = 5.426 (0.2155)
+푐1 = 0.1867 (0.0066)
+푐2 = -10.25 (0.2994) kcal/mol
+휒2
+휈 = 2.63
+Fanning Arrhenius
+(FA)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓퐹퐴
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(FA4)
+ln
+(
+휏푟푗−1
+)
+=
+[ln (1 − 푟푗−1
+) − 푐0
+] [
+1
+푅푇푗 − 푐3
+]
+푐1
++ 푐2
+(FA5)
+푐0 = -8.518 (1.072)
+푐1 = 0.1266 (0.0191) mol/kcal
+푐2 = -20.99 (5.81)
+푐3 = 0.2985 (0.1026) mol/kcal
+휒2
+휈 = 1.66
+Fanning curvilinear
+(FC)
+ln (1 − 푟푗−1
+) = 1 − exp
+[
+푓퐹퐶
+(
+Δt푗 + 휏푟푗−1, 푇푗
+)]
+(FC4)
+ln
+(
+휏푟푗−1
+)
+=
+[ln (1 − 푟푗−1
+) − 푐0
+] [
+ln
+(
+1
+푅푇푗
+)
+− 푐3
+]
+푐1
++ 푐2
+(FC5)
+푐0 = -9.449 (1.480)
+푐1 = 0.1627 (0.0298)
+푐2 = -24.58 (7.75)
+푐3 = -0.8626 (0.1549)
+휒2
+휈 = 1.88
+Notes: 1. For each fission-track annealing model (Eqs. (PA1), (PC1), (FA1), (FC1)), the length reduction 푟 (Eqs. (PA4),
+(PC4), (FA4), (FC4)) and equivalent time 휏 (Eqs. (PA5), (PC5), (FA5), (FC5)) were obtained using 푔(푟) = ln(1 − 푟).
+휏푟푗−1
+= (1 − 푟푗−1)−1∕푐1 exp
+(
+− 1
+푅푇푗
+푐2
+푐1 + 푐0
+푐1
+)
+(27b)
+The formulas for the PA, PC, CM, FA, and FC models are presented in Table A1.
+Acknowledgements
+This work has been funded by grant 308192/2019-2 by the National Council for Scientific and Technological
+Development (Brazil).
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+Page 16 of 15
+
diff --git a/W9E1T4oBgHgl3EQfvgXD/content/tmp_files/load_file.txt b/W9E1T4oBgHgl3EQfvgXD/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf,len=1110
+page_content='A reappraisal of the principle of equivalent time based on  physicochemical methods  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufinoa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filhoa and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedesa,∗  aDepartamento de Raios Cósmicos e Cronologia, Grupo de Cronologia, Instituto de Física “Gleb Wataghin", Universidade Estadual de  Campinas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Sérgio Buarque de Holanda, 777 - Cidade Universitária, Campinas - SP, 13083-859, Brazil  A R T I C L E I N F O  Keywords:  Fission-track thermochronology  Equivalent time  Effective rate constant  Physicochemical techniques  A B S T R A C T  The main feature of the Fission-Track Thermochronology is its ability to infer the thermal histo-  ries of mineral samples in regions of interest for geological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The ingredients that make  the thermal history inference possible are the annealing models, which capture the annealing ki-  netics of fission tracks for isothermal heating experiments, and the Principle of Equivalent Time  (PET), which allows the application of the annealing models to variable temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' It turns  out that the PET only applies to specific types of annealing models describing single activation  energy annealing mechanisms (parallel models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' However, the PET has been extensively applied  to models related to multiple activation energy mechanisms (fanning models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' This procedure is  an approximation that has been overlooked due to the lack of a suitable alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' To deal with  this difficult, a formalism, based on physicochemical techniques, that allows to quantify the ef-  fects of annealing on the fission tracks for variable temperatures, is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' It is independent  of the annealing mechanism and, therefore, is applicable to any annealing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' In the cases  in which the PET is valid, parallel models, the proposed method and the PET predict the same  degrees of annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' However, deviations appear when the methods are applied to the fanning  models, with the PET underestimating annealing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The consequences for the inference of  thermal histories are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Introduction  The Principle of Equivalent Time (PET) is one of the basic ingredients for the inference of thermal histories in  Fission-Track Thermochronology (FTT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The PET states that the rate of track shortening due to temperature is inde-  pendent of its previous thermal history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Thus, any thermal history may be replaced with a constant temperature heating  for an equivalent time resulting in the current fission-track length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Since the models that constrain the annealing ki-  netics are only applicable to constant temperature heating events, it is the PET that allows for the inference of variable  thermal histories from the fission-track age and from the distribution of fission-track lengths measured in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The PET was first proposed by (Goswami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Later on, Duddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (1988) established a practical method  of finding thermal histories, applying the PET, that has been mostly unchanged since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' They also demonstrated  that the PET is only valid for the single activation energy Arrhenius annealing equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Such equation is represented  as parallel straight isoretention (same fission-track length) curves on the pseudo-Arrhenius space (logarithm of time as  a function of inverse temperature, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 1a) and is called Parallel Arrhenius equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' However, they applied the PET to  the Fanning Arrhenius (FA) model (Laslett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1987), in which the isoretention curves diverge from a single point  with different slopes, implying different activation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' They recognized that this procedure is an approximation,  since the PET only applies to parallel models, but argued that their fanning model deviated only slightly from a parallel  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Annealing models continued to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Carlson (1990) presented a modified version of the parallel model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Crowley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (1991) proposed versions of the Parallel and Fanning equations that are curved in the pseudo-Arrhenius  space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 1b), implying activation energies that vary with the temperature of annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The Fanning Curvilinear  (FC) model has been shown to produce better geological extrapolations than the other models for the apatite (Ketcham  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2007) and zircon (Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2013) fission-track systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' It is currently the model of choice in most geological  studies using the FTT (Ketcham, 2019) in thermal history codes relying on the PET to apply the annealing equations for  ∗Principal corresponding author  rufino@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='br (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' allfilho@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='br (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' sguedes@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='br (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Guedes)  ORCID(s): 0000-0003-4871-5120 (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 0000-0002-8343-8942 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 0000-0002-7753-8584 (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes)  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 1 of 15  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='03401v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='geo-ph]  6 Jan 2023  athe inference of thermal histories (Ketcham, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Gallagher, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The joint application of the PET and FC equation  is an approximation that has been overlooked due to the lack of an alternative to deal with the variable temperature  thermal histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Recently, Rufino and Guedes (2022) applied a physicochemical technique to the Fission-Tack Arrhenius equations  and were able to formulate the annealing kinetics in terms of the reaction rate constant, which is the fundamental  quantity related to the activation energy (Arrhenius, 1889;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The reaction for the annealing process  is the recombination of displaced atoms and vacant sites that form the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Once the rate constant is determined for  the annealing equations, they can be represented in the Arrhenius space (logarithm of the rate constant as a function  of the inverse temperature) and their trends can be used to retrieve the general mechanisms underlying the Arrhenius  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The rate constant encodes the most fundamental features of annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Once it is determined, the shortening  of the fission tracks may in principle be quantified not only for constant temperatures but also for varying ones, being  an alternative to the PET, without the single activation energy restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The Arrhenius annealing equations can be derived from the rate constant for the case of constant temperature an-  nealing (Rufino and Guedes, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The next step is to apply the physicochemical approach to the variable temperature  annealing of the fission tracks and compare the results to the ones obtained with the PET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The length shortening of  fission tracks is calculated for cooling temperature-time (T-t) paths with different slopes using the parallel, fanning and  Carlson models as they are the representations of different activation energy mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The temperature indexes  Closure and Total Annealing temperatures, calculated using the PET and the rate constant techniques, are presented  and compared to illustrate the differences between both approaches for the geological extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Method  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The physicochemical perspective of fission track annealing  The kinetics of chemical reactions can be described by the Arrhenius equation (Arrhenius, 1889), which relates  the temperature derivative of the reaction rate 푘, the universal gas constant 푅 and a constant 푞, related to a change in  the standard internal energy (Laidler, 1984, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='494):  d ln 푘(푇 )  d푇  =  푞  푅푇 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (1)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (1) can be solved for the reaction rate as a function of temperature 푘(푇 ) using a pre-exponential factor 퐴 and  the Arrhenius activation energy 퐸푎:  푘(푇 ) = 퐴 exp (−퐸푎∕푅푇 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (2)  Chemical processes that obey Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (2) result in straight lines with slope −퐸푎∕푅 in Arrhenius plots (ln 푘 × 1∕푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Among the fission-track annealing models, the Parallel Arrhenius is the only one that actually fits this formulation  of a single constant activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Deviations from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (2) are quite common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' To enable a more complete study  of chemical reactions, the International Union of Pure and Applied Chemistry (IUPAC) has defined the Arrhenius  activation energy (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2007):  퐸푎 = −푅 d ln(푘)  d(1∕푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (3)  The Arrhenius activation energy, as defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (3), is an empirical quantity aimed to be a kinetic parameter that  can vary with the temperature of the reaction medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Its determination depends on the previous knowledge of the rate  constant, the quantity that encodes the reaction kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Thus, for application to the fission-track system, the reaction  rate constant associated with the annealing mechanisms must be found, which can be done using the formalism of  studies in solid state processes (Vyazovkin, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The annealing kinetics of fission tracks is described by empirical (Laslett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Crowley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Laslett  and Galbraith, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2021) or semi-empirical (Carlson, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2006, 2013) equations relating  the reduced track length, 푟 = 퐿∕퐿0 (where 퐿 is the length of the fission track after heating and 퐿0 is the unannealed  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 2 of 15  fission-track length), with the duration, 푡, of the constant temperature (푇 ) heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The general form of the annealing  equations is:  푔(푟) = 푓(푡, 푇 ),  (4)  in which 푔(푟) is a transformation of 푟 and 푓(푡, 푇 ) defines the geometrical characteristics of the isoretention curves  in the pseudo-Arrhenius space (ln 푡 × 1∕푇 , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The Parallel Arrhenius (PA, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA1)) and Fanning Arrhenius  (FA, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FA1)) equations (Laslett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1987), the Parallel Curvilinear (PC, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PC1)) and Fanning Curvilinear  (FC, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FC1)) models (Crowley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1991) as well as the Carlson Model (CM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (CM1)) that mixes the Parallel  Arrhenius and Parallel Curvilinear models in the same equation (Carlson, 1990), are used in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The trans-  formation function 푔(푟) = ln(1 − 푟) was chosen because it carries no fitting parameters and was shown to produce  good fits to annealing data (Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' In addition, it arises naturally from the physicochemical formulation  of the fission track annealing (Rufino and Guedes, 2022), as will be shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' More comprehensive descriptions  of the annealing models can be found elsewhere (Carlson, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Ketcham, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (a)  (b)  Figure 1: Representation of the Arrhenius fission-track annealing models in the pseudo-Arrhenius plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (a) Fanning  Arrhenius, Parallel Arrhenius, and Carlson models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (b) Fanning Curvilinear and Parallel Curvilinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Laboratory  annealing data are c-axis projected reduced fission-track lengths from Durango apatite (Carlson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Data from  the geological benchmark KTB (Wauschkuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2015) are included only for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The models are represented as  isoretention curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Points on these curves are the temperature and time of constant temperature heating resulting in the  same reduced length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The annealing data set on the c-axis projected fission tracks for Durango apatite (Carlson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1999) was used for  model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Durango apatite annealing data was chosen because Durango is a well-known standard sample often used  in methodological studies (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Carlson, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Ketcham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1999, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino and Guedes, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The fitting parameters for PA, PC, FA and FC models are the same presented in  Rufino and Guedes (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' They were numerically determined using the function nlsLM of the package minpack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='lm  (Elzhov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2016) written in R language, which applies the Levenberg– Marquardt algorithm to minimize the residual  sum of square (RSS), using the squared inverse of 푟 uncertainties as weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' With the same method, fitting parameters  were also obtained for the CM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The fitting parameters are presented in the last column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 3 of 15  40  Model  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='55  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9  ParallelArrhenius  In(time), time in hours  Carlson Model  20  Fanning Arrhenius  88880  0  Length reduction exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' data  20  r> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8 < r <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='7 < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8  40  口  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='6  5 < r<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='7  0  1  2  3  4  1000/T, K-140  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='55  Model  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9  Parallel Curvilinear  In(time), time in hours  AM  Fanning Curvilinear  20  88880  0  Length reduction exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' data  20  r> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8 < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='7 < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8  40  口  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='6  5 < r<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='7  0  1  2  3  4  1000/T, K-1Table 1  Effective reaction rate constant, 푟 reduction and equivalent time equations associated with  the fission-track annealing models  FT models  Equations  Parameters (standard error)  Parallel Arrhenius  (PA)  푓푃퐴(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐2  푅푇  (PA1)  푘푒푓(푇 )푃퐴 = 푐1푒푐0∕푐1 exp  (푐2∕푐1  푅푇  )  (PA2)  퐸푃퐴  푎  = −푐2  푐1  (PA3)  푐0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='631 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='220)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1865 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0066)  푐2 = -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='31) kcal/mol  휒2  휈 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='65  Parallel curvilinear  (PC)  푓푃퐶(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐2 ln  ( 1  푅푇  )  (PC1)  푘푒푓(푇 )푃퐶 = 푐1푒푐0∕푐1 (푅푇 )−푐2∕푐1  (PC2)  퐸푃퐶  푎 (푇 ) = −푐2  푐1  푅푇  (PC3)  푐0 = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='910 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='096)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1944 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0060)  푐2 = -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='610 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='244)  휒2  휈 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='12  Carlson Model  (CM)  푓퐶푀(푡, 푇 ) = 푐0 + 푐1 ln(푡) + 푐1 ln(푅푇 ) + 푐2  푅푇  (CM1)  푘푒푓(푇 )퐶푀 = 푐1푒푐0∕푐1 exp  (푐2∕푐1  푅푇  )  푅푇  (CM2)  퐸퐶푀  푎  (푇 ) = −푐2  푐1  + 푅푇  (CM3)  푐0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='426 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2155)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1867 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0066)  푐2 = -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='25 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2994) kcal/mol  휒2  휈 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='63  Fanning Arrhenius  (FA)  푓퐹퐴(푡, 푇 ) = 푐0 + 푐1  ln(푡) − 푐2  1  푅푇 − 푐3  (FA1)  푘푒푓(푡, 푇 )퐹퐴 =  푐1 exp[(1 − 푛)푓퐹퐴(푡, 푇 )]  푡(1 − 푐3푅푇 )  (FA2)  퐸퐹퐴  푎 (푡, 푇 ) =  (푅푇 )2 [푐1(푛 − 1)(푐2 − ln 푡) − 푐3 + 1∕푅푇 ]  (푐3푅푇 − 1)2  (FA3)  푐0 = -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='518 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='072)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1266 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0191) mol/kcal  푐2 = -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='99 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='81)  푐3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2985 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1026) mol/kcal  휒2  휈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='5 ≤ 푛 < 1  Fanning curvilinear  (FC)  푓퐹퐶(푡, 푇 ) = 푐0 + 푐1  ln(푡) − 푐2  ln  (  1  푅푇  )  − 푐3  (FC1)  푘푒푓(푡, 푇 )퐹퐶 =  푐1 exp[(1 − 푛)푓퐹퐶(푡, 푇 )]  푡  (  ln  (  1  푅푇  )  − 푐3  )  (FC2)  퐸퐹퐶  푎 (푡, 푇 ) =  푅푇  (  푐1푐2푛 − 푐1푐2 + (푐1 − 푐1푛) ln(푡) − 푐3 + ln  (  1  푅푇  ))  (  푐3 − ln  (  1  푅푇  ))2  (FC3)  푐0 = -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='449 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='480)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1627 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0298)  푐2 = -24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='58 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='75)  푐3 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8626 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1549)  휒2  휈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='5 ≤ 푛 < 1  Notes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For each fission-track annealing model (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA1), (PC1), (CM1) (FA1), (FC1)), the reaction rate constants,  푘푒푓 and Arrhenius Activation energies were obtained using 푔(푟) = ln(1 − 푟) and 푓푟(푟) = (1 − 푟)푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rate constants (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (PA2), (PC2), (CM2) (FA2), (FC2)) were calculated after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Arrhenius activation energies (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA3), (PC3),  (CM3) (FA3), (FC3)) were obtained by the application of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (3), and are average values for constant heating  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 4 of 15  Fission tracks are formed by displaced atoms and vacant sites, in concentrations high enough to change the structure  of the mineral in a volume of about 2-10 nm in diameter and around 20 휇m in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The annealing process is the  recombination of defects and vacancies, which also changes the neighbor structure and consequently the recombination  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' This kind of solid-state reaction is described by the conversion rate equation (Vyazovkin, 2015):  d훼  d푡 = 푘(푇 )푓훼(훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (5)  The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (5) relates the rate of conversion of the reactant 훼 with the constant rate and with the reaction function  푓훼(훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For the fission tracks, 훼 is the concentration of recombined atoms, and 푓훼(훼) describes how the recombination  process changes the surrounding structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The track length can be used as a proxy for the concentration of displaced  atoms (Rufino and Guedes, 2022) and:  훼 = 퐿0 − 퐿  퐿0  = 1 − 푟  (6)  and with this change of variable:  d푟  d푡 = −푘푒푓(푡, 푇 )푓푟(푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (7)  The rate constant has been replaced with an effective rate constant, 푘푒푓(푡, 푇 ), which may depend on time and  temperature and is suitable to describe more complex reactions (Vyazovkin, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For the reaction function, the  reaction-order function has already been shown to produce consistent results mainly for the single activation energy  mechanisms of annealing (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino and Guedes, 2022):  푓푟 = (1 − 푟)푛  (8)  in which 푛 is the reaction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (7) is a differential equation that can be solved by the separation of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' To define the limits of the integral,  consider that at the beginning of the thermal history (푡 = 0), the track is unannealed (푟 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' After a heating duration  푡, the track length has been shortened to 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Then:  ∫  1  푟  d푟  푓푟(푟) = ∫  푡  0  −푘푒푓(푡, 푇 )d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (9)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9) is the basic equation from which the annealing kinetics can be studied from a physicochemical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Once the reaction function and the rate constant are chosen, the dependence of the reduced fission-track length can be  calculated over any T-t path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Let’s start with the known case of constant temperature heating, from which the annealing  equations should be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For the single activation energy models, PA, PC, and CM, the rate constants are given  by:  푘푒푓(푇 )푃 퐴 = 퐴1 exp  (−푄1  푅푇  )  ,  (10a)  푘푒푓(푇 )푃 퐶 = 퐴2(푅푇 )푚,  (10b)  푘푒푓(푇 )퐶푀 = 퐴3(푅푇 ) exp  (  − 푄3  푅푇  )  ,  (10c)  where 퐴푖, 푄푖, and 푚 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (10a) is the original Arrhenius equation from which the PA equation is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  푄1 can be directly identified with the activation energy only in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (10b) generates the PC equation with  a temperature-dependent activation energy (Table 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PC3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (10c) generates the Carlson Model, also with a  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 5 of 15  temperature-dependent activation energy (Table 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (CM3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' It is the product of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (10a) and (10b), with 푚 = 1,  and has been proposed soon after the original Arrhenius equation to deal with reactions that deviate from the expected  Arrhenius behavior (Kooij, 1893).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Note that although the activation energies in the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (10b) and (10c) depend on  temperature, they still fall into the category of single activation energy processes, meaning that all recombination events  at a given temperature have the same activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Annealing experiments are isothermal heating procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Then, substituting the effective rate constants (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (10))  into the integral equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 9) together with the reaction-order function defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (8) and solving it considering  the temperature as a constant results in:  ln(1 − 푟) =  ln [퐴1(1 − 푛)]  1 − 푛  +  1  1 − 푛 ln(푡) −  푄1  1 − 푛  1  푅푇 ,  (11a)  ln(1 − 푟) = ln[퐴2(1 − 푛)]  1 − 푛  +  1  1 − 푛 ln(푡) −  푚  1 − 푛 ln  ( 1  푅푇  )  ,  (11b)  ln(1 − 푟) =  ln [퐴3(1 − 푛)]  1 − 푛  +  1  1 − 푛 ln(푡) −  푄3  1 − 푛  1  푅푇 −  1  1 − 푛 ln  ( 1  푅푇  )  ,  (11c)  which are the equations for the PA (11a), PC (11b), CM (11c) models with 푔(푟) = ln(1 − 푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For the chosen reaction  function, the integral only has a real solution if 푛 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Comparing the right sides of these equations respectively with  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA1), (PC1), and (CM1), one can find out that the rate constant parameters are related to the fitting parameters  of the annealing equations as  PA ∶ 푛 = 푐1 − 1  푐1  푄1 = −푐2  푐1  퐴1 = 푐1 exp(푐0∕푐1  )  (12)  PC ∶ 푛 = 푐1 − 1  푐1  푚 = −푐2  푐1  퐴2 = 푐1 exp(푐0∕푐1  )  (13)  CM ∶ 푛 = 푐1 − 1  푐1  푄3 = −푐2  푐1  퐴3 = 푐1 exp(푐0∕푐1  )  (14)  In this way, the rate constants can be expressed in terms of the fitting parameters of the annealing models as shown  in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA2), (PC2), and (CM2) of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The values for the reaction order 푛 for the three models are 푛 ≈ −4, in  agreement with a similar analysis carried out by Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (1988) for the PA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Therefore, the parallel models  are not compatible with first-order annealing kinetics, meaning that the neighbor structure has a strong influence on  the rate of defect recombination during annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  There are no obvious expressions for the rate constant for the fanning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' A physicochemical analysis of their  trends indicates that multiple concurring processes with different activation energies are occurring during the annealing  of the fission tracks (Rufino and Guedes, 2022), in agreement with previous suggestions (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Tamer  and Ketcham, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino and Guedes (2022) derived an expression from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (7) to find the effective rate constant  from the annealing model:  푘푒푓(푡, 푇 ) = −  1  푓푟(푟)  [휕푔(푟)  휕푟  ]−1 휕푓(푡, 푇 )  휕푡  |||||푇  (15)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (15) provides a direct way to calculate this effective reaction rate constant from the model functions that fit the  experimental annealing data, 푓(푡, 푇 ) and 푔(푟), and from the reaction function 푓푟(푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The partial derivative in relation to  time is taken because the annealing models were designed to describe constant temperature experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' As a check,  before applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (15) to the fanning models, one can show that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA2), (PC2), and (CM2) are found by the  application of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (15) respectively to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA1), (PC1), and (CM1), with 푔(푟) = ln(1 − 푟) and 푓푟(푟) = (1 − 푟)푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The same procedure can be applied to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FA1) and (FC1) to find the effective reaction rates respectively for the  Fanning Arrhenius (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FA2)) and Fanning Curvilinear (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FC2)) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' An alternative way to infer the effective  rate constants for the FA and FC models is departing from the hypothesis that the Arrhenius activation energies and,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 6 of 15  therefore, the rate constants are dependent on the fission-track reduced length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Then, integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9), on the  isothermal condition, results in  − ∫  푟  1  d푟  푓푟(푟)푘푒푓(푟) = ∫  푡  0  d푡 = 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (16)  It can be shown that the primitive functions that make Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (16) true for the FA and FC models, with 푓푟(푟) = (1−푛)푛  and 푔(푟) = ln(1 − 푟) are the ones with the effective rate constants given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FA2) and (FC2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' This approach also  illustrate how the incorporation of the time in the rate constant and, therefore, in the activation energies for the fanning  models are implied from the dependence of the activation energies on the values of 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  To obtain the reaction order 푛 for the FA and FC models, the effective reaction rate constants given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FA2)  and (FC2) are integrated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9) considering constant temperatures (isothermal experiments):  ∫  푟  1  d푟  (1 − 푟)푛 = − ∫  푡  0  푐1  1  푅푇 − 푐3  1  푡 exp  ⎡  ⎢  ⎢⎣  −(푛 − 1)  ⎛  ⎜  ⎜⎝  푐0 + 푐1  ln 푡 − 푐2  1  푅푇 − 푐3  ⎞  ⎟  ⎟⎠  ⎤  ⎥  ⎥⎦  d푡  (17a)  ∫  푟  1  d푟  (1 − 푟)푛 = − ∫  푡  0  푐1  ln  (  1  푅푇  )  − 푐3  1  푡 exp  ⎡  ⎢  ⎢  ⎢⎣  −(푛 − 1)  ⎛  ⎜  ⎜  ⎜⎝  푐0 + 푐1  ln 푡 − 푐2  ln  (  1  푅푇  )  − 푐3  ⎞  ⎟  ⎟  ⎟⎠  ⎤  ⎥  ⎥  ⎥⎦  d푡  (17b)  With the necessary condition of 푛 < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' the solution of the integral equation (17a) is  (1 − 푟) = (−1)−1∕(푛−1) exp  ⎡  ⎢  ⎢⎣  푐0 + 푐1  ln(푡) − 푐2  1  푅푇 − 푐3  ⎤  ⎥  ⎥⎦  (18)  As the solution of this equation is to represent the shortening of the fission tracks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (1 − 푟) must be a real value  between 0 and 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' which is true only if −1∕(푛 − 1) is an even and positive integer value 2푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Then, the values of 푛 are  restricted to  푛 = 2푗 − 1  2푗  ,  (19)  where 푗 = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='. With this condition and 푔(푟) = ln(1 − 푟), the FA model (Table 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FA1)) is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The  solutions for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (17a) and (17b) are similar, differing only on the logarithm of 1∕푅푇 for the FC model instead  of the 1∕푅푇 for the FA model, which are both constants in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The previous analysis holds also for the FC  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The values of 푛 will be fractional for FA and FC (푛 = 1∕2, 3∕4, 5∕6, 7∕8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='), according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Fractional  reaction orders are characteristics of multiple-step reactions or some more complex kinetic mechanism, as it has been  explained for the decomposition of acetaldehyde (Laidler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1965), a well know example of fractional reaction order  in chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' However, for the fission tracks, where the displaced atoms and vacant sites take the role of reactants and  the deformed track structure is the reaction medium, explanations of the kinetics of a single reactant via a mean-  field approximation (MFA) may not be appropriate (Córdoba-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Thus, for the effective reaction  rate constant 푘푒푓 of the fanning annealing models, mechanistic modeling considering the intermediate steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=',  recognizing the reaction order of each mechanism involved in annealing, would be desirable to elucidate the meaning  of the fractional reaction order found (Koga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The rate constants for FA and FC are to be viewed as effective  equations constraining the general behavior of annealing but that does not allow the description of the specifics of the  annealing kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  In this physicochemical framework of the fission-track annealing, the effective reaction rate constant, 푘푒푓(푡, 푇 ),  and the reaction function, 푓푟(푟), are the fundamental building blocks from which the fission-track annealing kinetics  can be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The application to the constant temperature annealing made it possible to determine the rate constant  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 7 of 15  parameters from the empirically determined parameters of the annealing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The calculation of the Arrhenius  activation energies (퐸푎) for different models becomes possible through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The Arrhenius activation energies of  the parallel annealing models (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' PA3, PC3 and CM3) will be constants with respect to the variable 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' As for the  fanning equations, 퐸푎 will vary with time and temperature (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' FA3 and FC3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' However, the main advantage of this  approach is the possibility of calculating the fission-track length reduction over any T-t path using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9), without  recurring to the interactive application of the Principle of Equivalent Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Fission track annealing under variable temperature thermal histories  Fission-track thermal history inference is based on the Principle of Equivalent Time (PET) (Goswami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Duddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1988), which is an interactive method that allows the application of isothermal annealing models to  variable temperature T-t paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' It is detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' In general, a given variable temperature thermal history  is divided into finite time intervals Δ푡푖, centered at times 푡푖 and temperatures 푇푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' At the time interval in which the  population was born, a first reduced length is calculated by applying the annealing equation, using the temperature of  the T-t path and the duration of the interval (푇푖, Δ푡푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' In the next interval, at a different temperature on T-t path, the  annealing model is used to find an equivalent time capable of producing the same length shortening of the previous  interval but at the new temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' A new length shortening is then calculated by applying the annealing model to  the period of time that is the sum of the equivalent time and the length of the time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' This procedure is repeated  and at any given temperature, 푇푖 on the T-t path, an equivalent time, 휏푖, which reproduces the length shortening at the  previous interval, 푟푖−1, is determined, so that the new length shortening can be calculated as if the track had been at  the same constant temperature from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The reduced length is updated (푟푖) by calculating it as a result of  heating at 푇푖 for the duration 휏푖 + Δ푡푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The hypothesis that the annealing kinetics does not depend on the previous  thermal history of the track, but only on its current length so that any previous T-t path can be replaced with a constant  temperature heating resulting in this length, is the basis of this procedure and defines the Principle of Equivalent Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  This means, in practice, that the track will have no memory of the material conditions of time and temperature of its  previous shortenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The equations for the application of the PET with the PA, PC, CM, FA, and FC models can be  found in Table A1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The physicochemical tool presented in the previous section provides an alternative way to access variable temper-  ature annealing kinetics by solving the integral in the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9) over a T-t path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9) is solved as a line  integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' A suitable parameterization is:  푠 =  {  푇 = 푇 (푢)  푡 = 푢  ,  d푠 =  √  1 +  (d푇  d푢  )2  d푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (20)  Implementing the parameterized variables on the right side of the integral equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 9),  퐼 = ∫  푡  0  푘푒푓 (푡(푢), 푇 (푢))  d푠  √  1 +  (  d푇  d푢  )2  ⟹ 퐼 = ∫  푡  0  푘푒푓 (푡(푢), 푇 (푢)) d푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (21)  Solving the left side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9) for the 푓푟(푟) function given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (8) and the parameterized integral for the rate  constant (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (21)), the reduced length, after the track has experienced the thermal history given by the T-t path, is  푟 = 1 −  (  (1 − 푛) ∫  푡  0  푘푒푓 (푡(푢), 푇 (푢)) d푢  )1∕1−푛  ,  (22)  in which 푛 < 1 as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' At a first glance, the advantage of the Rate Constant path Integral (RCI, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (22)) is that it is a  one-shot calculation of the reduced track length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' In addition, there was no need to restrict the form of the rate constant  function and therefore the annealing mechanism it is related to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 8 of 15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Results and Discussion  The RCI Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (22) can be applied to calculate the shortening in the reduced length of a single fission-track population  submitted to any T-t path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The same calculation can be carried out using the interactive technique based on the Principle  of Equivalent Time (PET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' To compare the outcomes of the two methods, both calculations will be performed for the  parallel (including CM) and fanning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The case will be made for the linear cooling, 푇 (푡) = 푇0 − ̇푇 푡, where ̇푇  is the cooling rate and 푇0 is the temperature at the time the track was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The temperature at the end of the T-t  path (present time) was fixed to be 20 ◦C (293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='15 K) for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Parallel models  To solve the RCI, the effective rate constant functions for the parallel models (Table 1, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA2), (PC2), and  (CM2)) are inserted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (22) with the variable 푇 replaced with 푇 (푡) = 푇0 − ̇푇 푡 wherever it appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The analytical  solutions for the reduced length shortening calculated for the PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' PC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' and CM are  푟푃 퐴 = 1 −  ⎛  ⎜  ⎜  ⎜  ⎜⎝  푒푐0∕푐1  (  푐2Ei  (  푐2  푐1푅(푇0− ̇푇 푡)  )  − 푐2Ei  (  푐2  푐1푅푇0  )  + 푐1푅  (  ( ̇푇 푡 − 푇0)푒  푐2  푐1푅(푇0− ̇푇 푡) + 푇0푒  푐2  푐1푅푇0  ))  푐1 ̇푇 푅  ⎞  ⎟  ⎟  ⎟  ⎟⎠  푐1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (23a)  푟푃 퐶 = 1 −  ⎛  ⎜  ⎜  ⎜  ⎜⎝  푐1푒푐0∕푐1  (  ( ̇푇 푡 − 푇0)(푅(푇0 − ̇푇 푡))  − 푐2  푐1 + 푇0(푅푇0)  − 푐2  푐1  )  푐1 ̇푇 − 푐2 ̇푇  ⎞  ⎟  ⎟  ⎟  ⎟⎠  푐1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (23b)  푟퐶푀 = 1 − 2−푐1푒푐0 푐−2푐1  1  ( ̇푇 푅)푐1  [  푐1푅  (  푇0푒  푐2  푐1푅푇0 (푐1푅푇0 + 푐2) − (푇0 − ̇푇 푡)푒  푐2  푐1푅푇0−푐1 ̇푇 푅푡 (푐1푅(푇0 − ̇푇 푡) + 푐2)  )  +푐2  2  (  Ei  (  푐2  푐1푅푇0 − 푐1푅푡 ̇푇  )  − Ei  (  푐2  푐1푅푇0  ))]푐1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (23c)  where Ei is the exponential integral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (23a) - (23c) give the resulting reduced length 푟 for the parallel  models, as functions of the three variables that characterize the thermal history: the duration of the T-t path (푡), the  cooling rate ( ̇푇 ), and the temperature at the time when the track was born (푇0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The parameters 푐푖 are given in the last  column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The values of 푟 for the cooling path with the cooling rate ̇푇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0◦C/Ma calculated with the three  parallel models are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For each point, the value of 푟 is the length reduction after a linear cooling  duration 푡 and measured in the present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Values of 푟 = 0 mean that the tracks have been erased before the present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Values obtained by the RCI solutions (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (23a) - (23c)) are represented as red curves marked with red circles and the  values calculated using the PET are represented as blue curves marked with blue squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' RCI and PET calculations  produce very close values of 푟 for the three parallel models (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 2a, 2b, 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The temperature indexes Closure Temperature (푇퐶) and Total Annealing Temperature (푇퐴) were also calculated  for the three parallel models, applying both methods of calculation, for cooling T-t paths with cooling rates of 1, 10,  and 100 ◦C/Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 푇퐶 is, for a monotonic cooling thermal history, the temperature at the apparent sample age (Dodson,  1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 푇퐴 is the age of the oldest track that has not been erased and can be counted in the sample (Issler, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Details for the method of calculation of the index temperatures can be found in Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 푇퐶 and 푇퐴 are  meaningful quantities that allow quantifying the impact of using the RCI instead of the interactive PET calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The uncertainties in 푇퐶 and in 푇퐴 were estimated by simple error propagation of apparent (푇퐶) or retention (푇퐴) ages  and present time temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Setting the PET results as the reference values, given  that PET is the method established in the literature, a relative error analysis can be carried out to verify the internal  consistency between PET and RCI calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The relative error between PA PET and PA RCI is on average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='69%  for 푇퐶 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='19% 푇퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The same trend is found for calculations of 푇퐶 and 푇퐴 with PC and CM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='66% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='58% (PC)  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='7% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='19% (CM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' All errors are well inside the estimated uncertainties for temperature index values and are  most probably artifacts of the PET numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 9 of 15  ,  (a)  (b)  (c)  Figure 2: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0◦C/Ma) starting at time 푡 and ending at  the present time at a fixed temperature of 20◦C, calculated using the Parallel models: (a) Arrhenius, (b) Curvilinear, and  (c) Carlson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The points that form the curve in red (circle marks) were calculated by applying the RCI (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 23a, 23b, and  23c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The values calculated using the PET are in blue (square marks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Table 2  Comparison of thermal indexes for the models presented in this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' calculated using the PET and the RCI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='Principle of Equivalent Time (PET) thermal indexes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='Fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐶1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='푇퐶100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐴1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐴10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='PC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='105(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='121(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='CM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='130(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='143(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='157(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='168(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='184(9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='FA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='134(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='163(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='176(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='191(10)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='FC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='111(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='126(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='143(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='143(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='160(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='179(9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='Rate constant integral on a path (RCI) thermal indexes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='Fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐶1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐶10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐶100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐴1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐴10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='푇퐴100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='PA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='131(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='155(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='170(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='186(9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='PC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='106(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='122(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='172(9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='CM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='131(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='FA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='125(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+page_content='FC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='100(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='115(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='130(6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='130(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='148(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='165(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='Notes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Temperatures in ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The standard errors are given in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 푇퐶: Closure Temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 푇퐴:  Total Annealing Temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The numbers to the right of 푇퐶 and 푇퐴 are the cooling rates in ◦C/Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The PET was formulated under the hypothesis that the annealing of fission tracks is a single activation energy pro-  cess Duddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The internal consistency between PET and RCI values of 푟, 푇퐶, and 푇퐴 calculated with the  parallel models is a check for the robustness of the physicochemical approach to deal with variable temperature thermal  histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' It is to be noted that not only the PA model, in which the activation energy is temperature-independent (Ta-  ble 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA3)), but also the PC model, in which the activation energy is temperature-dependent (Table 1, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FC3)),  show such internal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The same agreement is observed for CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The CM activation energy may vary with  temperature but, with the parameters shown in Table 1, its Arrhenius activation energy is approximately constant since  the value of 푐2∕푐1 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9 kcal/mol) is much higher than typical values of 푅푇 (< 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0 kcal/mol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Although the activation  energies may vary with temperature, these models imply that at any given temperature, the recombination events are  taking place with the same activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' This is a sufficient condition for the applicability of the PET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 10 of 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  PA  S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8  Reduction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='6  Method  Length F  RCI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='4  PET  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  50  100  150  200  Thermal history duration (t)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  PC  S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8  Reduction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='6  Method  Length F  RCI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='4  PET  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  50  100  150  200  Thermal history duration (t)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  CM  C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8  Reduction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='6  Method  Length F  RCI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='4  PET  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  50  100  150  200  Thermal history duration (t)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Fanning models  The values of 푟, 푇퐶, and 푇퐴 for the cooling T-t path were also calculated for the fanning models, using both the  interactive PET and RCI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' However, the RCI (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (22)) could not be solved analytically for the FA and FC  rate constants (Table 1, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FA2) and (FC2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The integrals were then solved numerically with the Wolfram Mathe-  matica software (Wolfram-Research-Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For validation, the integrals for the parallel models were also solved  numerically resulting in exactly the same values obtained with the analytical solutions (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (23a)-(23c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Another  feature to be considered is that there are certain fractional values allowed for the reaction order 푛, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The analysis will be limited to 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='5, 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='75 and 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The numerical method breaks down when 푛 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='95  although its mathematical upper bound is 푛 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The reduced length calculation results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Values  calculated with the PET are shown in blue, with triangle marks, while values found by solving the RCI are shown,  in red (푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='5), purple (푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='75), and light purple (푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9), respectively with circle, square, and diamond marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  RCI 푟 curves are very close to each other but depart from the 푟 values calculated with the PET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Significant differences  between RCI and PET 푟 values are observed for the FC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 3b) and for the FA (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 3a) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (a)  (b)  Figure 3: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0◦C/Ma) starting at time 푡 and ending at  the present time at a fixed temperature of 20◦C, calculated using the Fanning models: (a) Arrhenius and (b) Curvilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The points that form the curves in red (circle marks), purple (square marks), and light purple (diamond marks) were  calculated by applying the RCI, respectively with 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='5, 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='75, and 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The values calculated using the PET are  in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The values of 푇퐶 and 푇퐴 for the fanning models, calculated using the PET and the RCI (푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='5) are in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For  the FA model, the mean relative errors between the PET and the RCI 푇퐶 and 푇퐴 calculations are respectively 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='25%  and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='68%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For the FC model, the same comparisons result in still more significant differences: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='57% (푇퐶) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='48%  (푇퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The deviations between the values calculated using the PET and the RCI are much more significant than the ones  found for the parallel model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' One major issue is that the fanning models do not fulfill the single activation  energy hypothesis on which the PET is founded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The fanning models emerge from multiple concurring processes with  different activation energies (Tamer and Ketcham, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino and Guedes, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The effective Arrhenius activation  energies incorporate a time dependence (Table 1, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (FA3) and (FC3)) that is the consequence of their dependence  on the current reduced fission-track length (different slopes of the isoretention curves in the pseudo-Arrhenius plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  On the other hand, the rate constant integral (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (22)) was obtained in a physicochemical framework developed to  deal with chemical reactions that did not fit the single activation energy Arrhenius law (Vyazovkin, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' It is by  design suitable for complex activation energy systems like the ones pictured by the fanning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Note also that  the presented figures are particular of the fitting parameters in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' A different set of parameters would result in  different values without changing the conclusion that RCI and PET predictions deviate from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 11 of 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  FA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8  Reduction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='6  Method  RCI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='5)  Length F  RCI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='75)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='4  RCI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9)  PET  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  50  100  150  200  Thermal history duration (t)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  FC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8  Reduction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='6  Method  RCI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='5)  Length F  RCI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='75)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='4  RCI (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='9)  PET  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  50  100  150  200  Thermal history duration (t)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Implications for the thermal history modeling  The fanning models, especially the Fanning Curvilinear, have been shown to produce better fits to laboratory data  and better geological extrapolation of annealing effects (Ketcham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' However, the  application of the FC along with the PET is an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Compared with the RCI formulation, it underestimates  the annealing effect in about 10%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', it predicts that higher temperatures are necessary for the same length shortening  as calculated with the RCI for the tested cooling histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' In the context of the inverse problem of inferring T-t paths  from the FT age and track length distribution of a mineral sample, it implies the requirement of a longer residence  time in the partial annealing zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For instance, compare, in Table 2, the FC 푇퐴 calculated with RCI for a cooling rate  of 10◦C/Ma (148◦C) with the FC 푇퐴 calculated with PET for a cooling rate of 1◦C/Ma (143◦C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The same analysis  applies to the FA model with less significant relative error figures (about 6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The Parallel models (PA, PC, and CM), which can be safely applied along with the PET, have long been ruled out for  FTT studies (Laslett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Ketcham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1999, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Ketcham, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Duddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (1988)  had argued that the FA deviated only slightly from the PA model and applied it along with the PET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The isoretention  curves for the two models follow approximately the same trends (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The same behavior is observed for the  curvilinear models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' FC and PC isoretention curves bend together towards lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Their argument  can be better appreciated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' All the PET and RCI predictions for reduced lengths after the track underwent the  cooling history are gathered in the same plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Note that the linear models (PA and FA) and the approximately linear CM  form a cluster, while the curvilinear models (PC and FC) form a separate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The predictions with the fanning models  and PET are closer to the predictions of the parallel models for track populations born when the sample passed through  intermediate temperatures (partial annealing zone), which results in closer 푇퐶 values (compare values in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For  populations born at higher temperatures, the fanning-PET predictions depart from the parallel model ones, resulting in  a more significant difference between calculated 푇퐴 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Calculations with RCI approximate fanning and parallel  model predictions for populations born at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Within this approximation, it could be possible to  engineer fanning model parameters to make the model even closer to a parallel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Figure 4: Values of the reduced track lengths (푟), after a linear cooling ( ̇푇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0◦C/Ma) starting at time 푡 and ending at  the present time at a fixed temperature of 20◦C, calculated for the parallel and fanning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Calculations using RCI  are shown as solid geometric forms, while calculations using PET are represented by empty geometric forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Concluding remarks  Departing from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9), a physicochemical framework was built to deal with the effects of annealing in variable tem-  perature T-t paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The basic building blocks are the reaction function 푓푟(푟) and the effective rate constant, 푘푒푓(푡, 푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The parallel models (PA, PC, and CM) were shown to be consistent with the single activation energy rate constants  given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (10a)-(10c) and with the reaction-order function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The fanning models (FA and FC) are the  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 12 of 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8  Reduction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='6  RCI Method  PET Method  Length F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='4  PA  0  PA  PC  FA  CM  CM  FA  口  PC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2  FC  FC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0  50  100  150  200  Thermal history duration (t)representation of multiple concurrent recombination processes with different activation energies (Rufino and Guedes,  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The 푘푒푓(푡, 푇 ) functions were built (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (15)) to be consistent with the reaction-order function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='(8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Obtain-  ing FA and FC rate constants from first principle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', a composition of rate constants for individual processes, and  validating them experimentally is still an open issue that has to be dealt with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (22) is the line integral related  to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (9) from which length shortening due to cooling T-t paths can be directly calculated, independently of whether  the rate constant represents a single or multiple activation energy mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The Principle of Equivalent Time, on the other hand, is only valid for single activation energy equations, which, for  the fission-track system, are the parallel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' In these cases, the RCI-based calculations are in agreement with the  PET ones (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 2), indicating the robustness of RCI formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For the fanning models, the use of the PET has long  been recognized as an approximation (Duddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=', 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Deviations have indeed been observed between RCI and  PET-based calculations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Compared to the application of RCI, the PET calculation underestimates annealing  effects in variable temperature T-t paths (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The PET along with FA or FC models is the calculation method used to infer most published thermal histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  This procedure introduces a systematic deviation that should be considered in the geological interpretation of the  thermal history modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Alternatively, the rate constant integral (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (22)) could be considered to substitute the  PET in inversion thermal history codes (Ketcham, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Gallagher, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Computationally, solving an integral,  even numerically, is a routine faster than the interactive steps necessary to apply the PET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' More importantly, if the  rate constants are representative of the track annealing kinetics, this framework results, in principle, in more accurate  predictions of the annealing effects in samples submitted to variable temperature thermal histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Equations for the application of the Principle of Equivalent Time  The proposed method to deal with annealing in thermal histories with variable temperatures is also based on the  Arrhenius equation and was first proposed by Goswami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The Principle of Equivalent Time (PET), on  which this method is founded, states that the annealing rate of a track does not depend on its previous thermal history,  but only on its current length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  The procedure to solve this problem is to infer the magnitude of annealing recursively, dividing the thermal history  into appropriate intervals Δ푡푖 and starting from a given value of 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For each step, annealing is carried out at constant  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' In the first step, centered at (푡1, 푇1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The procedure will be shown for the Parallel Arrhenius model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The  reduced fission-track length is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA1), along with 푔(푟) = ln(1 − 푟):  푟1 = 1 − (Δ푡1)푐1 exp  (  푐0 + 푐2  푅푇1  )  (24)  For the next step, the calculation of 푟2 incorporates the principle in the form that this new shortening is postulated  to be independent of the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Thus, one can find a length of time, 휏푟1, that will yield the length 푟1 but for  heating at the temperature of the second interval, 푇2:  휏푟1 = (1 − 푟1)−1∕푐1 exp  (  − 1  푅푇2  푐2  푐1  + 푐0  푐1  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (25)  The value of 푟2 is then found by the application of the annealing model to the interval 휏푟1 + Δ푡2:  푟2 = 1 − (휏푟1 + Δ푡2)푐1 exp  (  푐0 + 푐2  푅푇2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  (26)  This procedure is interactively repeated for the entire T-t path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' The last value of 푟 will be the reduced length of  the population born in the first interval after experiencing the entire thermal history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For each interval 푗, the formulas  above are:  푟푗  = 1 − (휏푟푗−1 + Δ푡푗)푐1 exp  (  푐0 +  푐2  푅푇푗  )  (27a)  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Rufino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Lixandrão-Filho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' Guedes  Page 13 of 15  Table A1  Equations for the application of the Principle of Equivalent Time  FT models  Equations  Parameters (standard error)  Parallel Arrhenius  (PA)  ln (1 − 푟푗−1  ) = 1 − exp  [  푓푃 퐴  (  Δt푗 + 휏푟푗−1, 푇푗  )]  (PA4)  ln  (  휏푟푗−1  )  =  ln (1 − 푟푗−1  )  푐1  −  푐2  푐1푅푇푗  − 푐0  푐1  (PA5)  푐0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='631 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='220)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1865 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0066)  푐2 = -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='31) kcal/mol  휒2  휈 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='65  Parallel curvilinear  (PC)  ln (1 − 푟푗−1  ) = 1 − exp  [  푓푃 퐶  (  Δt푗 + 휏푟푗−1, 푇푗  )]  (PC4)  ln  (  휏푟푗−1  )  =  ln (1 − 푟푗−1  )  푐1  − 푐2  푐1  ln  (  1  푅푇푗  )  − 푐0  푐1  (PC5)  푐0 = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='910 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='096)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1944 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0060)  푐2 = -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='610 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='244)  휒2  휈 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='12  Carlson Model  (CM)  ln (1 − 푟푗−1  ) = 1 − exp  [  푓퐶푀  (  Δt푗 + 휏푟푗−1, 푇푗  )]  (CM4)  ln  (  휏푟푗−1  )  =  ln (1 − 푟푗−1  )  푐1  −  푐2  푐1푅푇푗  − 푐0  푐1  − ln (푅푇푗  )  (CM5)  푐0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='426 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2155)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1867 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0066)  푐2 = -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='25 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2994) kcal/mol  휒2  휈 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='63  Fanning Arrhenius  (FA)  ln (1 − 푟푗−1  ) = 1 − exp  [  푓퐹퐴  (  Δt푗 + 휏푟푗−1, 푇푗  )]  (FA4)  ln  (  휏푟푗−1  )  =  [ln (1 − 푟푗−1  ) − 푐0  ] [  1  푅푇푗 − 푐3  ]  푐1  + 푐2  (FA5)  푐0 = -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='518 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='072)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1266 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0191) mol/kcal  푐2 = -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='99 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='81)  푐3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='2985 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1026) mol/kcal  휒2  휈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='66  Fanning curvilinear  (FC)  ln (1 − 푟푗−1  ) = 1 − exp  [  푓퐹퐶  (  Δt푗 + 휏푟푗−1, 푇푗  )]  (FC4)  ln  (  휏푟푗−1  )  =  [ln (1 − 푟푗−1  ) − 푐0  ] [  ln  (  1  푅푇푗  )  − 푐3  ]  푐1  + 푐2  (FC5)  푐0 = -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='449 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='480)  푐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1627 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='0298)  푐2 = -24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='58 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='75)  푐3 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='8626 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='1549)  휒2  휈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='88  Notes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' For each fission-track annealing model (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA1), (PC1), (FA1), (FC1)), the length reduction 푟 (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA4),  (PC4), (FA4), (FC4)) and equivalent time 휏 (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content=' (PA5), (PC5), (FA5), (FC5)) were obtained using 푔(푟) = ln(1 − 푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  휏푟푗−1  = (1 − 푟푗−1)−1∕푐1 exp  (  − 1  푅푇푗  푐2  푐1 + 푐0  푐1  )  (27b)  The formulas for the PA, PC, CM, FA, and FC models are presented in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
+page_content='  Acknowledgements  This work has been funded by grant 308192/2019-2 by the National Council for Scientific and Technological  Development (Brazil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf'}
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+Facilitating Contrastive Learning of Discourse Relational Senses by
+Exploiting the Hierarchy of Sense Relations
+Wanqiu Long†, and Bonnie Webber†
+† University of Edinburgh, Edinburgh, UK
+Wanqiu.long@ed.ac.uk, Webber.Bonnie@ed.ac.uk
+Abstract
+Implicit discourse relation recognition is a
+challenging task that involves identifying the
+sense or senses that hold between two adja-
+cent spans of text, in the absence of an explicit
+connective between them.
+In both PDTB-2
+(Prasad et al., 2008) and PDTB-3 (Webber
+et al., 2019), discourse relational senses are
+organized into a three-level hierarchy rang-
+ing from four broad top-level senses, to more
+specific senses below them.
+Most previous
+work on implicit discourse relation recogni-
+tion have used the sense hierarchy simply
+to indicate what sense labels were available.
+Here we do more — incorporating the sense
+hierarchy into the recognition process itself
+and using it to select the negative examples
+used in contrastive learning.
+With no addi-
+tional effort, the approach achieves state-of-
+the-art performance on the task.
+Our code
+is released in https://github.com/wanqiulong
+0923/Contrastive_IDRR.
+1
+Introduction
+Discourse relations are an important aspect of
+textual coherence. In some cases, a speaker or
+writer signals the sense or senses that hold between
+clauses and/or sentences in a text using an explicit
+connective. Recognizing the sense or senses that
+hold can be more difficult, in the absense of an
+explicit connective.
+Automatically identifying the sense or senses
+that hold between sentences and/or clauses can
+be useful for downstream NLP tasks such as text
+summarization (Cohan et al., 2018), machine trans-
+lation (Meyer et al., 2015) and event relation ex-
+traction (Tang et al., 2021). Recent studies on im-
+plicit discourse relation recognition have shown
+great success. Especially, pre-trained neural lan-
+guage models (Peters et al., 2018; Devlin et al.,
+2019; Liu et al., 2019) have been used and dramat-
+ically improved the performances of models (Shi
+and Demberg, 2019b; Liu et al., 2020; Kishimoto
+Level2
+Level1
+Temporal
+Asynchronous
+Level3
+Precedence
+Succession
+Root
+Comparison
+Contigency
+Expansion
+Synchronous
+Concession
+Contrast
+Cause
+...
+Purpose
+Cause+Belief
+...
+Reason
+Result
+Equivalence
+Manner
+Reason+Belief
+Result+Belief
+Figure 1: The PDTB-3 Sense Hierarchy
+et al., 2020). The senses available for labelling
+discourse relations in the PDTB-2 (and later in the
+PDTB-3) are arranged in a three-level hierarchy,
+with the most general senses at the top and more
+specific senses further down. In the PDTB-3, anno-
+tators could only choose senses at terminal nodes
+in the hierarchy – level-2 senses for symmetric re-
+lations such as EXPANSION.EQUIVALENCE and
+TEMPORAL.SYNCHRONOUS, and level-3 senses
+for asymmetric relations, with the direction of
+the relation encoded in its sense label such as
+SUBSTITUTION.ARG1-AS-SUBST (where the text
+labelled ARG1 substitutes for the denied text
+labelled ARG2) and SUBSTITUTION.ARG2-AS-
+SUBST (where the text labelled ARG2 substitutes
+for the denied text labelled ARG1). Early work on
+recognizing the implicit relations only used the hi-
+erarchy to choose a target for recognition (e.g., the
+senses at level-1 (classes) or those at level-2 (types).
+Recently, Wu et al. (2022) have tried to leverage the
+dependence between the level-1 and level-2 labels
+(cf. Section 2). The current work goes further, us-
+ing the whole three-level sense hierarchy to select
+the negative examples for contrastive learning.
+Contrastive learning, which aims to minimize
+the distance between similar instances (defined as
+positive examples) and widen the difference with
+dissimilar instances (negative examples), has been
+considered as effective in constructing meaning-
+ful representations (Kim et al., 2021; Zhang et al.,
+2021; Yan et al., 2021). Previous work on con-
+trastive learning indicates that it is critical to se-
+arXiv:2301.02724v1  [cs.CL]  6 Jan 2023
+
+lect good negative samples (Alzantot et al., 2018;
+Wu et al., 2020b; Wang et al., 2021). The insight
+underlying the current work is that the hierarchy
+of sense labels can enable the selection of good
+negative examples for contrastive learning.
+To
+see this, consider Examples 1-3 below from the
+PDTB-3. On the surface, they look somewhat sim-
+ilar, but in Examples 1 and 2, the annotators took
+the second sentence (Arg2) as providing more de-
+tail about the first sentence (Arg1) — the sense
+called EXPANSION.LEVEL-OF-DETAIL.ARG2-AS-
+DETAIL, while in Example 3, they took the sec-
+ond sentence as expressing a substitute for “Amer-
+ican culture” in terms of what is relevant – the
+sense called EXPANSION.SUBSTITUTION.ARG2-
+AS-SUBST.
+(1) “Valley National ”“isn’t out of the woods yet
+”. The key will be whether Arizona real es-
+tate turns around or at least stabilizes..
+(2) The House appears reluctant to join the sena-
+tors. A key is whether House Republicans
+are willing to acquiesce to their Senate col-
+leagues’ decision to drop many pet provi-
+sions..
+(3) Japanese culture vs. American culture is ir-
+relevant. The key is how a manager from
+one culture can motivate employees from
+another..
+In this work, we use a multi-task learning frame-
+work, which consists of classification tasks and a
+contrastive learning task. Unlike most previous
+work using one benchmark dataset (usually PDTB-
+2 or PDTB-3), we evaluate our systems on both
+PDTB-2 and PDTB-3. Besides, Wang et al. (2021)
+have shown that data augmentation can make rep-
+resentations be more robust, thereby enriching the
+data used in training. We thus follow Ye et al.
+(2021) and Khosla et al. (2020) in identifying a rel-
+evant form of data augmentation for our contrastive
+learning approach to implicit relation recognition.
+The main contributions of our work are as fol-
+lows:
+• We leveraged the sense hierarchy to get con-
+trastive learning representation, learning an
+embedding space in which examples from
+same types at level-2 or level-3 stay close to
+each other while sister types are far apart.
+• We explored and compared different methods
+of defining the negatives based on the sense
+hierarchies in PDTB-2 and PDTB-3, finding
+the approach which leads to the greatest im-
+provements.
+• Our proposed data augmentation method to
+generate examples is helpful to improve the
+overall performance of our model.
+• We demonstrate that implicit relation recogni-
+tion can benefit from a deeper understanding
+of the sense labels and their organization.
+2
+Related Work
+Implicit discourse relation recognition
+For
+this task, Dai and Huang (2018) considered
+paragraph-level context and inter-paragraph de-
+pendency. Recently, Shi and Demberg (2019b)
+showed that using the bidirectional encoder repre-
+sentation from BERT (Devlin et al., 2019) is more
+accurately to recognize Temporal.Synchrony, Com-
+parison.Contrast, Expansion.Conjunction and Ex-
+pansion.Alternative. Liu et al. (2020) showed that
+different levels of representation learning are all
+important to implicit relation recognition, and they
+combined three modules to better integrate con-
+text information, the interaction between two argu-
+ments and to understand the text in depth. How-
+ever, only two existing works leveraged the hier-
+archy in implicit relation recognition. Both Wu
+et al. (2020a) and Wu et al. (2022) first attempted
+to assign a Level-1 sense that holds between argu-
+ments, and then only considered as possible Level-
+2 senses, those that are daughters of the Level-1
+sense.
+Contrastive learning
+Recently, there has been
+a growing interest in applying contrastive learn-
+ing in both the pre-training and fine-tuning objec-
+tives of pre-trained language models. Gao et al.
+(2021) used a contrastive objective to fine-tune pre-
+trained language models to obtain sentence embed-
+dings, and greatly improves state-of-the-art sen-
+tence embeddings on semantic textual similarity
+tasks. Suresh and Ong (2021) proposed label-aware
+contrastive loss in the presence of larger number
+and/or more confusable classes, and helps models
+to produce more differentiated output distributions.
+Besides, many works have demonstrated that se-
+lecting good negative examples are very important
+for using contrastive learning (Schroff et al., 2015;
+Joshua et al., 2021; Cao et al., 2022). In our work,
+we integrate contrastive learning loss with super-
+
+vised losses and we use the structure of the sense
+hierarchy to guide the selection of negative exam-
+ples.
+3
+Learning Loss
+3.1
+Supervised Learning Loss
+The standard approach today for classification task
+is to use a standard cross-entropy loss:
+Lsup = 1
+N
+N
+�
+i=1
+−log
+eW T
+yisi
+�
+j eW T
+j si
+(1)
+Where N denotes the number of training examples,
+yi is the ground-truth class of the i-th class and Wj
+is the weight vector of the j-th class.
+3.2
+Contrastive Learning Loss
+In contrastive learning, each example can be treated
+as an anchor to get its positive and negative ex-
+amples. Contrastive learning can pull the anchor
+and its positive example together in the embedding
+space, while the anchor and negative samples are
+pushed apart. The contrastive learning loss was
+used by Chen et al. (2020); Suresh and Ong (2021)
+before. A set of N randomly sampled label pairs
+is defined as xk, yk, where x and y represent sam-
+ples and labels, respectively, k = 1, ..., N. Let i
+be the index of anchor sample and j is the index
+of a positive sample. where iϵ{1, ..., N}, i ̸= j.
+Contrastive loss is defined as:
+Lscl = −
+N
+�
+i=1
+esim(hj,hi)τ
+�
+i̸=k esim(hk,hi)τ
+(2)
+Here, h denotes the feature vector in the em-
+bedding space, and τ is the temperature parameter.
+Intuitively, the numerator computes the inner dot
+product between the anchor points i and its positive
+sample j. The denominator computes the inner dot
+product between all i and the inner dot product be-
+tween all negative samples. where a total of N − 1
+samples are computed.
+Supervised contrastive learning (Gunel et al.,
+2021) extends the equation.2 to the supervised sce-
+nario. In particular, given the presence of labels,
+the positive examples are all examples with the
+same label. The loss is defined as:
+Lscl =
+N
+�
+i=1
+−
+1
+Nyi − 1
+N
+�
+j=1
+1i̸=j1yi=yj
+log
+esim(hj,hi)τ
+�N
+k=1 1i̸=kesim(hk,hi)/τ
+(3)
+Nyjindicates the number of examples in a batch
+that have the same label as i, τ is the temperature
+parameter and h denotes the feature vector that is
+from the l2 normalized final encoder hidden layer
+before the softmax projection.
+4
+Our Approach
+Figure 2 shows the overall architecture of our
+method. As figure 2 illustrates, we firstly use a
+simple multi-task model based on RoBERTa-base
+(Liu et al., 2019), and then we develop a contrastive
+learning algorithm where the sense hierarchy is
+used to select positive and negative examples. De-
+tailed descriptions of our framework and our data
+augmentation method are given below.
+4.1
+Sentence Encoder
+Every annotated discourse relation consists of two
+sentences or clauses (its arguments) and one or
+more relational senses that the arguments bear to
+each other. We concatenate the two arguments
+of each example and input them into RoBERTa.
+Following standard practices, we add two special
+tokens to mark the beginning ([CLS]) and the end
+([SEP]) of sentences. We use the representation of
+[CLS] in the last layer as the representation of the
+whole sentences.
+4.2
+Data Augmentation
+To increase the number of training examples, we
+take advantage of meta-data recorded with each Im-
+plicit Discourse Relation in the PDTB (cf. (Webber
+et al., 2019), Section 8]). For each sense taken to
+hold between the arguments of that relation, anno-
+tators have recorded in the meta-data, an explicit
+connective that could have signalled that sense. In
+the past, this meta-data was used in implicit relation
+recognition by both Patterson and Kehler (2013)
+and Rutherford and Xue (2015). We have used
+it in a different way, shown in Figure 3, to create
+an additional training example for each connective
+that appears in the meta-data. In the added training
+example, this added connective becomes part of
+the second argument of the relation (i.e., appearing
+after the [SEP] character)
+Since there is at least one explicit connective
+recorded in the meta-data for each implicit dis-
+course relation and at most two 1, for a training
+batch of N tokens, there will be at least another
+1This is because the PDTB only allows for one or two
+senses per relation.
+
+...
+Encoder La�er
+... ...
+T�a��f���e�
+�a�e�
+...
+Encoder La�er
+Encoder La�er
+I����
+...
+...
+Le�e�-2
+C�a���f�e�
+Le�e�-1
+C�a���f�e�
+P���
+P���
+Se��e H�e�a�c��
+a�c���
+�������e
+�e�a���e
+[CLS]
+�1
+1
+�1
+�
+[SEP]
+�2
+1
+�2
+�
+[SEP]
+[CLS]
+�1
+1
+�1
+�
+[SEP]
+�2
+1
+�2
+�
+[SEP]
+Figure 2: The overall architecture of our model. When given an anchor, we search the positive and negative
+examples in a training batch based on the sense hierarchy of the PDTB. We narrow the distances among examples
+from the same types at level-2 or level-3 and enlarge the distances among examples from different types at level-2
+and level-3.
+N tokens introduced by this data augmentation
+method, increasing the training batch to at least 2N
+tokens.
+𝑒���:������������������������������������������������������������������������������
+𝑒���
+∗
+��������������������������������������������������������������������������������������������
+Figure 3: An example with inserted connective: the
+connective word is “In contrast”.
+4.3
+Positive Pair and Negative Pair
+Generation
+We use the structure of the sense hierarchy to iden-
+tify the positive and negative examples needed for
+contrastive learning. The only senses used in anno-
+tating discourse relations are ones at terminal nodes
+of the sense hierarchy. This is Level 2 for symmet-
+ric senses and Level 3 for asymmetric senses (i.e.,
+where the inverse of the sense that holds between
+Arg1 and Arg2 is what holds between Arg2 and
+Arg1. For example, CONTRAST and SIMILARITY
+are both symmetric senses, while MANNER and
+CONDITION are asymmetric, given that there is a
+difference between Arg2 being the manner of do-
+ing Arg1 or Arg1 being the manner of doing Arg2).
+In our work, when the lowest level of the senses
+is level-3, we directly used the level-3 labels in-
+stead of their parent at level-2. For example, under
+the level-2 label Temporal.asynchronous, there are
+two labels which are precedence and succession
+at level-3. For this case, we replaced the level-2
+label Temporal.asynchronous with the two labels
+precedence and succession at level-3.
+Although supervised contrastive learning in Eq.
+3 can be valid for different classes of positive ex-
+ample pairs, its negative examples come from any
+examples inside a batch except itself. We defined
+l1, l2, l3 as the first, second, and third level in the
+hierarchical structure respectively, and lϵli refers
+to the labels from level i.
+Instance e ∼ Same sub-level
+epos
+Given the representation of a sentence ei and its
+first, second and third level of label li
+1, li
+2, li
+3, we
+searched the set of examples with the same sec-
+ond level labels or the same third level labels (if
+the lowest level is level-3) as epos in each training
+batch:
+ei
+pos = {e ∈ ei
+pos : le
+2 == li
+2
+or
+le
+3 == li
+3}
+(4)
+E.g.
+If the label of the anchor is Tempo-
+ral.asynchronous.precedence, its positive examples
+would be the examples with the same label.
+Instance e ∼ Batch instance
+eneg
+Here, we would like to help the model discriminate
+the sister types at level-2 and level-3 (if the lowest
+level is level-3). We searched the set of examples
+with different level-2 labels or level-3 labels as eneg
+in each training batch.
+E.g.
+If the label of the anchor is Tempo-
+ral.asynchronous.precedence, its negative exam-
+ples would be its sister types at level-2 and level-
+3, namely Temporal.asynchronous.succession and
+Temporal.synchronous.
+ei
+neg = {e ∈ ei
+neg : le
+1 == li
+1
+&
+(le
+2 ̸= li
+2
+&
+le
+3 ̸= li
+3)}
+(5)
+
+4.4
+Loss Algorithms
+As described above, given the query ei with its
+positive pairs and negative pairs and based on the
+general contrastive learning loss (see Equation 2),
+the contrastive learning loss for our task and ap-
+proach is:
+Lscl =
+N
+�
+i=1
+−
+1
+|eipos| − 1
+2N
+�
+j=1
+1i̸=j1j∈eipos
+log
+wjesim(hj,hi)τ
+�2N
+k=1 1i̸=k1k∈eineg+eiposwkesim(hk,hi)/τ
+(6)
+where wj and wj are weight factors for differ-
+ent positive pairs and negative pairs respectively,
+sim(hi, hj) is cosine similarity and τ is a tempera-
+ture hyperparameter.
+Our overall training goal is:
+L = Ll1
+sup + Ll2
+sup + βLscl
+(7)
+As our classifications are done in the first level
+and second level for the same inputs, we used a
+standard cross-entropy loss to get supervised loss
+LL1
+sup and LL2
+sup. And β is the weighting factor for
+the contrastive loss.
+5
+Experiment Setting
+5.1
+Datasets
+Besides providing a sense hierarchy, the Penn Dis-
+course TreeBank (PDTB) also frequently serves as
+a dataset for evaluating the recognition of discourse
+relations. The earlier corpus, PDTB-2 (Prasad et al.,
+2008) included 40,600 annotated relations, while
+the later version, PDTB-3 (Webber et al., 2019)
+includes an additional 13K annotations, primarily
+intra-sentential, as well as correcting some incon-
+sistencies in the PDTB-2. The sense hierarchy used
+in the PDTB-3 differs somewhat from that used in
+the PDTB-2, with additions motivated by the needs
+of annotating intra-sentential relations and changes
+motivated by difficulties that annotators had in con-
+sistently using some of the senses in the PDTB-2
+hierarchy.
+Because of the differences in these two hierar-
+chies, we use the PDTB-2 hierarchy for PDTB-2
+data and the PDTB-3 hierarchy for PDTB-3 data
+respectively. We follow earlier work (Ji and Eisen-
+stein, 2015; Bai and Zhao, 2018; Liu et al., 2020;
+Xiang et al., 2022) using Sections 2-20 of the cor-
+pus for Training, Sections 0-1 for Validation, and
+Sections 21-22 for testing. With regard to those
+instances with multiple annotated labels, we also
+follow previous work (Qin et al., 2016). They are
+treated as separate examples during training. At
+test time, a prediction matching one of the gold
+types is taken as the correct answer. Implicit rela-
+tion recognition is usually treated as a classifica-
+tion task. While 4-way (Level-1) classification was
+carried out on both PDTB-2 and PDTB-3, more
+detailed 11-way (Level 2) classification was done
+only on the PDTB-2 and 14-way (Level 2) classifi-
+cation, only on the PDTB-3.
+5.2
+Baselines
+To exhibit the effectiveness of our proposed
+method, we compare our method with strong base-
+lines. As previous work usually used one dataset
+(PDTB-2 or PDTB-3) for evaluation, we use dif-
+ferent baselines for PDTB-2 and PDTB-3. Since
+PDTB-3 was not released until 2019, the baselines
+for PDTB-3 from 2016 and 2017 are from (Xiang
+et al., 2022). They reproduced those models which
+were originally used on PDTB-2 on PDTB-3.
+Baselines for PDTB-2:
+• (Dai and Huang, 2019): a neural model lever-
+aging external event knowledge and corefer-
+ence relations.
+• (Shi and Demberg, 2019a): a neural model
+that leverages the inserted connectives to learn
+better argument representations.
+• (Nguyen et al., 2019): a neural model which
+predicts the labels and connectives. simulta-
+neously.
+• (Guo et al., 2020): a knowledge-enhanced
+Neural Network framework.
+• (Kishimoto et al., 2020): a model applying
+three additional training tasks.
+• (Liu et al., 2020): a RoBERTa-based model
+which consists of three different modules.
+• (Jiang et al., 2021): a method that recognizes
+the relation label and generates the target sen-
+tence simultaneously.
+• (Dou et al., 2021): a method using conditional
+VAE to estimate the risk of erroneous sam-
+pling.
+• (Wu et al., 2022): a label dependence-aware
+sequence generation model.
+Baselines for PDTB-3:
+• (Liu and Li, 2016): a model that combines
+two arguments’ representation for stacked in-
+teractive attention.
+
+�1
+Acc
+β
+β
+(a) Top-level label classification 
+(b) Second-level label classification 
+PDTB3
+PDTB2
+Figure 4: Effects of β on the validation set.
+• (Chen et al., 2016a): a mixed generative-
+discriminative framework.
+• (Lan et al., 2017): a multi-task attention neu-
+ral network.
+• (Ruan et al., 2020): a propagative attention
+learning model.
+• (Xiang et al., 2022): a model that uses a Dual
+Attention Network (DAN).
+5.3
+Parameters Setting
+In our experiments,
+we use the pre-trained
+RoBERTa-base (Liu et al., 2019) as our Encoder.
+We adopt Adam (Kingma and Ba, 2015) with the
+learning rate of 3e−5 and the batch size of 256 to
+update the model. The maximum training epoch is
+set to 25 and the wait patience for early stopping
+is set to 10 for all models. We clip the gradient
+L2-norm with a threshold 2.0. For contrast learn-
+ing, the weight of positive examples is set to 1.6
+and the weight of negative examples is set to 1. All
+experiments are performed with 1× 80GB NVIDIA
+A100 GPU.
+5.4
+Evaluation Metrics
+We used Accuracy and Macro-F1 score as evalu-
+ation metrics, because PDTB datasets are imbal-
+anced and Macro-F1 score has been said to be an
+more appropriate assessment measure for imbal-
+anced datasets (Akosa, 2017; Bekkar et al., 2013).
+5.5
+Effects of the Coefficient β
+As shown in Equation 7, the coefficient β is an
+important hyperparameter that controls the relative
+importance of supervised loss and contrastive loss.
+Thus, we vary β from 0 to 2.4 with an increment of
+0.2 each step, and inspect the performance of our
+model using different β on the validation set.
+From Figure 4, we can find that, compared with
+the model without contrastive learning (β = 0), the
+performance of our model at any level is always
+improved via contrastive learning. For PDTB-2,
+when β exceeds 1.0, the performance of our model
+tends to be stable and declines finally. Thus, we
+directly set β = 1.0 for all PDTB-2 related exper-
+iments thereafter. For PDTB-3, the Acc and F1
+of the validation set reach the highest point at β =
+2.0. Therefore we choose β = 2.0 for all related
+experiments.
+We have considered three ways of investigat-
+ing why there is such a difference in the optimal
+weighting coefficient. First, compared with PDTB-
+2, the PDTB-3 contains about 6000 more implicit
+tokens annotated for discourse relations. Secondly,
+although the sense hierarchies of both the PDTB-
+2 and the PDTB-3 have three levels and have the
+same senses at level- 1, but many changes at level-2
+and level-3 due to difficulties found in annotating
+certain senses. Moreover, the intra-sentential im-
+plicit relations might be another reason. In PDTB-
+3, many more discourse relations are annotated
+within sentences. Liang et al. (2020) report quite
+striking difference in the distribution of sense re-
+lations inter-sententially vs. intra-sententially be-
+tween PDTB-2 and PDTB-3. Therefore, these ma-
+jor differences in the PDTB-3 and the PDTB-2
+might cause the fluctuation of the coefficient value.
+6
+Results and Analysis
+The results on PDTB-2 and PDTB-3 for Level-1
+and Level-2 are presented in Table 1 and Table 2
+respectively, where the best results are highlighted
+in bold. Classification performance on PDTB-2 in
+terms of Macro-F1 for the four general sense types
+at Level-1 and 11 sense types at Level-2 is shown
+in Table 3 and Table 4.
+These results demonstrate better performance
+than previous systems for both Level-1 and Level-2
+classification on both PDTB-2 and PDTB-3. In
+particular, the results clearly demonstrate benefits
+to be gained from contrastive learning. But there is
+more to be said: In Section 6.1, we discuss different
+ways of defining negative examples with respect to
+the sense hierarchy, and in Section 6.2, we discuss
+the relative value of the particular form of data
+augmentation we have used (cf. Section 4.2) as
+compared with our method of contrastive learning.
+6.1
+Comparisons with Other Negatives
+Selecting Methods
+There is not only one way to select negative ex-
+amples for contrastive learning based on PDTB
+
+75
+70
+65
+60
+55
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.460
+55
+50
+45
+40
+35
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.465
+62
+59
+56
+53
+50
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.475
+73
+71
+69
+67
+65
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4Model
+PDTB-2
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+Dai and Huang (2019)
+59.66
+52.89
+48.23
+33.41
+Shi and Demberg (2019a)
+61.42
+46.40
+47.83
+-
+Nguyen et al. (2019)
+-
+53.00
+49.95
+-
+Guo et al. (2020)
+57.25
+47.90
+-
+-
+Kishimoto et al. (2020)
+65.26
+58.48
+52.34
+-
+Liu et al. (2020)
+69.06
+63.39
+58.13
+-
+Jiang et al. (2021)
+-
+57.18
+-
+37.76
+Dou et al. (2021)
+70.17
+65.06
+-
+-
+Wu et al. (2022)
+71.18
+63.73
+60.33
+40.49
+Ours
+72.18
+69.60
+61.69
+49.66
+Table 1: Experimental results on PDTB-2.
+Model
+PDTB-3
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+Liu and Li (2016)
+57.67
+46.13
+-
+-
+Chen et al. (2016b)
+57.33
+45.11
+-
+-
+Lan et al. (2017)
+57.06
+47.29
+-
+-
+Ruan et al. (2020)
+58.01
+49.45
+-
+-
+Xiang et al. (2022)
+60.45
+53.14
+-
+-
+(BiLSTM)
+Xiang et al. (2022)
+64.04
+56.63
+-
+-
+(BERT)
+Ours
+75.31
+70.05
+64.68
+57.62
+Table 2: Experimental results on PDTB-3.
+Model
+Comp.
+Cont
+Exp.
+Temp.
+Nguyen et al. (2019)
+48.44
+56.84
+73.66
+38.60
+Guo et al. (2020)
+43.92
+57.67
+73.45
+36.33
+Liu et al. (2020)
+59.44
+60.98
+77.66
+50.26
+Jiang et al. (2021)
+55.40
+57.04
+74.76
+41.54
+Dou et al. (2021)
+55.72
+63.39
+80.34
+44.01
+Ours
+65.84
+63.55
+79.17
+69.86
+Table 3: The results for relation types at level-1 on
+PDTB-2 in terms of F1 (%) (top-level multi-class clas-
+sification).
+hierarchical structures. In addition to the method
+we adopt, we have explored another 4 different
+methods of defining positive and negative exam-
+ples by using the sense hierarchies, which can be
+shown in Figure 5. One can choose the level against
+which to select negative examples: method 2 be-
+low uses examples with different labels at level-2,
+while methods 1, 3 and 4 use examples with dif-
+ferent labels at level-1. With regard to the use of
+weight for method 3 and method 4, we aim to give
+more weight to more similar (potentially) positive
+examples based on the hierarchy. Specifically, we
+give more weight to the examples from the same
+level-2/level-3 type than their sister types at level-
+2/level-3 when all of the examples from the same
+level-1 are positive examples. Besides, method 4
+leverages level-3 labels, while method 1 to 3 only
+consider level-1 and level-2 labels. In our exper-
+iments for other negatives defining methods, we
+use the same hyperparameters as the experimental
+setup of our methods.For method 3 and method 4,
+Second-level Label
+Liu et al. (2020)
+Wu et al. (2022)
+Ours
+Temp.Asynchronous
+56.18
+56.47
+59.79
+Temp.Synchrony
+0.00
+0.00
+78.26
+Cont.Cause
+59.60
+64.36
+65.58
+Cont.Pragmatic cause
+0.0
+0.0
+0.00
+Comp.Contrast
+59.75
+63.52
+62.63
+Comp.Concession
+0.0
+0.0
+0.00
+Exp.Conjunction
+60.17
+57.91
+58.35
+Exp.Instantiation
+67.96
+72.60
+73.04
+Exp.Restatement
+53.83
+58.06
+60.00
+Exp.Alternative
+60.00
+63.46
+53.85
+Exp.List
+0.0
+8.98
+34.78
+Table 4: The results for relation types at level-2 on
+PDTB-2 in terms of F1 (%) (second-level multi-class
+classification).
+the weight of positive examples is set to 1.6 and
+1.3 and the weight of negative examples still is 1.
+It can be seen from table 5 and table 6 that our
+method is better than the above methods in both
+datasets for both level-1 and level-2 classification
+tasks. Compared with method 2, we utilize level-3
+labels, which indicated the level-3 label informa-
+tion is helpful for the approach. The greatest differ-
+ence between our method and other three methods
+is that our negative examples are only those sis-
+ter types at level-2 or level-3, not including the
+examples from different level-1. On the contrary,
+the negative examples in those three methods are
+examples from other level-1 types. We suppose
+that this might make a too strong assumption that
+examples from different level-1 are very dissimilar.
+In PDTB datasets, some examples have been anno-
+tated with multiple labels. We found that among all
+examples with multiple annotated labels, there are
+99.26% examples whose multiple labels are under
+different level-1. Moreover, some level-1 types of
+relation might be overlapped even if the annotators
+just annotate one label. For example, some exam-
+ples annotated as Temporal.asynchronous might
+have the sense of Contingency.cause as well. And
+Moens and Steedman (1988) have pointed out that
+when-clauses do not simply predicate a temporal re-
+lation, but a causal one as well, which can be called
+contingency. This shows up in the PDTB in terms
+of the variation in how particular tokens of when
+clauses have been annotated. But it also means
+that in choosing Negative examples, relations la-
+belled TEMPORAL.SYNCHRONOUS or TEMPO-
+RAL.ASYNCHRONOUS may closely resemble those
+labelled CONTINGENCY.CAUSE and therefore not
+be effective as negative examples. Specifically, for
+the following example:
+(4) when [they built the 39th Street bridge]1,
+[they solved most of their traffic problems]2.
+
+Model
+PDTB-2
+PDTB-3
+Top Level
+Second Level
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+Acc
+Macro-F1
+Acc
+Macro-F1
+Method 1
+68.91
+65.04
+58.61
+46.27
+73.25
+68.00
+61.17
+55.58
+Method 2
+69.39
+63.95
+58.33
+44.80
+73.53
+68.36
+61.93
+54.85
+Method 3
+69.39
+66.53
+58.61
+39.20
+72.49
+67.49
+60.77
+54.33
+Method 4
+69.10
+65.30
+57.07
+47.46
+71.26
+66.47
+59.53
+47.24
+Ours
+72.18
+69.60
+61.69
+49.66
+75.31
+70.05
+64.48
+57.62
+Table 5: Comparisons with other negatives defining methods.
+Positve: examples with same label at level-1.
+Negative: examples with different labels at level-1.
+(a) method 1
+Positve: examples with same label at level-2.
+Negative: examples with different labels at level-2.
+(b) method 2
+Positve: Examples with same label at level-1.     
+              More weight are given to the examples 
+              with same label at level-2.
+Negative: Examples with different labels at level-1.
+(c) method 3
+Positve: Examples with same label at level-1, 
+              more weight are given to   the examples 
+              with same label at level-2 or level-3. 
+Negative: Examples with different labels at level-1.
+(d) method 4
+Figure 5: Another four negative examples selected
+methods. orange ball represent anchor, green ball rep-
+resent negative examples, and blue ball represent posi-
+tive examples. Darker blue ball means more weight is
+given to more similar (potentially) positive examples.
+If the connective “when” is replaced with “be-
+cause”, the sentence still sounds not strange. There-
+fore, regarding all examples from different level-1
+as negative examples might have some negative
+impacts on learning the representations.
+6.2
+Ablation Study
+We wanted to know how useful our data augmen-
+tation method and our contrastive learning method
+are, so we have undertaken ablation studies for this.
+Model
+Comp.
+Cont
+Exp.
+Temp.
+Method 1
+63.26
+60.42
+76.78
+59.74
+Method 2
+60.78
+60.82
+77.89
+56.30
+Method 3
+59.85
+65.18
+76.43
+64.67
+Method 4
+57.25
+61.73
+77.30
+64.90
+Ours
+65.84
+63.55
+79.17
+69.86
+Table 6: The results of relation types at level-1 on
+PDTB-2 in terms of F1 (%) (top-level multi-class clas-
+sification).
+Effects
+of
+contrastive
+learning
+algorithm
+From Table 7, it can be seen that multi-task
+learning
+method
+where
+level-1
+and
+level-2
+labels are predicted simultaneously by using the
+same [CLS] representation perform better than
+separately predicting level-1 and level-2 labels,
+which verifies the dependency between different
+levels.
+Compared with the multi-task learning
+method, our model with a contrastive loss has
+better performance in PDTB-2 and PDTB-3, which
+means that our contrasting learning method is
+indeed helpful.
+Effects of data augmentation
+Table 8 compares
+the results with and without data augmentation for
+both PDTB-2 and PDTB-3. From the comparisons,
+it is clear that the data augmentation method is
+helpful to generate useful examples. Khosla et al.
+(2020) showed that having a large number of hard
+positives/negatives in a batch leads to better perfor-
+mance. Since we have many classes at the second
+level, 11 types for PDTB-2 and 14 types for PDTB-
+3. In a batch with the size of 256, it is difficult to
+guarantee that there are enough positive examples
+for each class to take full advantage of contrast
+learning. Therefore, without data augmentation,
+the performance of our method degrades consider-
+ably.
+7
+Limitations and Future work
+With regard to PDTB-2 and PDTB-3 annotation,
+there are two cases: (1) Annotators can assign mul-
+tiple labels to an example when they believe more
+than one relation holds simultaneously; (2) An-
+notators can be told (in the Annotation Manual)
+to give precedence to one label if they take more
+than one to hold. For example, they are told in
+the Manual (Webber et al., 2019) that examples
+that satisfy the conditions for both Contrast and
+Concession, should be labelled as concession. We
+over-simplified the presence of multiple labels by
+following Qin et al. (2017) in treating each label as
+
+Datasets
+Model
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+PDTB-2
+RoBERTa
+68.14
+64.87
+58.33
+48.37
+RoBERTa-MTL
+69.87
+65.39
+58.22
+45.21
+Ours
+72.18
+69.60
+61.69
+49.66
+PDTB-3
+RoBERTa
+72.02
+67.44
+60.56
+57.12
+RoBERTa-MTL
+72.63
+68.23
+60.56
+57.16
+Ours
+75.31
+70.05
+64.68
+57.62
+Table 7: Ablation study on PDTB-2 and PDTB-3.
+Model
+Top Level
+Second Level
+Acc
+Macro-F1
+Acc
+Macro-F1
+PDTB-2
+Ours
+72.18
+69.60
+61.69
+49.66
+-augmentation
+71.70
+67.85
+59.19
+45.54
+PDTB-3
+Ours
+75.31
+70.05
+64.68
+57.62
+-augmentation
+73.32
+69.02
+63.24
+51.80
+Table 8: Effects of data augmentation.
+a separate example and did not consider the second
+case. Thus, our approach might be inadequate for
+dealing with the actual distribution of the data and
+can be extended or modified. It is worth exploring
+how to extend our approach to allow for examples
+with multiple sense labels and cases where one la-
+bel takes precedence over another. We believe that
+this will be an important property of the work.
+Another limitation is that we only use English
+datasets. There are PDTB-style datasets in other
+languages including a Chinese TED dicourse bank
+corpus (Long et al., 2020), a Turkish discourse
+Tree bank corpus (Zeyrek and Kurfalı, 2017) and
+an Italian Discourse Treebank (Pareti and Prodanof,
+2010). Moreover, Zeyrek et al. (2019) proposed
+a TED Multilingual Discourse Bank (TED-MDB)
+corpus, which has 6 languages. These datasets al-
+low us to assess the approach in languages other
+than English. Besides, there are datasets similar
+to PDTB-Style like Prague Dependency Treebank
+(Mírovský et al., 2014). The different datasets use
+essentially similar sense hierarchy, but two things
+need to be investigated (i) whether there are compa-
+rable differences between tokens that realise “sis-
+ter” relations, or (ii) whether tokens often have
+multiple sense labels, which would change what
+could be used as negative examples if leveraging
+our approach on them.
+In the future, we can also assess whether con-
+trastive learning could help in separating out En-
+tRel relations and AltLex relations from implicit
+relations or whether other methods would perform
+better.
+8
+Conclusions
+In this paper, we leverage the sense hierarchy to
+select the negative examples needed for contrastive
+learning for the task of implicit discourse relation
+recognition. Our method has better overall perfor-
+mance than achieved by previous systems, and com-
+pared with previous work, our method is better at
+learning minority labels. Moreover, we compared
+different methods of selecting the negative exam-
+ples based on the hierarchical structures, which
+shows some potential negative impacts might be
+produced when negative examples include those
+from other level-1 types. Moreover, we conduct ab-
+lation studies to investigate the effects of our data
+augmentation method and our contrastive learning
+method. Besides, the limitations and the future
+work are discussed.
+Acknowledgments
+This work was supported in part by the UKRI
+Centre for Doctoral Training in Natural Lan-
+guage Processing, funded by the UKRI (grant
+EP/S022481/1), the University of Edinburgh. The
+authors also gratefully acknowledge University of
+Edinburgh Huawei Laboratory for their support.
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+
+A
+Appendix
+A.1
+PDTB Hierarchy
+The hierarchies of both PDTB 2.0 and PDTB 3.0
+consist of three levels, but for implicit relation
+recognition, so far no classification for third level
+labels has been done. We also focus on the hier-
+archy between level-1 and level 2. The PDTB-3
+relation hierarchy simplifies and extends the PDTB-
+2 relation hierarchy. The PDTB 3.0 hierarchy not
+only simplifies the PDTB-2 relation hierarchy by
+restricting Level-3 relations to differences in di-
+rectionality and eliminating rare and/or difficult-
+to-annotate senses, but also augments the relation
+hierarchy. Figure 6 and Figure 7 show PDTB 2.0
+relation hierarchy and PDTB 3.0 relation hierarchy
+respectively.
+Asynchronous
+Synchronous
+Figure 6: The PDTB 2.0 Senses Hierarchy.
+Figure 7: The PDTB 3.0 Senses Hierarchy. The left-
+most column contains the Level-1 senses and the mid-
+dle column, the Level-2 senses. For asymmetric rela-
+tions, Level-3 senses are located in the rightmost col-
+umn.
+Model
+Comp.
+Cont
+Exp.
+Temp.
+Liu and Li (2016)
+29.15
+63.33
+65.10
+41.03
+Lan et al. (2017)
+30.10
+60.91
+64.03
+33.71
+Ruan et al. (2020)
+30.37
+61.95
+64.28
+34.74
+Chen et al. (2016b)
+27.34
+62.56
+64.71
+38.91
+Xiang et al. (2022)
+34.16
+65.48
+67.82
+40.22
+(BiLSTM)
+Xiang et al. (2022)
+35.83
+66.77
+70.00
+42.13
+(BERT)
+Ours
+63.30
+78.60
+79.91
+58.39
+Table 9: The results of different relations on PDTB-3
+in terms of F1 (%) (top-level multi-class classification).
+Second-level Label
+Ours
+Temp.Asynchronous
+66.35
+Temp.Synchrony
+41.38
+Cont.Cause
+71.38
+Cont.Cause+Belief
+0.0
+Cont.Condition
+74.07
+Cont.Purpose
+96.05
+Comp.Contrast
+56.91
+Comp.Concession
+60.11
+Exp.Conjunction
+61.70
+Exp.Equivalence
+11.43
+Exp.Instantiation
+69.83
+Exp.Level-of-detail
+55.34
+Exp.Manner
+78.43
+Exp.Substitution
+63.77
+Table 10: The results of different relations on PDTB-3
+in terms of F1 (%) (second-level multi-class classifica-
+tion).
+A.2
+The results on relation types on PDTB-3
+We also examine the classification performance on
+PDTB-3 in terms of Macro-F1 for the four main
+relation types at level-1 and 14 sense types at level-
+2. The results can be seen in Table 9 and Table 10.
+Our model has significantly better performance for
+all level-1 relations.
+As for level-2 sense types, because there are no
+results of previous systems, we just show the result
+of 14 level-2 sense types in PDTB-3 in terms of F1.
+
+Synchronous
+Temporal
+Precedence
+Asynchronous
+Succession
+Reason
+Cause
+Result
+Conjunction
+Disjunction
+Negative-result*
+Equivalence
+-
+Arg1-as-cond
+Condition
+Arg1-as-instance
+Contingency
+Arg2-as-cond
+Instantiation
+Arg2-as-instance
+Arg1-as-negcond
+Negative condition
+Arg1-as-detail
+Arg2-as-negcond
+Level-of-detail
+Expansion
+Arg2-as-detail
+Arg1-as-goal
+Arg1-as-subst
+Purpose
+Arg2-as-goal
+Substitution
+Arg2-as-negGoal
+Arg2-as-subst
+Arg1-as-excpt
+Exception
+Contrast
+Arg2-as-excpt
+Similarity
+Manner
+Arg1-as-manner
+Comparison
+Arg1-as-denier*
+Arg2-as-manner
+Concession
+Arg2-as-denierTEMPORAL
+COMPARISON
+ Asynchronous
+→Contrast
+→ Synchronous
+ juxtaposition
+> opposition
+precedence
+ succession
+Pragmatic Contrast
+→ Concession
+expectation
+contra-expectation
+CONTINGENCY
+Cause
+ Pragmatic Concession
+reason
+result
+EXPANSION
+Conjunction
+Pragmatic Cause
+Instantiation
+justification
+Restatement
+Condition
+ specification
+ hypothetical
+ equivalence
+general
+unreal present
+ generalization
+unreal past
+Alternative
+factual present
+ conjunctive
+ factual past
+→ disjunctive
+→ chosen alternative
+ Pragmatic Condition
+Exception
+relevance
+implicit assertion
+List
\ No newline at end of file
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf,len=1124
+page_content='Facilitating Contrastive Learning of Discourse Relational Senses by  Exploiting the Hierarchy of Sense Relations  Wanqiu Long†, and Bonnie Webber†  † University of Edinburgh, Edinburgh, UK  Wanqiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='long@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='uk, Webber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Bonnie@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='uk  Abstract  Implicit discourse relation recognition is a  challenging task that involves identifying the  sense or senses that hold between two adja-  cent spans of text, in the absence of an explicit  connective between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  In both PDTB-2  (Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2008) and PDTB-3 (Webber  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019), discourse relational senses are  organized into a three-level hierarchy rang-  ing from four broad top-level senses, to more  specific senses below them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Most previous  work on implicit discourse relation recogni-  tion have used the sense hierarchy simply  to indicate what sense labels were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Here we do more — incorporating the sense  hierarchy into the recognition process itself  and using it to select the negative examples  used in contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  With no addi-  tional effort, the approach achieves state-of-  the-art performance on the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Our code  is released in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='com/wanqiulong  0923/Contrastive_IDRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  1  Introduction  Discourse relations are an important aspect of  textual coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In some cases, a speaker or  writer signals the sense or senses that hold between  clauses and/or sentences in a text using an explicit  connective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Recognizing the sense or senses that  hold can be more difficult, in the absense of an  explicit connective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Automatically identifying the sense or senses  that hold between sentences and/or clauses can  be useful for downstream NLP tasks such as text  summarization (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2018), machine trans-  lation (Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2015) and event relation ex-  traction (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Recent studies on im-  plicit discourse relation recognition have shown  great success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Especially, pre-trained neural lan-  guage models (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019) have been used and dramat-  ically improved the performances of models (Shi  and Demberg, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Kishimoto  Level2  Level1  Temporal  Asynchronous  Level3  Precedence  Succession  Root  Comparison  Contigency  Expansion  Synchronous  Concession  Contrast  Cause  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Purpose  Cause+Belief  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Reason  Result  Equivalence  Manner  Reason+Belief  Result+Belief  Figure 1: The PDTB-3 Sense Hierarchy  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The senses available for labelling  discourse relations in the PDTB-2 (and later in the  PDTB-3) are arranged in a three-level hierarchy,  with the most general senses at the top and more  specific senses further down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In the PDTB-3, anno-  tators could only choose senses at terminal nodes  in the hierarchy – level-2 senses for symmetric re-  lations such as EXPANSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='EQUIVALENCE and  TEMPORAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='SYNCHRONOUS, and level-3 senses  for asymmetric relations, with the direction of  the relation encoded in its sense label such as  SUBSTITUTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='ARG1-AS-SUBST (where the text  labelled ARG1 substitutes for the denied text  labelled ARG2) and SUBSTITUTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='ARG2-AS-  SUBST (where the text labelled ARG2 substitutes  for the denied text labelled ARG1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Early work on  recognizing the implicit relations only used the hi-  erarchy to choose a target for recognition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', the  senses at level-1 (classes) or those at level-2 (types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Recently, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2022) have tried to leverage the  dependence between the level-1 and level-2 labels  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The current work goes further, us-  ing the whole three-level sense hierarchy to select  the negative examples for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Contrastive learning, which aims to minimize  the distance between similar instances (defined as  positive examples) and widen the difference with  dissimilar instances (negative examples), has been  considered as effective in constructing meaning-  ful representations (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Previous work on con-  trastive learning indicates that it is critical to se-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='02724v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='CL]  6 Jan 2023  lect good negative samples (Alzantot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The insight  underlying the current work is that the hierarchy  of sense labels can enable the selection of good  negative examples for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  To  see this, consider Examples 1-3 below from the  PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' On the surface, they look somewhat sim-  ilar, but in Examples 1 and 2, the annotators took  the second sentence (Arg2) as providing more de-  tail about the first sentence (Arg1) — the sense  called EXPANSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='LEVEL-OF-DETAIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='ARG2-AS-  DETAIL, while in Example 3, they took the sec-  ond sentence as expressing a substitute for “Amer-  ican culture” in terms of what is relevant – the  sense called EXPANSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='SUBSTITUTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='ARG2-  AS-SUBST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (1) “Valley National ”“isn’t out of the woods yet  ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The key will be whether Arizona real es-  tate turns around or at least stabilizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='.  (2) The House appears reluctant to join the sena-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' A key is whether House Republicans  are willing to acquiesce to their Senate col-  leagues’ decision to drop many pet provi-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='.  (3) Japanese culture vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' American culture is ir-  relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The key is how a manager from  one culture can motivate employees from  another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='.  In this work, we use a multi-task learning frame-  work, which consists of classification tasks and a  contrastive learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Unlike most previous  work using one benchmark dataset (usually PDTB-  2 or PDTB-3), we evaluate our systems on both  PDTB-2 and PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Besides, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2021)  have shown that data augmentation can make rep-  resentations be more robust, thereby enriching the  data used in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We thus follow Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (2021) and Khosla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020) in identifying a rel-  evant form of data augmentation for our contrastive  learning approach to implicit relation recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  The main contributions of our work are as fol-  lows:  We leveraged the sense hierarchy to get con-  trastive learning representation, learning an  embedding space in which examples from  same types at level-2 or level-3 stay close to  each other while sister types are far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  We explored and compared different methods  of defining the negatives based on the sense  hierarchies in PDTB-2 and PDTB-3, finding  the approach which leads to the greatest im-  provements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Our proposed data augmentation method to  generate examples is helpful to improve the  overall performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  We demonstrate that implicit relation recogni-  tion can benefit from a deeper understanding  of the sense labels and their organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  2  Related Work  Implicit discourse relation recognition  For  this task, Dai and Huang (2018) considered  paragraph-level context and inter-paragraph de-  pendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Recently, Shi and Demberg (2019b)  showed that using the bidirectional encoder repre-  sentation from BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019) is more  accurately to recognize Temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Synchrony, Com-  parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Contrast, Expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Conjunction and Ex-  pansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020) showed that  different levels of representation learning are all  important to implicit relation recognition, and they  combined three modules to better integrate con-  text information, the interaction between two argu-  ments and to understand the text in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' How-  ever, only two existing works leveraged the hier-  archy in implicit relation recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Both Wu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020a) and Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2022) first attempted  to assign a Level-1 sense that holds between argu-  ments, and then only considered as possible Level-  2 senses, those that are daughters of the Level-1  sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Contrastive learning  Recently, there has been  a growing interest in applying contrastive learn-  ing in both the pre-training and fine-tuning objec-  tives of pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (2021) used a contrastive objective to fine-tune pre-  trained language models to obtain sentence embed-  dings, and greatly improves state-of-the-art sen-  tence embeddings on semantic textual similarity  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Suresh and Ong (2021) proposed label-aware  contrastive loss in the presence of larger number  and/or more confusable classes, and helps models  to produce more differentiated output distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Besides, many works have demonstrated that se-  lecting good negative examples are very important  for using contrastive learning (Schroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Joshua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In our work,  we integrate contrastive learning loss with super-  vised losses and we use the structure of the sense  hierarchy to guide the selection of negative exam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  3  Learning Loss  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='1  Supervised Learning Loss  The standard approach today for classification task  is to use a standard cross-entropy loss:  Lsup = 1  N  N  �  i=1  −log  eW T  yisi  �  j eW T  j si  (1)  Where N denotes the number of training examples,  yi is the ground-truth class of the i-th class and Wj  is the weight vector of the j-th class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  Contrastive Learning Loss  In contrastive learning, each example can be treated  as an anchor to get its positive and negative ex-  amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Contrastive learning can pull the anchor  and its positive example together in the embedding  space, while the anchor and negative samples are  pushed apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The contrastive learning loss was  used by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Suresh and Ong (2021)  before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' A set of N randomly sampled label pairs  is defined as xk, yk, where x and y represent sam-  ples and labels, respectively, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Let i  be the index of anchor sample and j is the index  of a positive sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' where iϵ{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', N}, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Contrastive loss is defined as:  Lscl = −  N  �  i=1  esim(hj,hi)τ  �  i̸=k esim(hk,hi)τ  (2)  Here, h denotes the feature vector in the em-  bedding space, and τ is the temperature parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Intuitively, the numerator computes the inner dot  product between the anchor points i and its positive  sample j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The denominator computes the inner dot  product between all i and the inner dot product be-  tween all negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' where a total of N − 1  samples are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Supervised contrastive learning (Gunel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=',  2021) extends the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2 to the supervised sce-  nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In particular, given the presence of labels,  the positive examples are all examples with the  same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The loss is defined as:  Lscl =  N  �  i=1  −  1  Nyi − 1  N  �  j=1  1i̸=j1yi=yj  log  esim(hj,hi)τ  �N  k=1 1i̸=kesim(hk,hi)/τ  (3)  Nyjindicates the number of examples in a batch  that have the same label as i, τ is the temperature  parameter and h denotes the feature vector that is  from the l2 normalized final encoder hidden layer  before the softmax projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  4  Our Approach  Figure 2 shows the overall architecture of our  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' As figure 2 illustrates, we firstly use a  simple multi-task model based on RoBERTa-base  (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019), and then we develop a contrastive  learning algorithm where the sense hierarchy is  used to select positive and negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' De-  tailed descriptions of our framework and our data  augmentation method are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='1  Sentence Encoder  Every annotated discourse relation consists of two  sentences or clauses (its arguments) and one or  more relational senses that the arguments bear to  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We concatenate the two arguments  of each example and input them into RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Following standard practices, we add two special  tokens to mark the beginning ([CLS]) and the end  ([SEP]) of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We use the representation of  [CLS] in the last layer as the representation of the  whole sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  Data Augmentation  To increase the number of training examples, we  take advantage of meta-data recorded with each Im-  plicit Discourse Relation in the PDTB (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (Webber  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019), Section 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For each sense taken to  hold between the arguments of that relation, anno-  tators have recorded in the meta-data, an explicit  connective that could have signalled that sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In  the past, this meta-data was used in implicit relation  recognition by both Patterson and Kehler (2013)  and Rutherford and Xue (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We have used  it in a different way, shown in Figure 3, to create  an additional training example for each connective  that appears in the meta-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In the added training  example, this added connective becomes part of  the second argument of the relation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', appearing  after the [SEP] character)  Since there is at least one explicit connective  recorded in the meta-data for each implicit dis-  course relation and at most two 1, for a training  batch of N tokens, there will be at least another  1This is because the PDTB only allows for one or two  senses per relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Encoder La�er  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  T�a��f���e�  �a�e�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Encoder La�er  Encoder La�er  I����  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Le�e�-2  C�a���f�e�  Le�e�-1  C�a���f�e�  P���  P���  Se��e H�e�a�c��  a�c���  �������e  �e�a���e  [CLS]  �1  1  �1  �  [SEP]  �2  1  �2  �  [SEP]  [CLS]  �1  1  �1  �  [SEP]  �2  1  �2  �  [SEP]  Figure 2: The overall architecture of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' When given an anchor, we search the positive and negative  examples in a training batch based on the sense hierarchy of the PDTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We narrow the distances among examples  from the same types at level-2 or level-3 and enlarge the distances among examples from different types at level-2  and level-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  N tokens introduced by this data augmentation  method, increasing the training batch to at least 2N  tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  𝑒���:������������������������������������������������������������������������������  𝑒���  ∗  ��������������������������������������������������������������������������������������������  Figure 3: An example with inserted connective: the  connective word is “In contrast”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='3  Positive Pair and Negative Pair  Generation  We use the structure of the sense hierarchy to iden-  tify the positive and negative examples needed for  contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The only senses used in anno-  tating discourse relations are ones at terminal nodes  of the sense hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' This is Level 2 for symmet-  ric senses and Level 3 for asymmetric senses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=',  where the inverse of the sense that holds between  Arg1 and Arg2 is what holds between Arg2 and  Arg1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For example, CONTRAST and SIMILARITY  are both symmetric senses, while MANNER and  CONDITION are asymmetric, given that there is a  difference between Arg2 being the manner of do-  ing Arg1 or Arg1 being the manner of doing Arg2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  In our work, when the lowest level of the senses  is level-3, we directly used the level-3 labels in-  stead of their parent at level-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For example, under  the level-2 label Temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='asynchronous, there are  two labels which are precedence and succession  at level-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For this case, we replaced the level-2  label Temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='asynchronous with the two labels  precedence and succession at level-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Although supervised contrastive learning in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  3 can be valid for different classes of positive ex-  ample pairs, its negative examples come from any  examples inside a batch except itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We defined  l1, l2, l3 as the first, second, and third level in the  hierarchical structure respectively, and lϵli refers  to the labels from level i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Instance e ∼ Same sub-level  epos  Given the representation of a sentence ei and its  first, second and third level of label li  1, li  2, li  3, we  searched the set of examples with the same sec-  ond level labels or the same third level labels (if  the lowest level is level-3) as epos in each training  batch:  ei  pos = {e ∈ ei  pos : le  2 == li  2  or  le  3 == li  3}  (4)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  If the label of the anchor is Tempo-  ral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='asynchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='precedence, its positive examples  would be the examples with the same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Instance e ∼ Batch instance  eneg  Here, we would like to help the model discriminate  the sister types at level-2 and level-3 (if the lowest  level is level-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We searched the set of examples  with different level-2 labels or level-3 labels as eneg  in each training batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  If the label of the anchor is Tempo-  ral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='asynchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='precedence, its negative exam-  ples would be its sister types at level-2 and level-  3, namely Temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='asynchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='succession and  Temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='synchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  ei  neg = {e ∈ ei  neg : le  1 == li  1  &  (le  2 ̸= li  2  &  le  3 ̸= li  3)}  (5)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='4  Loss Algorithms  As described above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' given the query ei with its  positive pairs and negative pairs and based on the  general contrastive learning loss (see Equation 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  the contrastive learning loss for our task and ap-  proach is:  Lscl =  N  �  i=1  −  1  |eipos| − 1  2N  �  j=1  1i̸=j1j∈eipos  log  wjesim(hj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='hi)τ  �2N  k=1 1i̸=k1k∈eineg+eiposwkesim(hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='hi)/τ  (6)  where wj and wj are weight factors for differ-  ent positive pairs and negative pairs respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  sim(hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' hj) is cosine similarity and τ is a tempera-  ture hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Our overall training goal is:  L = Ll1  sup + Ll2  sup + βLscl  (7)  As our classifications are done in the first level  and second level for the same inputs, we used a  standard cross-entropy loss to get supervised loss  LL1  sup and LL2  sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' And β is the weighting factor for  the contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  5  Experiment Setting  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='1  Datasets  Besides providing a sense hierarchy, the Penn Dis-  course TreeBank (PDTB) also frequently serves as  a dataset for evaluating the recognition of discourse  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The earlier corpus, PDTB-2 (Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=',  2008) included 40,600 annotated relations, while  the later version, PDTB-3 (Webber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019)  includes an additional 13K annotations, primarily  intra-sentential, as well as correcting some incon-  sistencies in the PDTB-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The sense hierarchy used  in the PDTB-3 differs somewhat from that used in  the PDTB-2, with additions motivated by the needs  of annotating intra-sentential relations and changes  motivated by difficulties that annotators had in con-  sistently using some of the senses in the PDTB-2  hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Because of the differences in these two hierar-  chies, we use the PDTB-2 hierarchy for PDTB-2  data and the PDTB-3 hierarchy for PDTB-3 data  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We follow earlier work (Ji and Eisen-  stein, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Bai and Zhao, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2022) using Sections 2-20 of the cor-  pus for Training, Sections 0-1 for Validation, and  Sections 21-22 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' With regard to those  instances with multiple annotated labels, we also  follow previous work (Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' They are  treated as separate examples during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' At  test time, a prediction matching one of the gold  types is taken as the correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Implicit rela-  tion recognition is usually treated as a classifica-  tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' While 4-way (Level-1) classification was  carried out on both PDTB-2 and PDTB-3, more  detailed 11-way (Level 2) classification was done  only on the PDTB-2 and 14-way (Level 2) classifi-  cation, only on the PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  Baselines  To exhibit the effectiveness of our proposed  method, we compare our method with strong base-  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' As previous work usually used one dataset  (PDTB-2 or PDTB-3) for evaluation, we use dif-  ferent baselines for PDTB-2 and PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Since  PDTB-3 was not released until 2019, the baselines  for PDTB-3 from 2016 and 2017 are from (Xiang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' They reproduced those models which  were originally used on PDTB-2 on PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Baselines for PDTB-2:  (Dai and Huang, 2019): a neural model lever-  aging external event knowledge and corefer-  ence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Shi and Demberg, 2019a): a neural model  that leverages the inserted connectives to learn  better argument representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019): a neural model which  predicts the labels and connectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' simulta-  neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020): a knowledge-enhanced  Neural Network framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Kishimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020): a model applying  three additional training tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020): a RoBERTa-based model  which consists of three different modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2021): a method that recognizes  the relation label and generates the target sen-  tence simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2021): a method using conditional  VAE to estimate the risk of erroneous sam-  pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2022): a label dependence-aware  sequence generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Baselines for PDTB-3:  (Liu and Li, 2016): a model that combines  two arguments’ representation for stacked in-  teractive attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  �1  Acc  β  β  (a) Top-level label classification  (b) Second-level label classification  PDTB3  PDTB2  Figure 4: Effects of β on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2016a): a mixed generative-  discriminative framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2017): a multi-task attention neu-  ral network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020): a propagative attention  learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2022): a model that uses a Dual  Attention Network (DAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='3  Parameters Setting  In our experiments,  we use the pre-trained  RoBERTa-base (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019) as our Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  We adopt Adam (Kingma and Ba, 2015) with the  learning rate of 3e−5 and the batch size of 256 to  update the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The maximum training epoch is  set to 25 and the wait patience for early stopping  is set to 10 for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We clip the gradient  L2-norm with a threshold 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For contrast learn-  ing, the weight of positive examples is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='6  and the weight of negative examples is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' All  experiments are performed with 1× 80GB NVIDIA  A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='4  Evaluation Metrics  We used Accuracy and Macro-F1 score as evalu-  ation metrics, because PDTB datasets are imbal-  anced and Macro-F1 score has been said to be an  more appropriate assessment measure for imbal-  anced datasets (Akosa, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Bekkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='5  Effects of the Coefficient β  As shown in Equation 7, the coefficient β is an  important hyperparameter that controls the relative  importance of supervised loss and contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Thus, we vary β from 0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='4 with an increment of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2 each step, and inspect the performance of our  model using different β on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  From Figure 4, we can find that, compared with  the model without contrastive learning (β = 0), the  performance of our model at any level is always  improved via contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For PDTB-2,  when β exceeds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0, the performance of our model  tends to be stable and declines finally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Thus, we  directly set β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0 for all PDTB-2 related exper-  iments thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For PDTB-3, the Acc and F1  of the validation set reach the highest point at β =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Therefore we choose β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0 for all related  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  We have considered three ways of investigat-  ing why there is such a difference in the optimal  weighting coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' First, compared with PDTB-  2, the PDTB-3 contains about 6000 more implicit  tokens annotated for discourse relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Secondly,  although the sense hierarchies of both the PDTB-  2 and the PDTB-3 have three levels and have the  same senses at level- 1, but many changes at level-2  and level-3 due to difficulties found in annotating  certain senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Moreover, the intra-sentential im-  plicit relations might be another reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In PDTB-  3, many more discourse relations are annotated  within sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020) report quite  striking difference in the distribution of sense re-  lations inter-sententially vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' intra-sententially be-  tween PDTB-2 and PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Therefore, these ma-  jor differences in the PDTB-3 and the PDTB-2  might cause the fluctuation of the coefficient value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  6  Results and Analysis  The results on PDTB-2 and PDTB-3 for Level-1  and Level-2 are presented in Table 1 and Table 2  respectively, where the best results are highlighted  in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Classification performance on PDTB-2 in  terms of Macro-F1 for the four general sense types  at Level-1 and 11 sense types at Level-2 is shown  in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  These results demonstrate better performance  than previous systems for both Level-1 and Level-2  classification on both PDTB-2 and PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In  particular, the results clearly demonstrate benefits  to be gained from contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' But there is  more to be said: In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='1, we discuss different  ways of defining negative examples with respect to  the sense hierarchy, and in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2, we discuss  the relative value of the particular form of data  augmentation we have used (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2) as  compared with our method of contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='1  Comparisons with Other Negatives  Selecting Methods  There is not only one way to select negative ex-  amples for contrastive learning based on PDTB  75  70  65  60  55  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='460  55  50  45  40  35  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
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+page_content='475  73  71  69  67  65  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
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+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='4Model  PDTB-2  Top Level  Second Level  Acc  Macro-F1  Acc  Macro-F1  Dai and Huang (2019)  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='66  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='89  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='23  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='41  Shi and Demberg (2019a)  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='42  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='40  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='83 Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2019) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='95 Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020)  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='25  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='90 Kishimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='26  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='48  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='34 Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020)  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='06  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='39  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='13 Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2021) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='18 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='76  Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2021)  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='17  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='06 Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2022)  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='18  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='73  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='33  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='49  Ours  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='18  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='60  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='69  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='66  Table 1: Experimental results on PDTB-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Model  PDTB-3  Top Level  Second Level  Acc  Macro-F1  Acc  Macro-F1  Liu and Li (2016)  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='67  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='13 Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2016b)  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='33  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='11 Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2017)  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='06  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='29 Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020)  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='01  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='45 Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2022)  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='45  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='14 (BiLSTM)  Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2022)  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='04  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='63 (BERT)  Ours  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='31  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='05  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='68  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='62  Table 2: Experimental results on PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Model  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Cont  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2019)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='44  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='84  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='66  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='60  Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020)  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='92  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='67  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='45  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='33  Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020)  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='44  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='98  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='66  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='26  Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2021)  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='40  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='04  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='76  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='54  Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2021)  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='72  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='39  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='34  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='01  Ours  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='84  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='55  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='17  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='86  Table 3: The results for relation types at level-1 on  PDTB-2 in terms of F1 (%) (top-level multi-class clas-  sification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  hierarchical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In addition to the method  we adopt, we have explored another 4 different  methods of defining positive and negative exam-  ples by using the sense hierarchies, which can be  shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' One can choose the level against  which to select negative examples: method 2 be-  low uses examples with different labels at level-2,  while methods 1, 3 and 4 use examples with dif-  ferent labels at level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' With regard to the use of  weight for method 3 and method 4, we aim to give  more weight to more similar (potentially) positive  examples based on the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Specifically, we  give more weight to the examples from the same  level-2/level-3 type than their sister types at level-  2/level-3 when all of the examples from the same  level-1 are positive examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Besides, method 4  leverages level-3 labels, while method 1 to 3 only  consider level-1 and level-2 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In our exper-  iments for other negatives defining methods, we  use the same hyperparameters as the experimental  setup of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='For method 3 and method 4,  Second-level Label  Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020)  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2022)  Ours  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Asynchronous  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='18  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='47  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='79  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Synchrony  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='26  Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Cause  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='60  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='36  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='58  Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Pragmatic cause  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Contrast  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='75  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='52  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='63  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Concession  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Conjunction  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='17  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='91  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='35  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Instantiation  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='96  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='60  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='04  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Restatement  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='83  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='06  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Alternative  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='46  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='85  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='List  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='98  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='78  Table 4: The results for relation types at level-2 on  PDTB-2 in terms of F1 (%) (second-level multi-class  classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  the weight of positive examples is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='6 and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='3 and the weight of negative examples still is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  It can be seen from table 5 and table 6 that our  method is better than the above methods in both  datasets for both level-1 and level-2 classification  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Compared with method 2, we utilize level-3  labels, which indicated the level-3 label informa-  tion is helpful for the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The greatest differ-  ence between our method and other three methods  is that our negative examples are only those sis-  ter types at level-2 or level-3, not including the  examples from different level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' On the contrary,  the negative examples in those three methods are  examples from other level-1 types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We suppose  that this might make a too strong assumption that  examples from different level-1 are very dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  In PDTB datasets, some examples have been anno-  tated with multiple labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We found that among all  examples with multiple annotated labels, there are  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='26% examples whose multiple labels are under  different level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Moreover, some level-1 types of  relation might be overlapped even if the annotators  just annotate one label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For example, some exam-  ples annotated as Temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='asynchronous might  have the sense of Contingency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='cause as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' And  Moens and Steedman (1988) have pointed out that  when-clauses do not simply predicate a temporal re-  lation, but a causal one as well, which can be called  contingency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' This shows up in the PDTB in terms  of the variation in how particular tokens of when  clauses have been annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' But it also means  that in choosing Negative examples, relations la-  belled TEMPORAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='SYNCHRONOUS or TEMPO-  RAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='ASYNCHRONOUS may closely resemble those  labelled CONTINGENCY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='CAUSE and therefore not  be effective as negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Specifically, for  the following example:  (4) when [they built the 39th Street bridge]1,  [they solved most of their traffic problems]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Model  PDTB-2  PDTB-3  Top Level  Second Level  Top Level  Second Level  Acc  Macro-F1  Acc  Macro-F1  Acc  Macro-F1  Acc  Macro-F1  Method 1  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='91  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='04  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='61  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='27  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='25  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='17  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='58  Method 2  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='39  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='95  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='33  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='80  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='53  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='36  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='93  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='85  Method 3  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='39  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='53  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='61  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='20  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='49  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='49  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='77  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='33  Method 4  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='10  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='30  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='07  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='46  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='26  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='47  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='53  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='24  Ours  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='18  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='60  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='69  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='66  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='31  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='05  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='48  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='62  Table 5: Comparisons with other negatives defining methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Positve: examples with same label at level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Negative: examples with different labels at level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (a) method 1  Positve: examples with same label at level-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Negative: examples with different labels at level-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (b) method 2  Positve: Examples with same label at level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  More weight are given to the examples  with same label at level-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Negative: Examples with different labels at level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (c) method 3  Positve: Examples with same label at level-1,  more weight are given to   the examples  with same label at level-2 or level-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Negative: Examples with different labels at level-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (d) method 4  Figure 5: Another four negative examples selected  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' orange ball represent anchor, green ball rep-  resent negative examples, and blue ball represent posi-  tive examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Darker blue ball means more weight is  given to more similar (potentially) positive examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  If the connective “when” is replaced with “be-  cause”, the sentence still sounds not strange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' There-  fore, regarding all examples from different level-1  as negative examples might have some negative  impacts on learning the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  Ablation Study  We wanted to know how useful our data augmen-  tation method and our contrastive learning method  are, so we have undertaken ablation studies for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Model  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Cont  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Method 1  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='26  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='42  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='78  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='74  Method 2  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='78  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='82  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='89  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='30  Method 3  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='85  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='18  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='43  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='67  Method 4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='25  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='73  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='30  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='90  Ours  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='84  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='55  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='17  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='86  Table 6: The results of relation types at level-1 on  PDTB-2 in terms of F1 (%) (top-level multi-class clas-  sification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Effects  of  contrastive  learning  algorithm  From Table 7, it can be seen that multi-task  learning  method  where  level-1  and  level-2  labels are predicted simultaneously by using the  same [CLS] representation perform better than  separately predicting level-1 and level-2 labels,  which verifies the dependency between different  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Compared with the multi-task learning  method, our model with a contrastive loss has  better performance in PDTB-2 and PDTB-3, which  means that our contrasting learning method is  indeed helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Effects of data augmentation  Table 8 compares  the results with and without data augmentation for  both PDTB-2 and PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' From the comparisons,  it is clear that the data augmentation method is  helpful to generate useful examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Khosla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  (2020) showed that having a large number of hard  positives/negatives in a batch leads to better perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Since we have many classes at the second  level, 11 types for PDTB-2 and 14 types for PDTB-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In a batch with the size of 256, it is difficult to  guarantee that there are enough positive examples  for each class to take full advantage of contrast  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Therefore, without data augmentation,  the performance of our method degrades consider-  ably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  7  Limitations and Future work  With regard to PDTB-2 and PDTB-3 annotation,  there are two cases: (1) Annotators can assign mul-  tiple labels to an example when they believe more  than one relation holds simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2) An-  notators can be told (in the Annotation Manual)  to give precedence to one label if they take more  than one to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For example, they are told in  the Manual (Webber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2019) that examples  that satisfy the conditions for both Contrast and  Concession, should be labelled as concession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We  over-simplified the presence of multiple labels by  following Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2017) in treating each label as  Datasets  Model  Top Level  Second Level  Acc  Macro-F1  Acc  Macro-F1  PDTB-2  RoBERTa  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='14  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='87  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='33  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='37  RoBERTa-MTL  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='87  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='39  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='22  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='21  Ours  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='18  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='60  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='69  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='66  PDTB-3  RoBERTa  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='02  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='44  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='56  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='12  RoBERTa-MTL  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='63  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='23  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='56  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='16  Ours  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='31  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='05  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='68  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='62  Table 7: Ablation study on PDTB-2 and PDTB-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Model  Top Level  Second Level  Acc  Macro-F1  Acc  Macro-F1  PDTB-2  Ours  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='18  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='60  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='69  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='66  augmentation  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='70  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='85  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='19  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='54  PDTB-3  Ours  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='31  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='05  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='68  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='62  augmentation  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='32  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='02  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='24  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='80  Table 8: Effects of data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  a separate example and did not consider the second  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Thus, our approach might be inadequate for  dealing with the actual distribution of the data and  can be extended or modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' It is worth exploring  how to extend our approach to allow for examples  with multiple sense labels and cases where one la-  bel takes precedence over another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We believe that  this will be an important property of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Another limitation is that we only use English  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' There are PDTB-style datasets in other  languages including a Chinese TED dicourse bank  corpus (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2020), a Turkish discourse  Tree bank corpus (Zeyrek and Kurfalı, 2017) and  an Italian Discourse Treebank (Pareti and Prodanof,  2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Moreover, Zeyrek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2019) proposed  a TED Multilingual Discourse Bank (TED-MDB)  corpus, which has 6 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' These datasets al-  low us to assess the approach in languages other  than English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Besides, there are datasets similar  to PDTB-Style like Prague Dependency Treebank  (Mírovský et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The different datasets use  essentially similar sense hierarchy, but two things  need to be investigated (i) whether there are compa-  rable differences between tokens that realise “sis-  ter” relations, or (ii) whether tokens often have  multiple sense labels, which would change what  could be used as negative examples if leveraging  our approach on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  In the future, we can also assess whether con-  trastive learning could help in separating out En-  tRel relations and AltLex relations from implicit  relations or whether other methods would perform  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  8  Conclusions  In this paper, we leverage the sense hierarchy to  select the negative examples needed for contrastive  learning for the task of implicit discourse relation  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Our method has better overall perfor-  mance than achieved by previous systems, and com-  pared with previous work, our method is better at  learning minority labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Moreover, we compared  different methods of selecting the negative exam-  ples based on the hierarchical structures, which  shows some potential negative impacts might be  produced when negative examples include those  from other level-1 types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Moreover, we conduct ab-  lation studies to investigate the effects of our data  augmentation method and our contrastive learning  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Besides, the limitations and the future  work are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Acknowledgments  This work was supported in part by the UKRI  Centre for Doctoral Training in Natural Lan-  guage Processing, funded by the UKRI (grant  EP/S022481/1), the University of Edinburgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The  authors also gratefully acknowledge University of  Edinburgh Huawei Laboratory for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
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+page_content=' Association for Computational Lin-  guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Wei Shi and Vera Demberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Next sentence pre-  diction helps implicit discourse relation classifica-  tion within and across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In Proceedings of  the 2019 Conference on Empirical Methods in Nat-  ural Language Processing and the 9th International  Joint Conference on Natural Language Processing  (EMNLP-IJCNLP), pages 5790–5796, Hong Kong,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Varsha Suresh and Desmond C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Ong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Not all  negatives are equal: Label-aware contrastive loss for  fine-grained text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  From discourse to narrative: Knowledge projection  for event relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In Proceedings of the  59th Annual Meeting of the Association for Compu-  tational Linguistics and the 11th International Joint  Conference on Natural Language Processing (Vol-  ume 1: Long Papers), pages 732–742, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' As-  sociation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Dong Wang, Ning Ding, Piji Li, and Haitao Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' CLINE: Contrastive learning with semantic  negative examples for natural language understand-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In Proceedings of the 59th Annual Meeting of  the Association for Computational Linguistics and  the 11th International Joint Conference on Natu-  ral Language Processing (Volume 1: Long Papers),  pages 2332–2342, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Association for Computa-  tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Bonnie Webber, Rashmi Prasad, Alan Lee, and Ar-  avind Joshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The penn discourse treebank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  annotation manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Changxing Wu, Liuwen Cao, Yubin Ge, Yang Liu, Min  Zhang, and Jinsong Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' A label dependence-  aware sequence generation model for multi-level im-  plicit discourse relation recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Changxing Wu, Chaowen Hu, Ruochen Li, Hongyu  Lin, and Jinsong Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Hierarchical multi-  task learning with crf for implicit discourse relation  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Knowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
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+page_content='  Zhuofeng Wu, Sinong Wang, Jiatao Gu, Madian  Khabsa, Fei Sun, and Hao Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Clear: Con-  trastive learning for sentence representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' ArXiv,  abs/2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='15466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Wei Xiang, Bang Wang, Lu Dai, and Yijun Mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Encoding and fusing semantic connection  and linguistic evidence for implicit discourse rela-  tion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  In Findings of the Association  for Computational Linguistics: ACL 2022, pages  3247–3257, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Association for Com-  putational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Yuanmeng Yan, Rumei Li, Sirui Wang, Fuzheng Zhang,  Wei Wu, and Weiran Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' ConSERT: A con-  trastive framework for self-supervised sentence rep-  resentation transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In Proceedings of the 59th An-  nual Meeting of the Association for Computational  Linguistics and the 11th International Joint Confer-  ence on Natural Language Processing (Volume 1:  Long Papers), pages 5065–5075, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Associa-  tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Seonghyeon Ye, Jiseon Kim, and Alice Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
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+page_content=' Ef-  ficient contrastive learning via novel data augmen-  tation and curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
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+page_content=' Association  for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Deniz Zeyrek and Murathan Kurfalı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
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+page_content=' TDB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='1:  Extensions on Turkish discourse bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In Proceed-  ings of the 11th Linguistic Annotation Workshop,  pages 76–81, Valencia, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Association for Com-  putational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Deniz Zeyrek, Amalia Mendes, Yulia Grishina, Mu-  rathan Kurfali, Samuel Gibbon, and Maciej Ogrod-  niczuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Ted multilingual discourse bank (ted-  mdb): a parallel corpus annotated in the pdtb style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Language Resources and Evaluation, pages 1–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Dejiao Zhang, Shang-Wen Li, Wei Xiao, Henghui Zhu,  Ramesh Nallapati, Andrew O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Arnold, and Bing Xi-  ang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Pairwise supervised contrastive learning  of sentence representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' In Proceedings of the  2021 Conference on Empirical Methods in Natural  Language Processing, pages 5786–5798, Online and  Punta Cana, Dominican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Association for  Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='1  PDTB Hierarchy  The hierarchies of both PDTB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0 and PDTB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  consist of three levels, but for implicit relation  recognition, so far no classification for third level  labels has been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' We also focus on the hier-  archy between level-1 and level 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The PDTB-3  relation hierarchy simplifies and extends the PDTB-  2 relation hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The PDTB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0 hierarchy not  only simplifies the PDTB-2 relation hierarchy by  restricting Level-3 relations to differences in di-  rectionality and eliminating rare and/or difficult-  to-annotate senses, but also augments the relation  hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' Figure 6 and Figure 7 show PDTB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  relation hierarchy and PDTB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0 relation hierarchy  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Asynchronous  Synchronous  Figure 6: The PDTB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0 Senses Hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Figure 7: The PDTB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0 Senses Hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The left-  most column contains the Level-1 senses and the mid-  dle column, the Level-2 senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' For asymmetric rela-  tions, Level-3 senses are located in the rightmost col-  umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Model  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Cont  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Liu and Li (2016)  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='15  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='33  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='10  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='03  Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2017)  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='10  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='91  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='03  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='71  Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2020)  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='37  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='95  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='28  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='74  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2016b)  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='34  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='56  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='71  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='91  Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2022)  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='16  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='48  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='82  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='22  (BiLSTM)  Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' (2022)  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='83  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='77  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='00  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='13  (BERT)  Ours  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='30  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='60  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='91  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='39  Table 9: The results of different relations on PDTB-3  in terms of F1 (%) (top-level multi-class classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Second-level Label  Ours  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Asynchronous  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='35  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Synchrony  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='38  Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Cause  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='38  Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Cause+Belief  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='0  Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Condition  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='07  Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Purpose  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='05  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Contrast  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='91  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Concession  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='11  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Conjunction  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='70  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Equivalence  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='43  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Instantiation  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='83  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Level-of-detail  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='34  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Manner  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='43  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Substitution  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='77  Table 10: The results of different relations on PDTB-3  in terms of F1 (%) (second-level multi-class classifica-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='2  The results on relation types on PDTB-3  We also examine the classification performance on  PDTB-3 in terms of Macro-F1 for the four main  relation types at level-1 and 14 sense types at level-  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content=' The results can be seen in Table 9 and Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  Our model has significantly better performance for  all level-1 relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  As for level-2 sense types, because there are no  results of previous systems, we just show the result  of 14 level-2 sense types in PDTB-3 in terms of F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Synchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Temporal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Precedence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Asynchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Succession  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Reason  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Cause  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Result  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Conjunction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Disjunction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Negative-result*  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Equivalence Arg1-as-cond  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg1-as-instance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Contingency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-cond  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Instantiation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-instance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg1-as-negcond  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Negative condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg1-as-detail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-negcond  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Level-of-detail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Expansion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-detail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg1-as-goal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg1-as-subst  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Purpose  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-goal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Substitution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-negGoal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-subst  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg1-as-excpt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Exception  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Contrast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-excpt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Similarity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Manner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg1-as-manner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Comparison  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg1-as-denier*  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-manner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Concession  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Arg2-as-denierTEMPORAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='COMPARISON  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Asynchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='→Contrast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='→ Synchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='juxtaposition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='> opposition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='precedence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='succession  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Pragmatic Contrast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='→ Concession  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='expectation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='contra-expectation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='CONTINGENCY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Cause  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Pragmatic Concession  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='reason  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='result  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='EXPANSION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Conjunction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Pragmatic Cause  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Instantiation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='justification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Restatement  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='specification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='hypothetical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='equivalence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='general  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='unreal present  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='generalization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='unreal past  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Alternative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='factual present  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='conjunctive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='factual past  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='→ disjunctive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='→ chosen alternative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Pragmatic Condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='Exception  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='relevance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='implicit assertion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
+page_content='List' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE0T4oBgHgl3EQf3QLX/content/2301.02724v1.pdf'}
diff --git a/Z9A0T4oBgHgl3EQfF_-o/content/tmp_files/2301.02041v1.pdf.txt b/Z9A0T4oBgHgl3EQfF_-o/content/tmp_files/2301.02041v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..48b1300be54605d5fbfd6e05721e920beb64b7a5
--- /dev/null
+++ b/Z9A0T4oBgHgl3EQfF_-o/content/tmp_files/2301.02041v1.pdf.txt
@@ -0,0 +1,383 @@
+arXiv:2301.02041v1  [math.RA]  5 Jan 2023
+A Note On Square-free Commuting Probabilities
+of Finite Rings
+Andrew Mendelsohn
+Department of EEE, Imperial College London, London, SW7 2AZ, United Kingdom.
+andrew.mendelsohn18@imperial.ac.uk
+Abstract. It is shown that the commuting probability of a finite ring
+cannot be a fraction with square-free denominator, resolving a conjecture
+of Buckley and MacHale.
+1
+Introduction
+Let G be a finite group and U denote the uniform distribution. Let the commut-
+ing probability of G, PG, be denoted
+PG := Pra,b←U(G)(ab = ba).
+An alternative characterisation is
+PG =
+1
+|G|2 {(x, y) ∈ G2 : xy = yx}.
+Joseph made the following conjectures, where G is the set of all commuting
+probabilities of finite groups [1]:
+A. All limit points of G are rational.
+B. The set G is well ordered by >.
+C. The set 0 ∪ G is closed (that is, contains all its accumulation points).
+In [2], Eberhard resolved conjectures A and B, and in [3] conjecture C was
+resolved.
+One can extend the definition of commuting probability to finite rings: set
+PR =
+1
+|R|2 |{(x, y) ∈ R2 : xy = yx}| =
+1
+|R|2 |{(x, y) ∈ R2 : xy − yx = 0}|.
+In [4], the following conjectures were made, where R is the set PR over all finite
+rings R:
+1. 1/n ̸∈ R when n ∈ N is square-free.
+2. R ⊂ G.
+3. R coincides with the set of values of PG as G ranges over all finite nilpotent
+groups of class at most 2.
+4. All limit points of R are rational.
+
+2
+A. Mendelsohn
+5. For each 0 < t ≤ 1, there exists ǫt > 0 such that R ∩ (t − ǫt, t) = ∅.
+6. R does not contain any of its accumulation points.
+Note conjectures 4, 5, and 6 correspond to Joseph’s conjectures. Moreover, con-
+jectures 4 and 5 would follow from the veracity of conjectures 2 or 3, since
+Eberhard showed G has rational limit points and is well-ordered. In [5], conjec-
+ture 2 was in fact resolved, and thus conjectures 4 and 5. Moreover, conjecture
+3 was partially resolved: the authors obtained that R is a subset of the set of
+values of PG as G ranges over all finite nilpotent groups of class at most 2. We
+conclude that conjectures 1, 3, and 6 are open.
+In this work, we resolve conjecture 1.
+2
+Preliminaries and Prior Results
+Definition 1. Two finite groups G, H are called isoclinic if G/Z(G) ∼= H/Z(H)
+and G′ ∼= H′, and if the diagram below commutes:
+G/Z(G) × G/Z(G)
+H/Z(H) × H/Z(H)
+G′
+H′
+Isoclinism preserves nilpotency class and commuting probability [10]. A stem
+group is a group in a given isoclinism class of minimal order. It is well known
+that if G is a stem group, then Z(G) ≤ G′. For more on isoclinism, see [11].
+Below we state existing results in the literature we will need below.
+Lemma 1. [5] R ⊂ Gn,2, where Gn,2 is the set of commuting probabilities of all
+finite nilpotent groups of class at most 2.
+This statement is proved as follows: let R be a finite ring. We can turn
+R⊕R into a nilpotent ring of class 3 by endowing it with the multiplication rule
+(a, x)(b, y) = (0, ab). This ring can be turned into a nilpotent group GR of class
+at most 2 by endowing it with the binary operation a ◦ b = a + b + ab. Both
+of these transformations preserve the commuting probability. Thus the values of
+R \ {1} are a subset of the values PG, running over nilpotent groups G of class
+equal to 2. Note that if R has size n, then the resulting group GR has order n2,
+and if R is noncommutative then the resulting group is nonabelian.
+Lemma 2. [7] Pr(G) =
+1
+|G′|
+�
+1 + |G′|−1
+|G:Z(G)|
+�
+if and only if G is nilpotent.
+Lemma 3. [6] If G is a nilpotent group, then PG ̸= 1
+p.
+3
+Results
+Theorem 1.
+1
+p ̸∈ R for all p ∈ N≥2.
+
+On the Commuting Probability of Finite Rings
+3
+Proof. By Lemma 1, R is contained within the set of commuting probabilities
+of finite nilpotent groups of class at most 2. By Lemma 3, this latter set does
+not contain 1
+p for any prime p.
+Denote the set of commuting probabilities of rings of prime power order for
+some prime p by Rp.
+Proposition 1.
+1
+n ̸∈ Rp for any prime p ∈ N≥2 and n ∈ N>1.
+Proof. By Lemma 1, we need only consider commuting probabilities of finite
+nilpotent groups of class at most 2. By Lemma 2, we know a fortiori the com-
+muting probability of finite nilpotent groups of class at most 2 in terms of derived
+subgroups and centers. Suppose that for some n ∈ N≥2 we have
+1
+|G′|
+�
+1 +
+|G′| − 1
+|G : Z(G)|
+�
+= 1
+n.
+By the construction of [5] considered above, wlog let |G| = pe for some even
+positive integer e. Then Z(G) = pf and G′ = pg with 0 < g ≤ f < e (since G is
+at most class 2). Then
+1
+n = p−g
+�
+1 + pg − 1
+pe−f
+�
+= p−g + pg − 1
+pe−f+g
+(1)
+= pe−f + pg − 1
+pe−f+g
+.
+(2)
+For this to hold, some power ph must divide the numerator, with h > 0; but this
+cannot hold; for if so, then one must have pe−f + pg − 1 = kp, for some integer
+k ̸= 0. But then −1 ≡ 0 mod p, a contradiction.
+Theorem 2.
+ℓ
+n ̸∈ R for any squarefree n ∈ N>1 and ℓ < n with gcd(ℓ, n) = 1.
+Proof. Any finite ring can be turned into a nilpotent group of class at most
+2, such that the commuting probability of the ring is equal to the commuting
+probability of the group. The construction (outlined above) turns a commutative
+ring into a group of class 1, and a noncommutative ring into a nonabelian group of
+class at most 2, therefore of class equal to 2. The order of the group is the square
+of the order of the ring, so the Sylow subgroups of the group have order at least
+the square of a prime. Since the group is nilpotent, it can be written as a product
+of its Sylow subgroups, which are all of class at most 2, and the commuting
+probability of the group is the product of the commuting probabilities of its
+Sylow subgroups. Thus it remains to analyse the equation
+ℓ
+n =
+m
+�
+i=1
+pei−fi
+i
++ pgi
+i − 1
+pei−fi+gi
+i
+,
+for m > 1 and the pi distinct, where the ei, fi, and gi are as before. Via isoclinism,
+we may replace GR by a class two nilpotent (stem) group G with identical com-
+muting probability and minimal order. Thus we may assume that Z(GR) = G′
+R
+
+4
+A. Mendelsohn
+(note isoclinism preserves nilpotency class and GR is class two), and moreover
+that none of the Sylow subgroups are abelian. The above equality simplifies to
+ℓ
+n =
+m
+�
+i=1
+pe′
+i−f ′
+i
+i
++ pf ′
+i
+i − 1
+p
+e′
+i
+i
+,
+where the exponents e′
+i, f ′
+i correspond to the group G. We now proceed by in-
+duction on the number of prime factors of |GR|, denoted m.
+By Lemma 14 of [9], if PGR = ℓ
+n in lowest terms, the prime factors of n are
+precisely the prime factors of |GR|. If m = 1, it is known that
+ℓ
+n ̸∈ Rq for any
+prime q and square-free n (in fact, we know this to hold for m ≤ 69 by Theorem
+9 of [4]). Suppose the statement is true up to n = k − 1, and consider the case
+n = k; suppose, for a contradiction, that the commuting probability is equal to
+ℓ
+n, for some square-free integer n and ℓ ≤ n with gcd(ℓ, n) = 1, and without loss
+of generality that n has prime factors equal to the set of pi, i = 1, ..., k:
+ℓ
+n =
+k
+�
+i=1
+pe′
+i−f ′
+i
+i
++ pf ′
+i
+i − 1
+p
+e′
+i
+i
+.
+Rearrange for the following:
+ℓ · pe′
+k
+k
+n · (p
+e′
+k−f ′
+k
+k
++ p
+f ′
+k
+k − 1)
+=
+k−1
+�
+i=1
+pe′
+i−f ′
+i
+i
++ pf ′
+i
+i − 1
+p
+e′
+i
+i
+.
+Writing the left hand side in lowest terms, we have
+ℓ · pe′′
+k
+k
+n′ · (p
+e′
+k−f ′
+k
+k
++ p
+f ′
+k
+k − 1)
+=
+k−1
+�
+i=1
+pe′
+i−f ′
+i
+i
++ pf ′
+i
+i − 1
+p
+e′
+i
+i
+,
+where n′ is not divisible by pk. We have a commuting probability on the right
+hand side with k−1 prime factors; so by the induction hypothesis, the denomina-
+tor of the left hand side has no square factors. But we also have pe′
+k−f ′
+k
+k
++ pf ′
+k
+k − 1
+on the denominator of the left hand side, which is not divisible by pk, and by
+Lemma 14 of [9] must have prime factors equal to the set of pi, for i = 1, ..., k−1;
+moreover, there can be no cancellation between these factors and ℓ, by assump-
+tion on ℓ. But then for at least one index j, n′ · (pe′
+k−f ′
+k
+k
++ pf ′
+k
+k − 1) has a prime
+factor pj with multiplicity at least two, which is a contradiction.
+Remark 1. Since 1
+p is an accumulation point of Rp, 1
+n is an accumulation point
+of R for all n. The above result thus means that many accumulation points of
+R are not contained in R. As well as resolving the first conjecture stated at the
+beginning of this note, the result also makes progress on the sixth conjecture
+stated.
+
+On the Commuting Probability of Finite Rings
+5
+References
+1. Joseph,
+K.
+Several
+Conjectures
+on
+Commutativity
+in
+Algebraic
+Struc-
+tures.
+The
+American
+Mathematical
+Monthly.
+84,
+550-551
+(1977),
+https://doi.org/10.1080/00029890.1977.11994411.
+2. Eberhard, S. Commuting probabilities of finite groups. Bulletin Of The London
+Mathematical Society. 47, 796-808 (2015).
+3. Browning, T. Limit Points of Commuting Probabilities of Finite Groups. (arXiv,
+2022), https://arxiv.org/abs/2201.09402.
+4. Buckley,
+S.
+&
+MacHale,
+D.
+Contrasting
+the
+commut-
+ing
+probabilities
+of
+groups
+and
+rings.
+(arXiv,
+2014),
+https://archive.maths.nuim.ie/staff/sbuckley/Papers/bm_g-vs-r.pdf.
+5. Jur´a˘s, M. & Ursul, M. On commuting probabilities in finite groups and rings.
+Journal Of Algebra Combinatorics Discrete Structures And Applications. (2022),
+https://jacodesmath.com/index.php/jacodesmath/article/view/148.
+6. Castelaz, A. Commutativity Degree of Finite Groups. (Wake Forest University,
+2010), https://books.google.co.uk/books?id=QYxBnQAACAAJ.
+7. Nath, R. & Das, A. On a lower bound of commutativity degree. Rendiconti Del
+Circolo Matematico Di Palermo. 59 pp. 137-142 (2010, 4).
+8. Buckley,
+S.,
+MacHale,
+D.
+&
+Sh´e,
+A.
+Finite
+rings
+with
+many
+commuting
+pairs
+of
+elements.
+(2014),
+https://archive.maths.nuim.ie/staff/sbuckley/Papers/bms.pdf.
+9. Buckley,
+S.
+&
+MacHale,
+D.
+Groups
+with
+Pr(G)
+=
+1/3,
+https://archive.maths.nuim.ie/staff/sbuckley/Papers/bm_GpCP_1_3.pdf.
+10. Lescot, P. Isoclinism Classes and Commutativity Degrees of Finite Groups. Journal
+Of Algebra. 177, 847-869 (1995).
+11. Berkovich, Y. Groups of Prime Power Order, Volume 1. (De Gruyter, 2008),
+https://doi.org/10.1515/9783110208221.
+
diff --git a/Z9A0T4oBgHgl3EQfF_-o/content/tmp_files/load_file.txt b/Z9A0T4oBgHgl3EQfF_-o/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='02041v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='RA]  5 Jan 2023  A Note On Square-free Commuting Probabilities  of Finite Rings  Andrew Mendelsohn  Department of EEE, Imperial College London, London, SW7 2AZ, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='mendelsohn18@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='uk  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' It is shown that the commuting probability of a finite ring  cannot be a fraction with square-free denominator, resolving a conjecture  of Buckley and MacHale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  1  Introduction  Let G be a finite group and U denote the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Let the commut-  ing probability of G, PG, be denoted  PG := Pra,b←U(G)(ab = ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  An alternative characterisation is  PG =  1  |G|2 {(x, y) ∈ G2 : xy = yx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Joseph made the following conjectures, where G is the set of all commuting  probabilities of finite groups [1]:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' All limit points of G are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The set G is well ordered by >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The set 0 ∪ G is closed (that is, contains all its accumulation points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  In [2], Eberhard resolved conjectures A and B, and in [3] conjecture C was  resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  One can extend the definition of commuting probability to finite rings: set  PR =  1  |R|2 |{(x, y) ∈ R2 : xy = yx}| =  1  |R|2 |{(x, y) ∈ R2 : xy − yx = 0}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  In [4], the following conjectures were made, where R is the set PR over all finite  rings R:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 1/n ̸∈ R when n ∈ N is square-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' R ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' R coincides with the set of values of PG as G ranges over all finite nilpotent  groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' All limit points of R are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Mendelsohn  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' For each 0 < t ≤ 1, there exists ǫt > 0 such that R ∩ (t − ǫt, t) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' R does not contain any of its accumulation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Note conjectures 4, 5, and 6 correspond to Joseph’s conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Moreover, con-  jectures 4 and 5 would follow from the veracity of conjectures 2 or 3, since  Eberhard showed G has rational limit points and is well-ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' In [5], conjec-  ture 2 was in fact resolved, and thus conjectures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Moreover, conjecture  3 was partially resolved: the authors obtained that R is a subset of the set of  values of PG as G ranges over all finite nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' We  conclude that conjectures 1, 3, and 6 are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  In this work, we resolve conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  2  Preliminaries and Prior Results  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Two finite groups G, H are called isoclinic if G/Z(G) ∼= H/Z(H)  and G′ ∼= H′, and if the diagram below commutes:  G/Z(G) × G/Z(G)  H/Z(H) × H/Z(H)  G′  H′  Isoclinism preserves nilpotency class and commuting probability [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' A stem  group is a group in a given isoclinism class of minimal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' It is well known  that if G is a stem group, then Z(G) ≤ G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' For more on isoclinism, see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Below we state existing results in the literature we will need below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' [5] R ⊂ Gn,2, where Gn,2 is the set of commuting probabilities of all  finite nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  This statement is proved as follows: let R be a finite ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' We can turn  R⊕R into a nilpotent ring of class 3 by endowing it with the multiplication rule  (a, x)(b, y) = (0, ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' This ring can be turned into a nilpotent group GR of class  at most 2 by endowing it with the binary operation a ◦ b = a + b + ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Both  of these transformations preserve the commuting probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Thus the values of  R \\ {1} are a subset of the values PG, running over nilpotent groups G of class  equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Note that if R has size n, then the resulting group GR has order n2,  and if R is noncommutative then the resulting group is nonabelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' [7] Pr(G) =  1  |G′|  �  1 + |G′|−1  |G:Z(G)|  �  if and only if G is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' [6] If G is a nilpotent group, then PG ̸= 1  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  3  Results  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  1  p ̸∈ R for all p ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  On the Commuting Probability of Finite Rings  3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' By Lemma 1, R is contained within the set of commuting probabilities  of finite nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' By Lemma 3, this latter set does  not contain 1  p for any prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Denote the set of commuting probabilities of rings of prime power order for  some prime p by Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  1  n ̸∈ Rp for any prime p ∈ N≥2 and n ∈ N>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' By Lemma 1, we need only consider commuting probabilities of finite  nilpotent groups of class at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' By Lemma 2, we know a fortiori the com-  muting probability of finite nilpotent groups of class at most 2 in terms of derived  subgroups and centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Suppose that for some n ∈ N≥2 we have  1  |G′|  �  1 +  |G′| − 1  |G : Z(G)|  �  = 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  By the construction of [5] considered above, wlog let |G| = pe for some even  positive integer e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Then Z(G) = pf and G′ = pg with 0 < g ≤ f < e (since G is  at most class 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Then  1  n = p−g  �  1 + pg − 1  pe−f  �  = p−g + pg − 1  pe−f+g  (1)  = pe−f + pg − 1  pe−f+g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  (2)  For this to hold, some power ph must divide the numerator, with h > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' but this  cannot hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' for if so, then one must have pe−f + pg − 1 = kp, for some integer  k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' But then −1 ≡ 0 mod p, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  ℓ  n ̸∈ R for any squarefree n ∈ N>1 and ℓ < n with gcd(ℓ, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Any finite ring can be turned into a nilpotent group of class at most  2, such that the commuting probability of the ring is equal to the commuting  probability of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The construction (outlined above) turns a commutative  ring into a group of class 1, and a noncommutative ring into a nonabelian group of  class at most 2, therefore of class equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The order of the group is the square  of the order of the ring, so the Sylow subgroups of the group have order at least  the square of a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Since the group is nilpotent, it can be written as a product  of its Sylow subgroups, which are all of class at most 2, and the commuting  probability of the group is the product of the commuting probabilities of its  Sylow subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Thus it remains to analyse the equation  ℓ  n =  m  �  i=1  pei−fi  i  + pgi  i − 1  pei−fi+gi  i  ,  for m > 1 and the pi distinct, where the ei, fi, and gi are as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Via isoclinism,  we may replace GR by a class two nilpotent (stem) group G with identical com-  muting probability and minimal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Thus we may assume that Z(GR) = G′  R  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Mendelsohn  (note isoclinism preserves nilpotency class and GR is class two), and moreover  that none of the Sylow subgroups are abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The above equality simplifies to  ℓ  n =  m  �  i=1  pe′  i−f ′  i  i  + pf ′  i  i − 1  p  e′  i  i  ,  where the exponents e′  i, f ′  i correspond to the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' We now proceed by in-  duction on the number of prime factors of |GR|, denoted m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  By Lemma 14 of [9], if PGR = ℓ  n in lowest terms, the prime factors of n are  precisely the prime factors of |GR|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' If m = 1, it is known that  ℓ  n ̸∈ Rq for any  prime q and square-free n (in fact, we know this to hold for m ≤ 69 by Theorem  9 of [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Suppose the statement is true up to n = k − 1, and consider the case  n = k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' suppose, for a contradiction, that the commuting probability is equal to  ℓ  n, for some square-free integer n and ℓ ≤ n with gcd(ℓ, n) = 1, and without loss  of generality that n has prime factors equal to the set of pi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=', k:  ℓ  n =  k  �  i=1  pe′  i−f ′  i  i  + pf ′  i  i − 1  p  e′  i  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Rearrange for the following:  ℓ · pe′  k  k  n · (p  e′  k−f ′  k  k  + p  f ′  k  k − 1)  =  k−1  �  i=1  pe′  i−f ′  i  i  + pf ′  i  i − 1  p  e′  i  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Writing the left hand side in lowest terms, we have  ℓ · pe′′  k  k  n′ · (p  e′  k−f ′  k  k  + p  f ′  k  k − 1)  =  k−1  �  i=1  pe′  i−f ′  i  i  + pf ′  i  i − 1  p  e′  i  i  ,  where n′ is not divisible by pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' We have a commuting probability on the right  hand side with k−1 prime factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' so by the induction hypothesis, the denomina-  tor of the left hand side has no square factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' But we also have pe′  k−f ′  k  k  + pf ′  k  k − 1  on the denominator of the left hand side, which is not divisible by pk, and by  Lemma 14 of [9] must have prime factors equal to the set of pi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=', k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  moreover, there can be no cancellation between these factors and ℓ, by assump-  tion on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' But then for at least one index j, n′ · (pe′  k−f ′  k  k  + pf ′  k  k − 1) has a prime  factor pj with multiplicity at least two, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Since 1  p is an accumulation point of Rp, 1  n is an accumulation point  of R for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' The above result thus means that many accumulation points of  R are not contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' As well as resolving the first conjecture stated at the  beginning of this note, the result also makes progress on the sixth conjecture  stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  On the Commuting Probability of Finite Rings  5  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Joseph,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Several  Conjectures  on  Commutativity  in  Algebraic  Struc-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  The  American  Mathematical  Monthly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  84,  550-551  (1977),  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='1080/00029890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='11994411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Eberhard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Commuting probabilities of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Bulletin Of The London  Mathematical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 47, 796-808 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Browning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Limit Points of Commuting Probabilities of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' (arXiv,  2022), https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='09402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Buckley,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  &  MacHale,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Contrasting  the  commut-  ing  probabilities  of  groups  and  rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  (arXiv,  2014),  https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='nuim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='ie/staff/sbuckley/Papers/bm_g-vs-r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Jur´a˘s, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' & Ursul, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' On commuting probabilities in finite groups and rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Journal Of Algebra Combinatorics Discrete Structures And Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' (2022),  https://jacodesmath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='com/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='php/jacodesmath/article/view/148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Castelaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Commutativity Degree of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' (Wake Forest University,  2010), https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='uk/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='id=QYxBnQAACAAJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Nath, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' & Das, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' On a lower bound of commutativity degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Rendiconti Del  Circolo Matematico Di Palermo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 59 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 137-142 (2010, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Buckley,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=',  MacHale,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  &  Sh´e,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Finite  rings  with  many  commuting  pairs  of  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  (2014),  https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='nuim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='ie/staff/sbuckley/Papers/bms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Buckley,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  &  MacHale,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  Groups  with  Pr(G)  =  1/3,  https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='nuim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='ie/staff/sbuckley/Papers/bm_GpCP_1_3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Lescot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Isoclinism Classes and Commutativity Degrees of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Journal  Of Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' 177, 847-869 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Berkovich, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' Groups of Prime Power Order, Volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content=' (De Gruyter, 2008),  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
+page_content='1515/9783110208221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9A0T4oBgHgl3EQfF_-o/content/2301.02041v1.pdf'}
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+Laser Field Initiation of Higher Order Poles of S-Matrix-Optical Realization of
+Field Theoretic Models*
+G.S. Agarwal1,2
+1Physical Research Laboratory, Navrangpura, Ahmedabad, India
+2Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore, India
+We discuss the possibility of converting a simple pole in the radiative decay of a state into a pole of
+higher order by using resonant electromagnetic fields. This process of creation of higher order pole
+is controllable by the intensity of the laser field. We use density matrix and Liouville space and
+present the modification of the Lorentzian line shapes (Breit-Wigner formula) for example to ones
+involving square of Lorentzian and derivatives of Lorentzians.
+INTRODUCTION
+In a classic paper Goldberger and Watson [1] con-
+sidered the possibility that the decay law for an un-
+stable particle can be more complex than a simple
+exponential. They showed the possibility of the ex-
+istence of the poles of S-matrix which were not nec-
+essarily simple poles. Since then, higher order poles
+have been extensively studied. Recently, there is re-
+vival [2,3] of interest in such studies and in particular
+Bhamathi and Sudarshan have analyzed several field
+theoretic models like Friedrich-Lee model, cascade
+model and their extensions. They examine the spec-
+trum (complex) of eigenvalues for such models. A
+related question is how the Breit-Wigner line shape
+formula is modified if S-matrix possess higher order
+poles.
+In this paper we examine the possibility of cre-
+ation of the higher order poles using laser fields. We
+consider the decay of say excited state of an atom.
+Normally this decay is described by the Wigner-
+Weisskopf theory which leads to exponential decay
+law.
+We next discuss the case when the excited
+state is coupled to another state by a resonant elec-
+tromagnetic field. In such a case we show that for
+appropriate value of the intensity of the laser field
+the corresponding spectral function has a pole of or-
+der two. We calculate the resulting line shape and
+discuss the line narrowing etc. We emphasize that
+we work within the framework of density matrices
+and hence we work in Liouville space rather than
+in Hilbert space. We present optical realization of
+various field theoretic models.
+Consider the decay of the state |1⟩ into the states
+|3⟩ and |2⟩ at the rates 2γ1 and 2γ2 respectively as
+shown in Fig.1 (with Gl = 0, △l = 0).
+It is well
+known that the rate of decay of the population in
+|1⟩ is given by
+ρ11(t) = ρ11(0)exp(−2(γ1 + γ2)t).
+(1)
+Here ρ is the density matrix of the atom. The spec-
+trum of the spontaneously emitted photons will con-
+sist of two Lorentzians centered at ω13 and ω12 with
+a half width (γ1 + γ2). Let us concentrate on the
+emission on the transition |1⟩ ↔ |3⟩. The spectrum
+will be described by the well-known form
+S(ω) =
+γ1/π
+(γ1 + γ2)2 + (ω − ω13)2 .
+(2)
+Note that γ2 will be zero if the decay channel |1⟩ →
+|2⟩ is not allowed.
+We will discuss how the laser
+fields could be used to modify significantly the re-
+sults predicted by (1) and (2).
+LIOUVILLE SPACE FORMULATION OF
+DECAY
+We next recall how the spectrum is calculated in
+the density matrix framework [4]. We have included
+this material for completeness so that our discussion
+in subsequent sections can be followed by the non-
+Quantum optics practitioners.
+Consider a system
+with two states |1⟩ and |3⟩ interacting with the vac-
+uum of the electromagnetic field. The Hamiltonian
+can be written in the form
+H = ¯hω13 |1⟩ ⟨1| +
+�
+ks
+¯hωksa†
+ksaks + V13
+V13 =
+�
+ks
+(¯hgksa†
+ks |1⟩ ⟨3| + h.c.).
+(3)
+The vacuum modes are characterized by the propa-
+gation index −→k and the polarization index s. The
+aks, a†
+ks represent annihilation and creation opera-
+tors for the mode −→k s. The V13 describes the decay
+of |1⟩ to |3⟩. The gks is the coupling constant be-
+tween the field mode and the atom.
+We use the
+weak coupling assumption and the flat nature of the
+density of states of the electromagnetic vacuum to
+eliminate the degrees of freedom associated with the
+field vacuum. We derive an equation for the den-
+sity matrix of the atomic system alone which can be
+written in the form
+∂ρ
+∂t = Lρ
+(4)
+or in terms of the components as
+˙ρ11 = −2γ1ρ11,
+˙ρ13 = −iω13ρ13 − γ1ρ13,
+˙ρ33 = 2γ1ρ11,
+etc., 2γ1 =
+�
+ks
+|gks|2δ(ω13 − ωks).
+(5)
+This yields steady state as well as transient behavior.
+The spectrum of radiation is related to the Fourier
+transform of the two time dipole correlation func-
+tion, for example in the above case to
+S(ω) = 1
+π Re[S(z)|z=+iω],
+(6)
+S(z) ≡
+� ∞
+0
+dτe−zτ ⟨A13(t + τ)A31(t)⟩ ,
+A13 = A†
+31 = |1⟩ ⟨3| .
+(7)
+The poles of S(z) determine the spectrum. For the
+standard problem S(z) has simple poles.
+The two time correlation function is calculated
+from the solution of (4) and by using the quantum
+———————————————————————————————————————————————————————
+*Published in “Frontiers of Quantum Optics and Laser Physics”, p.155-165, ed. S.Y. Zhu, M.S. Zubairy and M.O. Scully
+(Springer, 1997). This work on higher order poles of S matrix has close connection to the exceptional point physics. Thus this
+work brings out how the exceptional point physics in active systems can be manipulated by laser field. See also G. S. Agarwal,
+Quantum Optics, Cambridge University Press, 2012, Section 17.3.1.
+arXiv:2301.05179v1  [physics.optics]  12 Jan 2023
+
+2
+regression theorem. For completeness, we state what
+it means. We write the solution of (4) as
+ραβ(t + τ) =
+�
+m,n
+Gαβ,mn(τ)ρmn(t).
+(8)
+It should be borne in mind that in the Liouville space
+ραβ is an element of the column matrix.
+We can
+rewrite (8) as
+⟨Aβα(t + τ)⟩ =
+�
+m,n
+Gαβ,mn(τ) ⟨Anm(t)⟩ ,
+(9)
+then the quantum regression theorem leads to two
+time correlation function:
+⟨Aβα(t + τ)Apq(t)⟩ ≡
+�
+m,n
+Gαβ,mn(τ) ⟨Anm(t)Apq(t)⟩
+=
+�
+m,n
+Gαβ,mn(τ) ⟨Anq(t)⟩ δmp
+=
+�
+m,n
+Gαβ,mn(τ)δmpρqn(t).
+(10)
+On using (10) in (6) it is clear that S(z) is related to
+the Laplace transform of G(τ) or to (z −L)−1. Gen-
+erally, the Liouvilliean matrix relevant for the cal-
+culation of (10) decomposes in block diagonal form
+and only a part of L determines the decay or the
+spectral line shapes. For the two level example, the
+correlation function is essentially determined by a
+single equation for ρ13. If there is more than one
+decay channel, then additional terms appear in (5),
+for example, for the case shown in Fig.1, γ1 should
+be replaced by (γ1 +γ2) in the two first equations in
+(5).
+Figure 1. Schematic illustration of the scheme that leads
+to the creation of poles of order two in the decay of the
+state |1⟩; which could be pumped in two different ways ei-
+ther from the state |3⟩ or from a state outside the system.
+This provides the realization of the extended Friedrich-
+Lee model.
+CREATION OF A DOUBLE POLE
+We next demonstrate how by using external elec-
+tromagnetic fields we can convert simple poles of L
+into poles of higher order. For this purpose, we con-
+sider the application of an electromagnetic field that
+is tuned close to the transition frequency ω12 [Fig. 1.
+Λ0 = 0, Λ ̸= 0, Gl ̸= 0]. The Hamiltonian describing
+this system can be written as
+H = ¯hω13 |1⟩ ⟨1|+¯h(ω13−ω12) |2⟩ ⟨2|+Hext+V12+V13,
+(11)
+where Vαβ describes the decay on the transition
+|α⟩ → |β⟩ and where
+Hext = −¯h(Gle−iωlt |1⟩ ⟨2| + h.c.),
+(12)
+Gl = (−→d 12 · −→
+E l/¯h).
+(13)
+The parameter 2Gl is the Rabi frequency of the field
+and is a measure of the strength of the laser field
+applied on the transition |1⟩ ↔ |2⟩.
+The Hamil-
+tonian (11) is time-dependent.
+However one can
+make a canonical transformation to reduce it to a
+time-independent Hamiltonian. In the special case
+V12 → 0 the model (11) is equivalent to the ex-
+tended Friedrich-Lee model.
+We have thus pro-
+duced a realization of a field-theoretic model in the
+context of atoms interacting with laser fields.
+In
+our case lasers are used to control the decay pro-
+cess. Note that we have two control parameters ωl
+and Gl, to manipulate the nature of the poles of
+L. The situation shown in Fig. 1 is realizable in
+many atoms, molecules dopants in solid matrices,
+etc. For example, in 87Rb vapor, the states |1⟩, |2⟩
+and |3⟩ could be the states 5P 3
+2 , 5S 1
+2 , F = 2 and
+5S 1
+2 , F = 1, respectively.
+We eliminate the opti-
+cal frequencies by making canonical transformations
+ρ13 → ρ13e−iω13t, ρ12 → ρ12e−iωlt etc. After canon-
+ical transformations and after eliminating vacuum
+degrees of freedom using the master equation tech-
+niques the density matrix equations read [5]
+˙ρ11 = −2(γ1 + γ2 + Λ)ρ11 + 2Λρ33 + iGlρ21 − iG∗
+l ρ12,
+˙ρ22 = 2γ2ρ11 − iGlρ21 + iG∗
+l ρ12,
+˙ρ21 = −(Γ21 − i∆l)ρ21 − iG∗
+l ρ22 + iG∗
+l ρ11,
+˙ρ31 = −Γ31ρ31 − iG∗
+l ρ32,
+˙ρ32 = −(Γ32 + i∆l)ρ32 − iGlρ31.
+(14)
+Here we have also included a pumping parameter λ
+to pump the population from the level |3⟩ to |1⟩.
+The Γ
+′s
+αβ give the decay of off-diagonal elements ρ
+′s
+αβ
+of the density matrix and are given by
+Γ31 = γ1 + γ2 + 2Λ, Γ32 = Λ,
+Γ21 = γ1 + γ2 + Λ, ∆2 = ω12 − ωl.
+(15)
+From (14) and the quantum regression theorem we
+derive coupled equations for two time atomic corre-
+lation functions
+�
+d
+dτ +
+�
+Γ31
+iG∗
+l
+iGl
+Γ32 + i∆2
+�� �
+⟨A13(t + τ)A31(t)⟩
+⟨A23(t + τ)A31(t)⟩
+�
+= 0.
+(16)
+These are to be solved subject to initial conditions
+⟨A13A31⟩ = ρ11, ⟨A23A31⟩ = ρ12,
+(17)
+which in turn are determined from the steady state
+solution of (14). Clearly the poles of L that deter-
+mine the spectral characteristics are given by
+P(z) = (z + Γ31)(z + Γ32 + i∆l) + |Gl|2.
+(18)
+The zeroes of (18) for ∆l = 0 are shown in Fig. 2.
+The conditions under which P(z) has double zero
+are
+∆l = 0, (Γ32 − Γ31)2 = 4|Gl|2.
+(19)
+The double zero z0 occurs at the bifurcation point
+in Fig. 2
+z0 = −1
+2(Γ31 + Γ32).
+(20)
+We therefore conclude [6] that a simple pole can be
+converted into a double pole in a laboratory experi-
+ment by applying an electromagnetic field resonant
+with the transition |1⟩ ↔ |2⟩ and with Rabi fre-
+quency equal to |Γ31 − Γ32|.
+
+[1)
+V
+W1, G
+Y1
+[2)
+[3]3
+Figure 2. Motion of the zeroes of (18) for Γ31 = 1, Γ32 =
+0.2.
+Note the presence of the bifurcation point.
+This
+is precisely the point where we create a pole of order
+two. The solid curve represents Im(z) + 0.6 whereas the
+dashed curve gives Re(z).
+LINE SHAPES AND DOUBLE POLES
+The line shape can be calculated from the solution
+of (16) and (6):
+S(ω) ≡ ρ11Re[
+(γ2 + Γ32 − iδ)
+(Γ31 − iδ)(Γ32 − iδ) + |Gl|2 ]
+(21)
+which under the double pole condition 2|Gl| = |Γ31−
+Γ32| reduces to
+S(ω) = ρ11Re[γ2 + Γ32 − iδ
+(−iδ + γ0)2 ]
+= ρ11
+δ2(γ1 + 2Λ) + γ2
+0(γ2 + Λ)
+(δ2 + γ2
+0)2
+,
+γ0 = 1
+2(γ1 + γ2 + 3Λ).
+(22)
+This is the modification of the line shape formula.
+Note the double hump structure of the line shape.
+Note further the sensitiveness of S(ω) to the pump-
+ing parameter Λ. In the limit γ2 → 0 and Λ ≪ γ1,
+(22) reduces to
+S(ω) ≡ ρ11
+γ1(δ2 + γ1
+4 Λ)
+(δ2 + γ2
+1
+4 )2
+(23)
+It is also interesting to note, that the scale param-
+eter is now (γ1/2) rather than γ1. Thus the total
+line shape is a sum of (a) Square of the Lorentzian
+(b) derivative of the Lorentzian (ζ/(ζ + γ0)2 ≡
+−ζ ∂
+∂ζ (
+1
+ζ+γ )).
+It is possible to consider an alternate model of
+pumping obtained by setting Λ = 0 in Fig. 1. As-
+suming that γ2 = 0, one can show that instead of
+(23) the spectral line shape is now given by
+S(ω) ≡
+γ1ρ11δ2
+(δ2 + γ2
+1
+4 )2 = (−δ ∂
+∂δ ) (γ1/2)ρ11
+(δ2 + γ2
+1/4)
+(24)
+which is shown in Fig.
+3.
+The figure also shows
+for comparison the Breit-Wigner formula (2) Note
+the double hump structure of the line shape. The
+maxima now occur at δ = ±γ1/2. From Eq. (14)
+we can also compute the time dependence of ρ11(t)
+under the condition of a double pole. The result is
+ρ11(t) = (1 − γ1t
+2 )2e−γ1t
+(25)
+It is again interesting to note that the time scale
+is governed by γ1/2 rather than γ1.
+The basic idea presented above is easily extended
+to more complex situations.
+For example, two-
+photon decay in the system as shown in figure 4
+Figure 3.
+The modified line shape (24) (dashed) as
+a function of δ/γ1 and its comparison with the Breit-
+Wigner line shape (solid).
+which is easily realizable atoms and molecules. The
+full Hamiltonian for this system can be written as
+H = ¯hω13 |1⟩ ⟨1| + ¯hω23 |2⟩ ⟨2| + ¯hω43 |4⟩ ⟨4|
+− ¯h(Gle−iωlt |4⟩ ⟨2| + h.c.)
++
+�
+ks
+¯hωksa†
+kxaks + V12 + V23 + V42
+(26)
+where the meaning of different terms is obvious.
+Again a canonical transformation will change the
+above H into a time-independent H. For V42 → 0,
+the above Hamiltonian becomes identical to the one
+for the quantum field theoretic extended cascade
+field model. We thus have a simple atomic realiza-
+tion of the field-theoretic model. As shown recently
+[7], this system exhibits very interesting two photon
+absorption characteristics. Clearly, the electromag-
+netic coupling between the levels |2⟩ and |4⟩ can pro-
+duce a double pole in the decay of the system. It is
+interesting that a system equivalent to this has been
+studied by Bhamathi and Sudarshan [2].
+Figure 4. A scheme involving laser coupling the inter-
+mediate state |2⟩ which will create pole of order two in
+the two-photon decay. This provides an analog of the
+extended cascade model.
+DOUBLE POLES AND INTERFERENCE
+EFFECTS
+The existence of double poles and the possibility
+of a line shape which is a derivative of Lorentzian
+suggest that the quantum interferences must be
+crucial.
+This is indeed the case as can be seen
+from the following considerations. The electromag-
+netic coupling of |1⟩ and |2⟩ produced dressed states
+|ψ±⟩ =
+1
+√
+2(± |1⟩ + |2⟩) with eigenvalues ±Gl. Since
+Gl ∼ γ, the two states are within the radiative
+line width. We pump the population in the state
+|1⟩ which is equivalent to pumping in both |ψ±⟩ as
+
+2.0
+1.5
+1.0
+.......
+.=
+0.5
+0.0
+-0.5
+-1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Gt1.0
+1
+-
+-
+0.8
+-
+-
+-
+-
+-
+1
+1
+0.6
+-
+1
+1
+S
+1
+-
+1
+1
+1
+0.4
+1
+1
+1
+0.2
+I:
+0.0
+-2
+0
+2
+4
+S|1)
+14)
+Laser Coupling
+[2)
+ 3)4
+|1⟩ = (|ψ+⟩ + |ψ−⟩)/
+√
+2. Both states |ψ±⟩ can de-
+cay to |3⟩ as |ψ±⟩ involve admixtures of |1⟩ and |2⟩.
+These two decays will not be independent [8,9] as
+−→d +3 · −→d ∗
+−3 ̸= 0 and as Gl ∼ γ.
+EXPONENTIAL DECAY RECOVERED
+We also examine the initial conditions for our sys-
+tem which would result in exponential decay. From
+Eq. (16) it is seen that
+d
+dτ ⟨(A13(t + τ) + iA23(t + τ))A31(t)⟩
++ γ1
+2 ⟨(A13(t + τ) + iA23(t + τ))A31(t)⟩ = 0 (27)
+if Gl = γ1
+2 , γ2 → 0. Thus the correlation function
+defined in terms of the vector ˜ψ =
+1
+√
+2(|1⟩ + i |2⟩)
+obeys simple exponential decay law with a time scale
+governed by γ1/2 rather than γ1:
+�
+A ˜
+ψ3(t + τ)A3 ˜
+ψ(t)
+�
+= e−γ1τ/2 �
+A ˜
+ψ ˜
+ψ(t)
+�
+(28)
+Thus a pumping of the system to the state ˜ψ rather
+than |1⟩ will result in exponential decay[10].
+Thus, in conclusion, we have shown how higher-
+order poles in the decay of states can be produced
+by using resonant electromagnetic fields. We demon-
+strated this by creating a pole of order two. Clearly,
+the technique is quite versatile and by using combi-
+nations of electromagnetic fields we can create poles
+of higher order.
+I thank George Sudarshan for discussions on
+higher order poles of S-Matrix and R.P. Singh for
+help in preparation of this paper.
+[1] M.L. Goldberger and K.M. Watson, Phys. Rev. 136,
+B 1472 (1964); J, S. Bell and C.J. Goebel, Phys,
+Rev. 138, B 1198 (1965)
+[2] G. Bhamathi and E.C.G. Sudarshan, Int. J. Mod
+Phys. B 10, 1531 (1996); see also E.C.G Sudarshan,
+Phys. Rev. A 50, 2006 (1994); E.C.G. Sudarshan,
+C.B. Chiu and G. Bhamathi, Phys. Rev. D 46, 3508
+(1992).
+[3] A. Bohm, S. Maxson and M. Loewe, Physica A, in
+press; A. Mondragon and E. Her- nandez, J. Phys.
+A 26, 5595 (1993); C. Puntmann, paper presented
+at the International Colloquium on Group Theory,
+Goslar, Germany 1996.
+[4] G.S. Agarwal, Quantum Optics (Springer-Verlag,
+Berlin, 1974).
+[5] G.S. Agarwal, Phys. Rev. A 54, Rapid Commun.
+3734 (1996).
+[6] An almost trivial case occurs when Γ23 = 0, γ2 = 0
+and no pumping (Λ = 0). The atom can start in
+state |1⟩. Then one can work with a nonhermitian
+Hamiltonian
+�
+−iγ1 G
+G
+0
+�
+which has identical real
+eigenvalues if 2G = γ1. This case has been pre-
+viously considered in literature (H. Steudel, Ann.
+Physik 22, 113 (1969).
+[7] G.S. Agarwal and W. Harshawardhan, Phys. Rev,
+Lett. 77, 1039 (1996).
+[8] G.S. Agarwal, Quantum Optics (Springer-Verlag,
+Berlin, 1974) pp. 94-96.
+[9] A. Imamoglu, Phys. Rev. A 40, 2835 (1989); S.Y.
+Zhu and M.O. Scully, Phys. Rev. Lett. 76, 388
+(1996); D.A. Cardimona, M.G. Raymer and C.R.
+Stroud Jr., J. Phys. B 15, 55 (1982).
+[10] Pumping the system in the state ≫ is possible using
+an excitation pulse with phase switching at appro-
+priate instant (cf. Y.S. Bai, A.G. Yodh and T.W.
+Mossberg, Phys. Rev. Lett. 55, 1277 (1984)).
+
diff --git a/aNE4T4oBgHgl3EQfnw2m/content/tmp_files/load_file.txt b/aNE4T4oBgHgl3EQfnw2m/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,289 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf,len=288
+page_content='Laser Field Initiation of Higher Order Poles of S-Matrix-Optical Realization of  Field Theoretic Models*  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal1,2  1Physical Research Laboratory, Navrangpura, Ahmedabad, India  2Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore, India  We discuss the possibility of converting a simple pole in the radiative decay of a state into a pole of  higher order by using resonant electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' This process of creation of higher order pole  is controllable by the intensity of the laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We use density matrix and Liouville space and  present the modification of the Lorentzian line shapes (Breit-Wigner formula) for example to ones  involving square of Lorentzian and derivatives of Lorentzians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  INTRODUCTION  In a classic paper Goldberger and Watson [1] con-  sidered the possibility that the decay law for an un-  stable particle can be more complex than a simple  exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' They showed the possibility of the ex-  istence of the poles of S-matrix which were not nec-  essarily simple poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Since then, higher order poles  have been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Recently, there is re-  vival [2,3] of interest in such studies and in particular  Bhamathi and Sudarshan have analyzed several field  theoretic models like Friedrich-Lee model, cascade  model and their extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' They examine the spec-  trum (complex) of eigenvalues for such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A  related question is how the Breit-Wigner line shape  formula is modified if S-matrix possess higher order  poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  In this paper we examine the possibility of cre-  ation of the higher order poles using laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We  consider the decay of say excited state of an atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Normally this decay is described by the Wigner-  Weisskopf theory which leads to exponential decay  law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We next discuss the case when the excited  state is coupled to another state by a resonant elec-  tromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' In such a case we show that for  appropriate value of the intensity of the laser field  the corresponding spectral function has a pole of or-  der two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We calculate the resulting line shape and  discuss the line narrowing etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We emphasize that  we work within the framework of density matrices  and hence we work in Liouville space rather than  in Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We present optical realization of  various field theoretic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Consider the decay of the state |1⟩ into the states  |3⟩ and |2⟩ at the rates 2γ1 and 2γ2 respectively as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='1 (with Gl = 0, △l = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  It is well  known that the rate of decay of the population in  |1⟩ is given by  ρ11(t) = ρ11(0)exp(−2(γ1 + γ2)t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (1)  Here ρ is the density matrix of the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The spec-  trum of the spontaneously emitted photons will con-  sist of two Lorentzians centered at ω13 and ω12 with  a half width (γ1 + γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Let us concentrate on the  emission on the transition |1⟩ ↔ |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The spectrum  will be described by the well-known form  S(ω) =  γ1/π  (γ1 + γ2)2 + (ω − ω13)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (2)  Note that γ2 will be zero if the decay channel |1⟩ →  |2⟩ is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We will discuss how the laser  fields could be used to modify significantly the re-  sults predicted by (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  LIOUVILLE SPACE FORMULATION OF  DECAY  We next recall how the spectrum is calculated in  the density matrix framework [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We have included  this material for completeness so that our discussion  in subsequent sections can be followed by the non-  Quantum optics practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Consider a system  with two states |1⟩ and |3⟩ interacting with the vac-  uum of the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The Hamiltonian  can be written in the form  H = ¯hω13 |1⟩ ⟨1| +  �  ks  ¯hωksa†  ksaks + V13  V13 =  �  ks  (¯hgksa†  ks |1⟩ ⟨3| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (3)  The vacuum modes are characterized by the propa-  gation index −→k and the polarization index s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The  aks, a†  ks represent annihilation and creation opera-  tors for the mode −→k s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The V13 describes the decay  of |1⟩ to |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The gks is the coupling constant be-  tween the field mode and the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We use the  weak coupling assumption and the flat nature of the  density of states of the electromagnetic vacuum to  eliminate the degrees of freedom associated with the  field vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We derive an equation for the den-  sity matrix of the atomic system alone which can be  written in the form  ∂ρ  ∂t = Lρ  (4)  or in terms of the components as  ˙ρ11 = −2γ1ρ11,  ˙ρ13 = −iω13ρ13 − γ1ρ13,  ˙ρ33 = 2γ1ρ11,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=', 2γ1 =  �  ks  |gks|2δ(ω13 − ωks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (5)  This yields steady state as well as transient behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The spectrum of radiation is related to the Fourier  transform of the two time dipole correlation func-  tion, for example in the above case to  S(ω) = 1  π Re[S(z)|z=+iω],  (6)  S(z) ≡  � ∞  0  dτe−zτ ⟨A13(t + τ)A31(t)⟩ ,  A13 = A†  31 = |1⟩ ⟨3| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (7)  The poles of S(z) determine the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For the  standard problem S(z) has simple poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The two time correlation function is calculated  from the solution of (4) and by using the quantum  ———————————————————————————————————————————————————————  Published in “Frontiers of Quantum Optics and Laser Physics”, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='155-165, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Zhu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Zubairy and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Scully  (Springer, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' This work on higher order poles of S matrix has close connection to the exceptional point physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Thus this  work brings out how the exceptional point physics in active systems can be manipulated by laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' See also G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal,  Quantum Optics, Cambridge University Press, 2012, Section 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='05179v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='optics]  12 Jan 2023  2  regression theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For completeness, we state what  it means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We write the solution of (4) as  ραβ(t + τ) =  �  m,n  Gαβ,mn(τ)ρmn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (8)  It should be borne in mind that in the Liouville space  ραβ is an element of the column matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We can  rewrite (8) as  ⟨Aβα(t + τ)⟩ =  �  m,n  Gαβ,mn(τ) ⟨Anm(t)⟩ ,  (9)  then the quantum regression theorem leads to two  time correlation function:  ⟨Aβα(t + τ)Apq(t)⟩ ≡  �  m,n  Gαβ,mn(τ) ⟨Anm(t)Apq(t)⟩  =  �  m,n  Gαβ,mn(τ) ⟨Anq(t)⟩ δmp  =  �  m,n  Gαβ,mn(τ)δmpρqn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (10)  On using (10) in (6) it is clear that S(z) is related to  the Laplace transform of G(τ) or to (z −L)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Gen-  erally, the Liouvilliean matrix relevant for the cal-  culation of (10) decomposes in block diagonal form  and only a part of L determines the decay or the  spectral line shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For the two level example, the  correlation function is essentially determined by a  single equation for ρ13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' If there is more than one  decay channel, then additional terms appear in (5),  for example, for the case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='1, γ1 should  be replaced by (γ1 +γ2) in the two first equations in  (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Schematic illustration of the scheme that leads  to the creation of poles of order two in the decay of the  state |1⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' which could be pumped in two different ways ei-  ther from the state |3⟩ or from a state outside the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  This provides the realization of the extended Friedrich-  Lee model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  CREATION OF A DOUBLE POLE  We next demonstrate how by using external elec-  tromagnetic fields we can convert simple poles of L  into poles of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For this purpose, we con-  sider the application of an electromagnetic field that  is tuned close to the transition frequency ω12 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Λ0 = 0, Λ ̸= 0, Gl ̸= 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The Hamiltonian describing  this system can be written as  H = ¯hω13 |1⟩ ⟨1|+¯h(ω13−ω12) |2⟩ ⟨2|+Hext+V12+V13,  (11)  where Vαβ describes the decay on the transition  |α⟩ → |β⟩ and where  Hext = −¯h(Gle−iωlt |1⟩ ⟨2| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='),  (12)  Gl = (−→d 12 · −→  E l/¯h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (13)  The parameter 2Gl is the Rabi frequency of the field  and is a measure of the strength of the laser field  applied on the transition |1⟩ ↔ |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The Hamil-  tonian (11) is time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  However one can  make a canonical transformation to reduce it to a  time-independent Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' In the special case  V12 → 0 the model (11) is equivalent to the ex-  tended Friedrich-Lee model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We have thus pro-  duced a realization of a field-theoretic model in the  context of atoms interacting with laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  In  our case lasers are used to control the decay pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Note that we have two control parameters ωl  and Gl, to manipulate the nature of the poles of  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The situation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 1 is realizable in  many atoms, molecules dopants in solid matrices,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For example, in 87Rb vapor, the states |1⟩, |2⟩  and |3⟩ could be the states 5P 3  2 , 5S 1  2 , F = 2 and  5S 1  2 , F = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  We eliminate the opti-  cal frequencies by making canonical transformations  ρ13 → ρ13e−iω13t, ρ12 → ρ12e−iωlt etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' After canon-  ical transformations and after eliminating vacuum  degrees of freedom using the master equation tech-  niques the density matrix equations read [5]  ˙ρ11 = −2(γ1 + γ2 + Λ)ρ11 + 2Λρ33 + iGlρ21 − iG∗  l ρ12,  ˙ρ22 = 2γ2ρ11 − iGlρ21 + iG∗  l ρ12,  ˙ρ21 = −(Γ21 − i∆l)ρ21 − iG∗  l ρ22 + iG∗  l ρ11,  ˙ρ31 = −Γ31ρ31 − iG∗  l ρ32,  ˙ρ32 = −(Γ32 + i∆l)ρ32 − iGlρ31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (14)  Here we have also included a pumping parameter λ  to pump the population from the level |3⟩ to |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The Γ  ′s  αβ give the decay of off-diagonal elements ρ  ′s  αβ  of the density matrix and are given by  Γ31 = γ1 + γ2 + 2Λ, Γ32 = Λ,  Γ21 = γ1 + γ2 + Λ, ∆2 = ω12 − ωl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (15)  From (14) and the quantum regression theorem we  derive coupled equations for two time atomic corre-  lation functions  �  d  dτ +  �  Γ31  iG∗  l  iGl  Γ32 + i∆2  �� �  ⟨A13(t + τ)A31(t)⟩  ⟨A23(t + τ)A31(t)⟩  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (16)  These are to be solved subject to initial conditions  ⟨A13A31⟩ = ρ11, ⟨A23A31⟩ = ρ12,  (17)  which in turn are determined from the steady state  solution of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Clearly the poles of L that deter-  mine the spectral characteristics are given by  P(z) = (z + Γ31)(z + Γ32 + i∆l) + |Gl|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (18)  The zeroes of (18) for ∆l = 0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The conditions under which P(z) has double zero  are  ∆l = 0, (Γ32 − Γ31)2 = 4|Gl|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (19)  The double zero z0 occurs at the bifurcation point  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 2  z0 = −1  2(Γ31 + Γ32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (20)  We therefore conclude [6] that a simple pole can be  converted into a double pole in a laboratory experi-  ment by applying an electromagnetic field resonant  with the transition |1⟩ ↔ |2⟩ and with Rabi fre-  quency equal to |Γ31 − Γ32|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [1)  V  W1, G  Y1  [2)  [3]3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Motion of the zeroes of (18) for Γ31 = 1, Γ32 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Note the presence of the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  This  is precisely the point where we create a pole of order  two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The solid curve represents Im(z) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='6 whereas the  dashed curve gives Re(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  LINE SHAPES AND DOUBLE POLES  The line shape can be calculated from the solution  of (16) and (6):  S(ω) ≡ ρ11Re[  (γ2 + Γ32 − iδ)  (Γ31 − iδ)(Γ32 − iδ) + |Gl|2 ]  (21)  which under the double pole condition 2|Gl| = |Γ31−  Γ32| reduces to  S(ω) = ρ11Re[γ2 + Γ32 − iδ  (−iδ + γ0)2 ]  = ρ11  δ2(γ1 + 2Λ) + γ2  0(γ2 + Λ)  (δ2 + γ2  0)2  ,  γ0 = 1  2(γ1 + γ2 + 3Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  (22)  This is the modification of the line shape formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Note the double hump structure of the line shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Note further the sensitiveness of S(ω) to the pump-  ing parameter Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' In the limit γ2 → 0 and Λ ≪ γ1,  (22) reduces to  S(ω) ≡ ρ11  γ1(δ2 + γ1  4 Λ)  (δ2 + γ2  1  4 )2  (23)  It is also interesting to note, that the scale param-  eter is now (γ1/2) rather than γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Thus the total  line shape is a sum of (a) Square of the Lorentzian  (b) derivative of the Lorentzian (ζ/(ζ + γ0)2 ≡  −ζ ∂  ∂ζ (  1  ζ+γ )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  It is possible to consider an alternate model of  pumping obtained by setting Λ = 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' As-  suming that γ2 = 0, one can show that instead of  (23) the spectral line shape is now given by  S(ω) ≡  γ1ρ11δ2  (δ2 + γ2  1  4 )2 = (−δ ∂  ∂δ ) (γ1/2)ρ11  (δ2 + γ2  1/4)  (24)  which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The figure also shows  for comparison the Breit-Wigner formula (2) Note  the double hump structure of the line shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The  maxima now occur at δ = ±γ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' (14)  we can also compute the time dependence of ρ11(t)  under the condition of a double pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The result is  ρ11(t) = (1 − γ1t  2 )2e−γ1t  (25)  It is again interesting to note that the time scale  is governed by γ1/2 rather than γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The basic idea presented above is easily extended  to more complex situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  For example, two-  photon decay in the system as shown in figure 4  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  The modified line shape (24) (dashed) as  a function of δ/γ1 and its comparison with the Breit-  Wigner line shape (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  which is easily realizable atoms and molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The  full Hamiltonian for this system can be written as  H = ¯hω13 |1⟩ ⟨1| + ¯hω23 |2⟩ ⟨2| + ¯hω43 |4⟩ ⟨4|  − ¯h(Gle−iωlt |4⟩ ⟨2| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=')  +  �  ks  ¯hωksa†  kxaks + V12 + V23 + V42  (26)  where the meaning of different terms is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Again a canonical transformation will change the  above H into a time-independent H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' For V42 → 0,  the above Hamiltonian becomes identical to the one  for the quantum field theoretic extended cascade  field model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We thus have a simple atomic realiza-  tion of the field-theoretic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' As shown recently  [7], this system exhibits very interesting two photon  absorption characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Clearly, the electromag-  netic coupling between the levels |2⟩ and |4⟩ can pro-  duce a double pole in the decay of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' It is  interesting that a system equivalent to this has been  studied by Bhamathi and Sudarshan [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A scheme involving laser coupling the inter-  mediate state |2⟩ which will create pole of order two in  the two-photon decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' This provides an analog of the  extended cascade model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  DOUBLE POLES AND INTERFERENCE  EFFECTS  The existence of double poles and the possibility  of a line shape which is a derivative of Lorentzian  suggest that the quantum interferences must be  crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  This is indeed the case as can be seen  from the following considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The electromag-  netic coupling of |1⟩ and |2⟩ produced dressed states  |ψ±⟩ =  1  √  2(± |1⟩ + |2⟩) with eigenvalues ±Gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Since  Gl ∼ γ, the two states are within the radiative  line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We pump the population in the state  |1⟩ which is equivalent to pumping in both |ψ±⟩ as  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='=  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  Gt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='8 1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='6 1  1  S  1 1  1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='4  1  1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='2  I:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='0  2  0  2  4  S|1)  14)  Laser Coupling  [2)  3)4  |1⟩ = (|ψ+⟩ + |ψ−⟩)/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Both states |ψ±⟩ can de-  cay to |3⟩ as |ψ±⟩ involve admixtures of |1⟩ and |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  These two decays will not be independent [8,9] as  −→d +3 · −→d ∗  −3 ̸= 0 and as Gl ∼ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  EXPONENTIAL DECAY RECOVERED  We also examine the initial conditions for our sys-  tem which would result in exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' From  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' (16) it is seen that  d  dτ ⟨(A13(t + τ) + iA23(t + τ))A31(t)⟩  + γ1  2 ⟨(A13(t + τ) + iA23(t + τ))A31(t)⟩ = 0 (27)  if Gl = γ1  2 , γ2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Thus the correlation function  defined in terms of the vector ˜ψ =  1  √  2(|1⟩ + i |2⟩)  obeys simple exponential decay law with a time scale  governed by γ1/2 rather than γ1:  �  A ˜  ψ3(t + τ)A3 ˜  ψ(t)  �  = e−γ1τ/2 �  A ˜  ψ ˜  ψ(t)  �  (28)  Thus a pumping of the system to the state ˜ψ rather  than |1⟩ will result in exponential decay[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Thus, in conclusion, we have shown how higher-  order poles in the decay of states can be produced  by using resonant electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' We demon-  strated this by creating a pole of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Clearly,  the technique is quite versatile and by using combi-  nations of electromagnetic fields we can create poles  of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  I thank George Sudarshan for discussions on  higher order poles of S-Matrix and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Singh for  help in preparation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Goldberger and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Watson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 136,  B 1472 (1964);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' J, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bell and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Goebel, Phys,  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 138, B 1198 (1965)  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bhamathi and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Sudarshan, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Mod  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' B 10, 1531 (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' see also E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G Sudarshan,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A 50, 2006 (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Sudarshan,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Chiu and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bhamathi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' D 46, 3508  (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bohm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Maxson and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Loewe, Physica A, in  press;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Mondragon and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Her- nandez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  A 26, 5595 (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Puntmann, paper presented  at the International Colloquium on Group Theory,  Goslar, Germany 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal, Quantum Optics (Springer-Verlag,  Berlin, 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' A 54, Rapid Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  3734 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [6] An almost trivial case occurs when Γ23 = 0, γ2 = 0  and no pumping (Λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' The atom can start in  state |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Then one can work with a nonhermitian  Hamiltonian  �  −iγ1 G  G  0  �  which has identical real  eigenvalues if 2G = γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' This case has been pre-  viously considered in literature (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Steudel, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Agarwal and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content='  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Imamoglu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Zhu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Scully, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Cardimona, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content=' Raymer and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Stroud Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' B 15, 55 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  [10] Pumping the system in the state ≫ is possible using  an excitation pulse with phase switching at appro-  priate instant (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Bai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' Yodh and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content='  Mossberg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
+page_content=' 55, 1277 (1984)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfnw2m/content/2301.05179v1.pdf'}
diff --git a/cNFAT4oBgHgl3EQfYB2G/content/tmp_files/2301.08537v1.pdf.txt b/cNFAT4oBgHgl3EQfYB2G/content/tmp_files/2301.08537v1.pdf.txt
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+Towards Multi-robot Exploration:
+A Decentralized Strategy for UAV Forest Exploration
+Luca Bartolomei, Lucas Teixeira and Margarita Chli
+Vision For Robotics Lab, ETH Z¨urich, Switzerland
+Abstract— Efficient exploration strategies are vital in tasks
+such as search-and-rescue missions and disaster surveying.
+Unmanned Aerial Vehicles (UAVs) have become particularly
+popular in such applications, promising to cover large areas at
+high speeds. Moreover, with the increasing maturity of onboard
+UAV perception, research focus has been shifting toward higher-
+level reasoning for single- and multi-robot missions. However,
+autonomous navigation and exploration of previously unknown
+large spaces still constitutes an open challenge, especially when
+the environment is cluttered and exhibits large and frequent
+occlusions due to high obstacle density, as is the case of forests.
+Moreover, the problem of long-distance wireless communication
+in such scenes can become a limiting factor, especially when
+automating the navigation of a UAV swarm. In this spirit,
+this work proposes an exploration strategy that enables UAVs,
+both individually and in small swarms, to quickly explore
+complex scenes in a decentralized fashion. By providing the
+decision-making capabilities to each UAV to switch between
+different execution modes, the proposed strategy strikes a
+great balance between cautious exploration of yet completely
+unknown regions and more aggressive exploration of smaller
+areas of unknown space. This results in full coverage of forest
+areas of variable density, consistently faster than the state of the
+art. Demonstrating successful deployment with a single UAV as
+well as a swarm of up to three UAVs, this work sets out the basic
+principles for multi-root exploration of cluttered scenes, with
+up to 65% speed up in the single UAV case and 40% increase in
+explored area for the same mission time in multi-UAV setups.
+I. INTRODUCTION
+The growing interest in Unmanned Aerial Vehicles (UAVs)
+has led to their extensive deployment in tasks such as
+inspection and search-and-rescue missions. In these appli-
+cations, the capacity of the robot to quickly explore and
+map unknown environments autonomously is fundamental.
+The literature on this topic is extensive, and many different
+approaches have been proposed throughout the years [1]–
+[5]. However, one of the biggest challenges in the explo-
+ration of unknown environments is the capacity to achieve
+a good trade-off between the competing goals of shorter
+exploration times of an area of interest (i.e. pushing for
+high-speed navigation) and safety, which requires caps on
+the velocity of each robot. In fact, navigating in the vicinity
+of the boundaries between known and unknown space is
+challenging, as the robot can get stuck in dead ends, or
+needs to perform complex dodging maneuvers to avoid
+collisions. Consequently, to maintain the safety of both the
+platform and its surroundings, most path planners generate
+conservative start-and-stop motions, not fully exploiting the
+This work was supported by NCCR Robotics, the Amazon Research
+Awards, and the HILTI group.
+Fig. 1: 3D-view of the proposed system guiding safe and successful
+exploration of a UAV in a digital model of a real-world forest
+[6]. The planner is able to avoid collisions between the UAV and
+the obstacles, clearing frontiers on-the-go by balancing cautious
+navigation with aggressive exploitation of known, free space, in a
+bid to maximize the efficiency of the exploration.
+capacity of a UAV to fly at high speeds. This effect is
+exacerbated when the environment to explore is particularly
+cluttered, as is the case in forests, leading to inefficient and
+incomplete coverage. By design, these methods generally
+drive the exploration process by biasing exploration towards
+large areas of unexplored space. While this strategy could be
+advantageous in open and wide spaces, it can be detrimental
+when exploring cluttered scenes. In fact, the main pitfall of
+such strategies is that, while the exploration process attempts
+to cover as much unknown space as possible, when this
+is deployed in environments with many obstacles, thinner
+trails of unknown space are left unexplored (e.g. due to
+occlusions), imposing the need for a second sweep of the
+environment over mostly explored areas.
+Aiming to mitigate these issues, pushing for faster cov-
+erage of the areas of interest, multi-robot extensions for
+exploration have also been proposed [7]–[10]. However,
+these focus on the problem of coordination at the system-
+level and, and while they can perform better from a global
+planning point of view, they suffer from the same limitations
+as the single-UAV case in obstacle-dense environments.
+Motivated by these challenges, in this work we propose an
+exploration strategy for autonomous UAV robots aiming to
+explore forests of increasing tree density, as they pose some
+of the most difficult challenges for exploration planning. Our
+objective is to exploit the platform’s dynamics to the fullest
+despite the high density of obstacles, in order to achieve the
+complete coverage of the environment efficiently. To this end,
+the proposed strategy enables switching between two differ-
+arXiv:2301.08537v1  [cs.RO]  20 Jan 2023
+
+ent behaviors for each robot; namely, cautious exploration
+of unknown space and more aggressive maneuvers when
+navigating in already explored areas to clear smaller portions
+of unknown space caused by occlusions. We evaluate the
+proposed approach in a series of challenging experiments in
+simulation, both in randomly generated forests and in a 3D
+reconstruction of a real forest (Fig 1). Benchmarking against
+the state of the art reveals superior efficiency for the proposed
+approach achieving higher overall UAV speeds and lower
+exploration times. Finally, aiming to set out the scaffolding
+toward decentralized multi-robot exploration planning, we
+show how the proposed strategy can accommodate more than
+one UAV in the exploration mission, and we demonstrate that
+our method performs comparably to or better than a map-
+splitting centralized approach.
+In summary, the contributions of this work are as follows:
+• the design of an exploration strategy, able to strike
+an effective balance between cautious exploration and
+aggressive exploitation of the explored map,
+• the extension of the single-robot design to a multi-robot
+decentralized approach, and
+• extensive evaluations in simulation, demonstrating bet-
+ter performance than the state of the art.
+II. RELATED WORKS
+Autonomous exploration of unknown environments with
+UAVs has been an active field of research over the past
+few decades. The most popular approach to exploring an
+area of interest is to use frontiers, defined as the boundary
+between known and unknown space [11]. These can be
+utilized to identify potentially informative spatial regions
+in order to drive the exploration process efficiently until
+no new frontiers are found and the exploration process can
+be considered complete. There are different criteria used to
+decide which frontier to explore next, such as their proximity
+to the current field of view, following a greedy selection
+strategy, or having global planning dictate the selection [12].
+However, while frontier-based approaches have been proven
+to yield satisfactory performances, especially in terms of
+coverage [2], [3], they generally lead to inefficient motions
+and sub-optimal action selection. This is mostly caused by
+the sensing modalities used to generate the map of the
+environment to explore, as the most common sensors, such as
+RGB-D and stereo cameras, have a limited detection range.
+Consequently, UAVs need to fly cautiously to ensure safety.
+Cieslewski et al. [4] tackle this limitation, by proposing
+an exploration strategy that generates velocity commands
+based on newly detected frontiers, in a bid to maximize
+the UAV’s speed. This method is shown to outperform
+classical methods [11], but focuses only on local frontiers.
+Instead, FUEL [5] proposes a hierarchical planner which
+generates efficient global paths, while encouraging safe and
+agile local maneuvers for high-speed exploration. FUEL’s
+strategy performs better than [4] and [11] in scenes with
+low obstacle densities. However, it is more computationally
+demanding, as it needs to maintain a list of active frontiers,
+as well as to compute accurate distances between them. This
+additional bookkeeping becomes prohibitive and impractical
+in more cluttered and complex environments such as forests.
+In fact, in this type of scenery, the number of frontiers
+quickly increases due to occlusions caused by tree trunks,
+branches, and shrubs.
+Another line of research focuses instead on sampling-
+based path planning to generate viewpoints to explore the
+space [13], [14], by guiding the robot along possible trails
+of sampled configurations. The best path is generally found
+using a greedy approach [14], bringing to complete explo-
+ration or accurate surface reconstruction [2] depending on
+the information gain formulation. Nonetheless, these sampled
+routes may generate trajectories that deviate from the shortest
+paths, without taking into consideration the robot’s dynamics.
+Consequently, this causes the UAV to navigate in zigzag
+patterns, leading to inefficient, slow motions and conservative
+maneuvers.
+To tackle the limitations of frontier- and sampling-based
+methods, also hybrid approaches have been proposed [1],
+[15]. Such methods compute global paths towards the most
+informative frontiers while generating local trajectories using
+sampling-based planners. However, they do not exploit the
+full dynamics of the platform and generate sub-optimal
+routes.
+To boost the efficiency in exploration, various multi-robot
+cooperative frontier-based methods have also been proposed
+in the literature, both in centralized [16] and decentralized
+formats [17]. In this spirit, the work in [18] greedily assigns
+view configurations, while [19] distributes the workload
+between agents using a Voronoi-based partitioning of the
+area to explore. Nevertheless, these solutions suffer from
+the same limitations as in the single-robot case. Instead, the
+approach in [20] is able to generate efficient trajectories for
+3D reconstruction, tackling the multi-robot coordination with
+a centralized architecture. However, this method requires
+a prior overhead flight over the area of interest, making
+it unsuitable for the exploration of forests. The approach
+proposed in [7] puts more focus on the problem of navigating
+forests, but the emphasis is more on state estimation rather
+than on path planning.
+Motivated by these limitations, in this work, we propose
+a strategy that allows a robot to explore complex forest-
+like environments while flying at high speeds, thanks to
+the freedom and flexibility that our planner provides to
+each UAV to switch between different navigation modes
+online. While slower, cautious exploration is performed
+using a frontier-based approach, we efficiently clear trails of
+unexplored space caused by occlusions by employing a more
+aggressive local exploration strategy, boosting the efficiency
+of the mission and pushing the overall time to cover a given
+area of interest down. Moreover, we demonstrate that the
+proposed pipeline can also be extended to the multi-robot
+setting in a decentralized fashion.
+III. METHODOLOGY
+The overall problem considered in this work is to ex-
+plore unknown cluttered environments, such as forests, in
+
+(a) Time-instant 1
+(b) Time-instant 2
+(c) Time-instant 3
+(d) Time-instant 4
+Fig. 2: A schematic example demonstrating the problem with greedy frontier-based exploration, at progressive time-instants, generating
+islands of unknown space surrounded by free regions. The field of view of the robot is depicted as a light-gray shaded area delimited by
+black solid lines, while the obstacles and the unexplored space are in black and dark gray, respectively (a). The robot navigates towards
+the most informative frontiers (b); however, due to the limited sensor range, the space occluded by the obstacles is not cleared (c).
+Consequently, since the exploration process is biased towards larger, more informative frontiers, the na¨ıve planner flies the UAV robot
+ignoring the smaller portion of unexplored space (d).
+the minimum time possible. We assume that the robot is
+equipped with a front-looking depth camera with a limited
+sensing range and that the robot’s odometry information is
+available at a constant rate. However, forest-like scenes are
+characterized by a high number of obstacles in a variety
+of dimensions (e.g. trunks, leaves, and branches) that make
+standard frontier-based exploration approaches inefficient.
+In fact, during the exploration process, many islands of
+unknown space are usually left behind, as illustrated in
+Fig. 2, necessitating subsequent passes of exploration on a
+nearly completely explored map. To tackle this limitation,
+we propose an exploration pipeline that can change the ex-
+ploratory behavior of the robot depending on the frontiers in
+its vicinity. In particular, we propose to define two different
+modes of operation for the robot, namely the Explorer and
+the Collector modes. In the Explorer state, the robot is driven
+by frontiers and it is tasked to explore large unknown areas.
+Consequently, it predominantly operates on the most external
+boundaries between known and unknown areas. Conversely,
+the robot in the Collector mode clears small islands of
+unknown space generated by occlusions, that are left behind
+during the exploratory phase. The objective of a Collector is
+to clear these portions of space on the go, avoiding the need
+for subsequent revisits of the map, at the expense of short
+local detours. However, notice that these can be performed at
+high speed, since, when in Collector mode, the robot operates
+in mostly explored areas. By allowing a robot to switch
+between these two different modes and by finding the right
+trade-off between map exploration and exploitation, we can
+quickly reach full coverage of large cluttered environments.
+In the following, we first give an overview of the proposed
+system and our exploration strategies, and then we illustrate
+how it can be extended to multiple robots.
+A. System Overview
+As shown in Fig. 3, the pipeline is composed of three main
+components: a mapping system, a mode selector, and a path
+planner.
+Given input depth and odometry information, a voxel
+grid map M of the environment is generated. At every
+update, frontiers are extracted from M and clustered. For
+each cluster, we adopt the sampling strategy from [5] to
+generate viewpoints covering the frontiers, and we use them
+as possible target poses during the exploration process.
+Moreover, each cluster undergoes a binary classification
+step, where unconnected islands of frontiers, or trails, are
+identified. This is necessary to identify those regions that are
+likely to require an additional revisiting phase towards the
+end of the mission if a traditional frontier-based exploration
+method is utilized. Here, a cluster is considered a trail if its
+convex hull is surrounded by free space, or when it has only
+another neighboring cluster. This implies that most clusters
+at the corners of the area to be explored are classified as
+trails. We motivate this design choice by arguing that corners
+are generally problematic for exploration due to their low
+informative value. In fact, they are rarely covered in a first
+sweep of the map, implying the need for a revisiting step.
+The labeled clusters are then utilized by the Mode Selector to
+choose the best exploration strategy for the robot, deciding
+whether it has to persevere in its current mode, or transit
+to Explorer or Collector. The mode assignment is regulated
+according to the frontiers in the vicinity of the UAV. Given
+that our objective is to clear trails locally to avoid large
+detours on the map, we assign the role of Collector if a
+minimum number of trails is close to the robot. Instead, we
+adopt a more exploratory strategy once all smaller islands of
+unknown space are cleared, or when the trails are far away
+from the drone. Once a strategy is selected, the viewpoint of
+the most promising cluster is selected as the new target pose.
+This is fed to a path planner [21] that generates the trajectory
+flying the UAV toward its destination. We now describe the
+different exploration modes in more detail.
+B. Exploration Strategies
+1) Explorer: Driven by frontiers, the objective of an
+Explorer is to cover large areas of previously unknown space.
+Similarly to [4], we process the incoming clusters of frontiers
+C from the most recent map update and extract the one with
+the lowest cost JE. Notice that these clusters are mostly
+aligned with the direction of the UAV’s motion, implying
+that, if one of these is selected as the target, the robot
+avoids abrupt changes in the flight direction or aggressive
+
+Mapping
+3D Map
+Frontiers Classification
+Frontiers Extraction
+Odometry 
+Depth
+Target Pose Selector
+Trajectory Generation
+Path Planning
+Mode Selector
+Low-level 
+Controller
+Fig. 3: Schematic overview of the proposed exploration pipeline for a single agent. The inputs to the system are the robot’s odometry and
+depth information. These are used to generate a 3D grid-based map of the environment, from which frontiers are extracted and clustered.
+The trails of frontiers are identified and used to select the adequate exploration mode for the agent. Then, the next target pose is chosen
+and a trajectory towards the goal pose is generated using [21].
+maneuvers.
+The cost associated to the viewpoint ξc := {xc, γc}
+covering cluster c ∈ C is defined as
+JE(ξc) := ωDJD(ξc) + ωV JV (ξc) + ωLJL(c),
+(1)
+where xc ∈ R3 is the position of the viewpoint and γc ∈
+R its orientation. The cost JD is the length of the path in
+M between the current robot’s position and xc, and it is
+calculated using the A* algorithm. Instead, JV is associated
+with the change in direction of travel, while JL to the label
+of cluster c. The terms ωD, ωV and ωL are constant weights.
+The cost JV (ξc) is calculated as
+JV (ξc) := acos(vT
+R
+xc − xR
+||xc − xR||2
+),
+(2)
+where vR ∈ R3 and xR ∈ R3 are the robot’s current velocity
+and position, respectively. This cost is directly associated
+with the angle between the velocity and the direction vector
+towards the candidate position xc covering cluster c. How-
+ever, it may happen that the cluster is labeled as trails, e.g. in
+the case of occlusions caused by thin obstacles, such as tree
+trunks. Since an Explorer should focus on actual frontiers,
+we assign a penalty to these clusters:
+JL(c) =
+�
+0
+if c is frontier
+ptrail
+if c is trail
+,
+(3)
+where ptrail is the constant penalty associated with trails.
+We then select as target pose the next best viewpoint
+ξc∗ := {xc∗, γc∗} covering the cluster c∗ with the lowest
+cost:
+ξc∗ := arg min
+ξc ∀c∈C JE(ξc).
+(4)
+In case the UAV is trapped in a dead-end, or if no new
+clusters are available in front of the robot, we ignore the cost
+associated with the robot’s velocity and we employ a greedy
+approach to select the new target pose. We find the best
+cluster in the vicinity of the robot at a maximum distance
+dmax using the same cost function as in Eq. 1, with JV set
+to zero:
+ξc∗ := arg min
+ξc ∀c∈C ωDJD(ξc) + ωLJL(c)
+s.t. ||xc − xR||2 ≤ dmax.
+(5)
+2) Collector: The objective of a Collector is to clear as
+many trails as possible, in order to avoid the need for a
+revisiting step in poorly explored regions of the map at the
+end of the mission. Since this task implies a detour from
+the main direction of exploration, the UAV’s speed needs
+to be maximized in order to go back to Explorer mode as
+soon as possible. To reach this objective, we sort the set of
+trails Ctrails ⊆ C by associating a cost JC to each cluster
+c ∈ Ctrails:
+JC(ξc) := ωP JP (ξc) + ωAJA(ξc),
+(6)
+where JP is associated with the time to reach xc and JA with
+the time to cover the angular change between the current
+robot’s yaw and the viewpoint’s orientation. Instead, ωP and
+ωA are constant weights.
+Given the path πR
+c
+from xR to xc and the maximum
+allowed velocity vmax, JP is computed as
+JP (ξc) := length(πR
+c )
+vmax
+.
+(7)
+Similarly, given the robot current’s heading γR, the view-
+point’s orientation γc and the maximum allowed yaw rate
+˙γmax, JA is computed as
+JA(ξc) := ∠(γR, γc)
+˙γmax
+,
+(8)
+where ∠(γR, γc) indicates the angular difference between γR
+and γc.
+The robot then selects the target trail in a step-by-step
+greedy procedure and behaves as a Collector until all close-
+by trails are cleared. Since the trails are surrounded by free
+known space, we double the maximum velocity compared to
+when in Explorer mode. Consequently, the UAV is able to
+maximize its velocity, leading to fast motions that allow it
+to quickly cover all the viewpoints associated with the trails.
+C. Extension to Multi-Robot
+1) System Architecture: The proposed exploration strat-
+egy can be easily extended to the multi-robot case. The
+extended pipeline is shown in Fig. 4. Assuming that the
+agents can localize in a common reference frame, they
+exchange local sub-maps, as well as odometry informa-
+tion, current target pose, and execution mode. Notice that
+here we propose a decentralized architecture. Centralized
+approaches generally assume infinite-range communication
+between agents and with a ground station. However, standard
+Wi-Fi communications have a limited range and, when
+navigating in cluttered environments such as forests, com-
+munication lines can be potentially obstructed and signal
+can be lost. In this work, we propose to use a more flexible
+point-to-point strategy, assuming that there exists a maximum
+range of communication between each pair of agents. Our
+
+Mapping
+Path Planning
+Mode Selector
+Agent 1
+Odometry 
+Depth
+Mapping
+Path Planning
+Mode Selector
+Agent 2, ..., N
+Odometry 
+Depth
+Mode
+Target Poses
+Local 
+Sub-maps
+Mode
+Fig. 4: Overview of the pipeline in the multi-robot setting, where
+the UAVs are tasked to collaboratively build a complete map of the
+area of interest. To fulfill the objective, agents exchange odometry
+information and local sub-maps, as well as current execution mode
+and target poses. We assume there exists a maximum communi-
+cation range between robots. If the distance between two agents
+exceeds this limit, communication is lost and information is not
+exchanged anymore, leading to poor coordination.
+design targets to keep the agents at a valid communication
+distance. If the distance between agents is higher than the
+maximum range, communication is lost and information is
+not exchanged anymore, leading to poor inter-agent coor-
+dination and sub-optimal decision-making. We also assume
+that, when communication is lost and successively regained,
+agents can synchronize their maps.
+2) Multi-robot Coordination: We encourage coordination
+between pairs of agents only if they perform compatible
+actions from an exploration point of view. This implies that,
+if two UAVs are operating in the same mode, they can
+collaborate in order to either explore more unknown spaces
+(Explorers) or clear trails (Collectors). On the contrary, coor-
+dination between an Explorer and a Collector should not be
+encouraged, since their tasks are intrinsically different. In this
+situation, we propose a leader-follower paradigm, where the
+Explorer (leader) explores unknown areas regardless of the
+position of the second agent, while the Collector (follower)
+follows the leader and clears the trails left unexplored. This
+design choice allows more flexibility during the execution
+of a mission, thanks to the possibility to change execution
+modes online. Notice that, if communication between all
+agents is lost, their exploration strategy falls back to the
+single-robot case.
+Our collaboration strategy within a robotic team is encour-
+aged as a soft constraint by modifying the cost functions in
+Eq. 1 and Eq. 6. If we consider robot i with position xi
+R,
+given the positions X i
+R := {xk
+R}N−1
+k=0 of the other N − 1
+robots in the team with k ̸= i, and their current target
+positions GR = {xk
+c∗}N−1
+k=0 , we modify the cost functions
+JE and JC for robot i as follows:
+JE(ξc, X i
+R, GR) := ωDJD(ξc) + ωV JV (ξc)+
+ωLJL(c) + ωF JF (ξc, X i
+R, GR)
+(9)
+and
+JC(ξc, X i
+R, GR) := ωP JP (ξc) + ωAJA(ξc)+
+ωF JF (ξc, X i
+R, GR),
+(10)
+where
+ωF
+is
+a
+constant
+weight.
+The
+cost
+function
+JF (ξc, X i
+R, GR) is defined as
+JF (ξc, X i
+R, GR) := Jatt
+F (X i
+R) + Jrep
+F
+(ξc, X i
+R, GR),
+(11)
+where Jatt
+F
+aims at keeping the agents i and k in com-
+munication range, while Jrep
+F
+ensures a minimum distance
+between them to avoid collisions. Moreover, Jrep
+F
+encourages
+map splitting, by assigning a high cost to candidate target
+positions close to other agents’ current goals.
+In more details, the function Jatt
+F
+for agent i is defined as
+follows:
+Jatt
+F (X i
+R) :=
+N−1
+�
+k=0,k̸=i
+I(i, k) · 1
+2kA||xi
+R − xk
+R||2,
+(12)
+where kA is constant factor and I(i, k) is an indicator
+function that embeds our coordination strategy:
+I(i, k) :=
+�
+0
+if i Explorer and k Collector
+1
+otherwise
+.
+(13)
+This indicates that agent i is attracted toward agent k only if
+they are in a compatible execution mode. On the contrary, in
+leader-follower mode, i.e. when robot i is an Explorer and
+robot k is a Collector, the leader ignores the follower, and
+Jatt
+F
+goes to zero. Notice that instead, if i is Collector and
+k an Explorer, I(i, k) = 1 and Jatt
+F
+̸= 0.
+Instead, Jrep
+F
+is computed as follows:
+Jrep
+F
+(ξc, X i
+R, GR) :=
+N−1
+�
+k=0,k̸=i
+Jrep
+ik (xi
+R, xk
+R) + Jrep
+ik (xi
+c, xk
+c∗),
+(14)
+where, given the Euclidean distance dAB := ||xA − xB||2,
+Jrep
+AB(xA, xB) :=
+�
+�
+�
+�
+�
+kR(dc − d0)2 dcd0
+d0−dc
+if dAB ≤ d0
+kR(dAB − d0)2
+if dc ≤ dAB ≤ d0
+0
+otherwise
+.
+(15)
+The parameter kR is a constant weight, while d0 represents
+the minimum distance between positions A and B to have
+a collision. The parameter dc represents the distance after
+which the positions should not approach any closer. This
+can be selected on the basis of the safety distance required
+between the UAVs. Notice that Jrep
+ik
+is not influenced by
+the roles of agents i and k, as safety and minimum distance
+requirements need to be always met.
+IV. EXPERIMENTS
+We evaluate the proposed exploration pipeline in both
+single- and multi-UAV setups in simulation. In particular,
+we benchmark our method on a series of realistic, randomly
+generated forests of increasing tree densities [22], as well as
+on a 3D reconstruction of a real forest [6].
+In the single-agent setup, we compare the proposed
+method against FUEL [5], while in the experiments with
+multiple robots we test against a centralized strategy based
+on map-splitting. In all tests, we use grid map resolutions of
+0.10 m or 0.15 m depending on the map size, while we set the
+dynamic limits to vmax = 1.5 m/s and ˙γmax = 0.9 rad/s for
+all planners. We simulate a depth camera with a fixed range
+
+Ours
+FUEL [5]
+REAL FOREST
+Completion Time [s]
+500.7 ± 14.8
+757.7 ± 47.9
+Travelled Distance [m]
+645.0 ± 20.0
+533.2 ± 11.0
+Velocity [m/s]
+1.3 ± 0.5
+0.7 ± 0.4
+SPARSE FOREST (0.05 TREES / m2)
+Completion Time [s]
+665.4 ± 32.7
+1114.1 ± 97.4
+Travelled Distance [m]
+860.9 ± 34.0
+758.1 ± 57.6
+Velocity [m/s]
+1.3 ± 0.6
+0.7 ± 0.5
+AVERAGE-DENSITY FOREST (0.10 TREES / m2)
+Completion Time [s]
+779.6 ± 110.9
+954.1 ± 28.8
+Travelled Distance [m]
+910.3 ± 63.7
+713.5 ± 33.1
+Velocity [m/s]
+1.2 ± 0.6
+0.7 ± 0.5
+DENSE FOREST (0.15 TREES / m2)
+Completion Time [s]
+613.2 ± 16.2
+1130.2 ± 28.8
+Travelled Distance [m]
+789.7 ± 16.8
+791.0 ± 37.7
+Velocity [m/s]
+1.2 ± 0.6
+0.7 ± 0.5
+VERY DENSE FOREST (0.20 TREES / m2)
+Completion Time [s]
+658.2 ± 57.2
+904.1 ± 109.5
+Travelled Distance [m]
+802.0 ± 52.7
+680.6 ± 49.9
+Velocity [m/s]
+1.2 ± 0.6
+0.7 ± 0.5
+TABLE I: Results of the experiments for a single agent. We report
+the average completion time over 3 runs, as well as the average
+travelled distance and velocity. The best performance is in bold.
+The relatively high standard deviation in the timings to complete
+the missions, in particular in the cases of tree densities of 0.10 and
+0.20 trees/m2, is caused by the complex nature of the map and the
+large number of occlusions.
+0
+200
+400
+600
+Time [s]
+0
+2000
+Explored Volume [m3]
+Ours
+FUEL
+Fig. 5: The average exploration rate during the experiments in
+the model REAL FOREST. The shaded region shows the standard
+deviation. The proposed method reaches complete coverage in less
+time than FUEL [5].
+of 4.5 m using the Vulkan-based renderer of [23] and the
+same physical simulator as in [5]. We report the planners’
+performance in terms of the time needed to complete the
+exploration of the scene, the total travelled distance, and the
+average velocity of the UAVs during each experiment.
+A. Single-robot Experiments
+In the single-robot experiments, the models of the syn-
+thetic forests are of size 50 m × 50 m × 2 m, while the 3D
+reconstruction of the REAL FOREST has dimensions 40 m ×
+40 m×2 m. The map resolution is set to 0.10 m. As reported
+in Table I, the proposed planner outperforms FUEL [5] across
+all scenes in terms of the time taken to reach full coverage
+(Fig. 5), thanks to our adaptive exploration policy that leads
+0
+200
+400
+600
+Time [s]
+0.5
+1.0
+1.5
+Velocity [m/s]
+Ours
+FUEL
+Fig. 6: The average UAV velocity during the experiments in the
+REAL FOREST. The shaded region indicates the standard deviation.
+The proposed strategy is able to fly the UAV at higher velocities
+than FUEL leading to improved mission efficiency (i.e. time to
+mission completion). Towards the end of the mission, the UAV
+following the FUEL strategy also speeds up, as by then the map
+is mostly explored, and smaller trails are cleared, resembling the
+Collector mode in the proposed strategy.
+Fig. 7: Top view of an exploration mission, where a team of two
+UAVs is tasked to map a randomly generated forest with tree
+density 0.05 trees/m2, illustrated with the black dots here. The
+initial positions of the UAVs are denoted as colored blobs and the
+final positions as enlarged drone models for clearer visualization.
+The map is represented as a 2D occupancy grid obtained by slicing
+the 3D model at 1.5 m from the ground.
+to consistently higher UAV velocity throughout each mission,
+as illustrated in Fig. 6. These results demonstrate the benefit
+of using the proposed adaptive exploration strategy over
+a fixed-mode method. However, notice that the proposed
+design leads to longer travelled distances, albeit guaranteeing
+that there are no small unexplored areas left. In fact, decision-
+making both in Explorer and Collector modes is done on a
+local-map level, and this may cause the UAV to fly longer
+routes, deviating from the shortest path. Nonetheless, in the
+proposed strategy we compensate for this shortcoming by
+encouraging decisions leading to higher UAV velocities, and
+thus shorter mission times.
+B. Multi-robot Experiments
+In the multi-robot experiments, the proposed planning
+strategy is tested in a variety of models with fixed, homoge-
+neous obstacle density, as well as in a randomly generated
+forest with tree density varying across different regions of
+the map.
+1) Maps with a fixed tree density: The results of the multi-
+robot collaborative exploration strategy of maps with fixed
+tree densities with two agents are shown in Table II, where
+
+0
++Ours
+Split Map (FUEL [5])
+SPARSE FOREST (0.05 TREES / m2)
+Completion Time [s]
+780.3 ± 32.2
+834.3 ± 64.3
+Travelled Distance [m]
+958.8 ± 9.2
+595.7 ± 51.8
+958.7 ± 60.4
+Velocity [m/s]
+1.2 ± 0.6
+0.7 ± 0.4
+1.3 ± 0.7
+AVERAGE-DENSITY FOREST (0.10 TREES / m2)
+Completion Time [s]
+838.5 ± 64.4
+848.1 ± 98.5
+Travelled Distance [m]
+915.3 ± 68.4
+616.1 ± 40.7
+906.2 ± 58.3
+Velocity [m/s]
+1.2 ± 0.5
+0.8 ± 0.4
+1.1 ± 0.6
+DENSE FOREST (0.15 TREES / m2)
+Completion Time [s]
+786.3 ± 40.7
+754.0 ± 23.9
+Travelled Distance [m]
+839.0 ± 26.1
+586.7 ± 14.2
+882.2 ± 32.0
+Velocity [m/s]
+1.2 ± 0.7
+0.8 ± 0.4
+1.3 ± 0.7
+VERY DENSE FOREST (0.20 TREES / m2)
+Completion Time [s]
+803.7 ± 52.8
+705.8 ± 73.2
+Travelled Distance [m]
+873.8 ± 47.0
+580.3 ± 73.3
+912.6 ± 42.7
+Velocity [m/s]
+1.2 ± 0.7
+0.7 ± 0.5
+1.2 ± 0.8
+TABLE II: Results in randomly generated forests with fixed tree
+densities explored with two UAVs, averaged over 3 runs. For the
+proposed strategy we report the average travelled distance and
+velocity per agent. The best performance is highlighted in bold.
+0
+200
+400
+600
+800
+Time [s]
+0
+2500
+Explored Volume [m3]
+Agent 1
+Agent 2
+Fig. 8: The average exploration rate per agent with the proposed
+approach using two UAVs during the experiments in a random
+forest with density 0.10 trees/m2. The shaded region shows the
+standard deviation. The explored volume is shown to be consistently
+balanced across the two agents.
+a maximum connection distance of 50 m for data exchange
+between agents is assumed (Fig. 7). Here, experiments are
+performed in forest models of size 100 m×50 m×2 m with
+a map resolution of 0.10 m. The proposed approach is com-
+pared against a centralized strategy we devised, employing
+the FUEL planner [5] and assigning the UAVs to explore
+maps of equal sizes (i.e. using map-splitting). Note that this
+strategy assumes homogeneous forest maps, and knowledge
+of the original map size, which renders this unsuitable for
+realistic deployment unlike the proposed approach; however,
+comparisons are presented for the sake of benchmarking.
+The proposed strategy reaches comparable results with
+respect to the centralized approach using FUEL. Similarly
+to single-robot exploration, the proposed strategy flies the
+UAVs at higher speeds, incurring longer travel distances.
+Moreover, our strategy enables automatic load balancing of
+the exploration mission, yielding similar exploration rates
+per agent as illustrated in Fig. 8. However, in denser for-
+est models, the performance of the proposed approach is
+seen to degrade, as the UAVs are tasked to fly within a
+connection range to each other, resulting in limited freedom
+of movement. This is exacerbated by the increased number
+of obstacles and occlusions in smaller maps, leading to
+lower exploration rates. Nevertheless, as aforementioned, the
+proposed approach is realistically deployable in contrast to
+the baseline strategy using FUEL.
+2) Map with non-homogeneous tree density: The results
+of the experiments in a map with non-homogeneous obstacle
+densities with a team of two agents are shown in Table III.
+We utilize a model with size 100 m × 200 m × 2 m, with
+different tree densities (0.2, 0.3 and 0.5 trees/m2) across
+distinct map regions. The occupancy grip map resolution is
+set to 0.15 m, with a maximum inter-agent communication
+range of 200 m. We perform the same analysis as in Sec.
+IV-B.1, assuming a realistic maximum flight time of 1500 s
+for the UAVs. As none of the tested planners is able to fully
+explore the environment within the allowed time, we report
+the total team coverage at fixed timestamps. The proposed
+approach consistently outperforms the solution based on
+map-splitting using FUEL [5] as the planning back-end. The
+gain in performance is related to the capacity of our strategy
+to fly the UAVs faster in regions with lower obstacle density
+and to explore cautiously more cluttered areas while clearing
+smaller frontier trails on the go. This leads to a more efficient
+exploration process, able to better exploit the capacity of
+UAVs to perform highly dynamic flights.
+Finally, we report the performance of our method when
+a team of three robots is deployed. As shown in Table
+IV, a larger team size yields higher total coverage. This
+demonstrates that this work presents an effective strategy for
+peer-to-peer sharing of the responsibility of exploration of
+a large forest area and that the extension to larger teams
+of multiple UAVs can be realized following this paradigm,
+pushing for scalable, decentralized multi-robot planning for
+exploration.
+V. CONCLUSION AND FUTURE WORK
+In this work, we propose an exploration pipeline for
+autonomous UAVs operating in complex, cluttered environ-
+ments, with a particular focus on forests. We choose this type
+of environment as one of the inherently most challenging for
+effective planning due to the increased number of obstacles
+and occlusions that they exhibit. The proposed strategy
+allows each UAV to switch between different exploratory
+behaviors, autonomously balancing cautious exploration of
+unknown space and more aggressive maneuvers, exploiting
+already mapped space within a mission. This leads to faster
+completion times due to higher-speed flights and, conse-
+quently, to more efficient and faster map coverage than the
+state of the art. Moreover, we show how the proposed method
+can be extended to three, and potentially more robots in a
+decentralized fashion, demonstrating automatic and effective
+load balancing across the participating agents. Following the
+
+Ours
+Split Map (FUEL [5])
+Travelled Distance [m]
+1481.9 ± 467.0
+735.2 ± 500.1
+1487.0 ± 440.8
+697.1 ± 365.6
+Velocity [m/s]
+1.3 ± 0.6
+0.7 ± 0.4
+1.2 ± 0.7
+0.6 ± 0.5
+Explored Volume [m3] - 300 s
+542.3 ± 23.3
+514.6 ± 313.0
+565.9 ± 31.4
+632.0 ± 184.8
+Explored Volume [m3] - 600 s
+1062.8 ± 81.3
+840.0 ± 580.7
+1011.6 ± 65.3
+728.3 ± 163.5
+Explored Volume [m3] - 900 s
+1400.0 ± 265.3
+1145.3 ± 814.7
+1302.6 ± 110.5
+798.8 ± 146.2
+Explored Volume [m3] - 1200 s
+1626.7 ± 431.2
+1330.1 ± 905.3
+1466.5 ± 185.4
+910.4 ± 83.9
+Explored Volume [m3] - 1500 s
+1816.0 ± 550.8
+1437.8 ± 971.8
+1637.0 ± 299.1
+1040.5 ± 124.3
+TABLE III: Results in a randomly generated forest with non-
+homogeneous tree densities when explored with two UAVs, av-
+eraged over 3 runs. We report the average travelled distance and
+velocity per agent at 1500 s, as well as the total explored volume
+by the team at different timestamps. The best performance is
+highlighted in bold.
+Timestamp
+Two Agents
+Three Agents
+300 s
+1108.2 ± 54.7 m3
+1208.0 ± 91.6 m3
+600 s
+2074.3 ± 144.8 m3
+2098.4 ± 143.8 m3
+900 s
+2702.6 ± 375.9 m3
+2748.3 ± 111.9 m3
+1200 s
+3093.2 ± 616.6 m3
+3223.5 ± 127.9 m3
+1500 s
+3453.0 ± 849.9 m3
+3621.8 ± 321.0 m3
+TABLE IV: Results in a randomly generated forest with varying
+tree densities across different regions of the model, averaged over
+3 runs. Here, the map is explored with teams composed of two and
+three UAVs. We report the total volume covered by the team at
+different timestamps. The best performance is highlighted in bold.
+push for automating higher-level decision-making in robotic
+missions, this work constitutes a key milestone towards
+effective exploration planning for robotic teams.
+The natural next step for this work is to address the inte-
+gration and deployment of the proposed pipeline onboard real
+platforms, while further investigations will push advancing
+coordination strategies in larger multi-robot teams.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf,len=646
+page_content='Towards Multi-robot Exploration:  A Decentralized Strategy for UAV Forest Exploration  Luca Bartolomei, Lucas Teixeira and Margarita Chli  Vision For Robotics Lab, ETH Z¨urich, Switzerland  Abstract— Efficient exploration strategies are vital in tasks  such as search-and-rescue missions and disaster surveying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Unmanned Aerial Vehicles (UAVs) have become particularly  popular in such applications, promising to cover large areas at  high speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Moreover, with the increasing maturity of onboard  UAV perception, research focus has been shifting toward higher-  level reasoning for single- and multi-robot missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However,  autonomous navigation and exploration of previously unknown  large spaces still constitutes an open challenge, especially when  the environment is cluttered and exhibits large and frequent  occlusions due to high obstacle density, as is the case of forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Moreover, the problem of long-distance wireless communication  in such scenes can become a limiting factor, especially when  automating the navigation of a UAV swarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In this spirit,  this work proposes an exploration strategy that enables UAVs,  both individually and in small swarms, to quickly explore  complex scenes in a decentralized fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' By providing the  decision-making capabilities to each UAV to switch between  different execution modes, the proposed strategy strikes a  great balance between cautious exploration of yet completely  unknown regions and more aggressive exploration of smaller  areas of unknown space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This results in full coverage of forest  areas of variable density, consistently faster than the state of the  art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Demonstrating successful deployment with a single UAV as  well as a swarm of up to three UAVs, this work sets out the basic  principles for multi-root exploration of cluttered scenes, with  up to 65% speed up in the single UAV case and 40% increase in  explored area for the same mission time in multi-UAV setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' INTRODUCTION  The growing interest in Unmanned Aerial Vehicles (UAVs)  has led to their extensive deployment in tasks such as  inspection and search-and-rescue missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In these appli-  cations, the capacity of the robot to quickly explore and  map unknown environments autonomously is fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The literature on this topic is extensive, and many different  approaches have been proposed throughout the years [1]–  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, one of the biggest challenges in the explo-  ration of unknown environments is the capacity to achieve  a good trade-off between the competing goals of shorter  exploration times of an area of interest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' pushing for  high-speed navigation) and safety, which requires caps on  the velocity of each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In fact, navigating in the vicinity  of the boundaries between known and unknown space is  challenging, as the robot can get stuck in dead ends, or  needs to perform complex dodging maneuvers to avoid  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Consequently, to maintain the safety of both the  platform and its surroundings, most path planners generate  conservative start-and-stop motions, not fully exploiting the  This work was supported by NCCR Robotics, the Amazon Research  Awards, and the HILTI group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 1: 3D-view of the proposed system guiding safe and successful  exploration of a UAV in a digital model of a real-world forest  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The planner is able to avoid collisions between the UAV and  the obstacles, clearing frontiers on-the-go by balancing cautious  navigation with aggressive exploitation of known, free space, in a  bid to maximize the efficiency of the exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  capacity of a UAV to fly at high speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This effect is  exacerbated when the environment to explore is particularly  cluttered, as is the case in forests, leading to inefficient and  incomplete coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' By design, these methods generally  drive the exploration process by biasing exploration towards  large areas of unexplored space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' While this strategy could be  advantageous in open and wide spaces, it can be detrimental  when exploring cluttered scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In fact, the main pitfall of  such strategies is that, while the exploration process attempts  to cover as much unknown space as possible, when this  is deployed in environments with many obstacles, thinner  trails of unknown space are left unexplored (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' due to  occlusions), imposing the need for a second sweep of the  environment over mostly explored areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Aiming to mitigate these issues, pushing for faster cov-  erage of the areas of interest, multi-robot extensions for  exploration have also been proposed [7]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However,  these focus on the problem of coordination at the system-  level and, and while they can perform better from a global  planning point of view, they suffer from the same limitations  as the single-UAV case in obstacle-dense environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Motivated by these challenges, in this work we propose an  exploration strategy for autonomous UAV robots aiming to  explore forests of increasing tree density, as they pose some  of the most difficult challenges for exploration planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Our  objective is to exploit the platform’s dynamics to the fullest  despite the high density of obstacles, in order to achieve the  complete coverage of the environment efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' To this end,  the proposed strategy enables switching between two differ-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='08537v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='RO]  20 Jan 2023  ent behaviors for each robot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' namely, cautious exploration  of unknown space and more aggressive maneuvers when  navigating in already explored areas to clear smaller portions  of unknown space caused by occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We evaluate the  proposed approach in a series of challenging experiments in  simulation, both in randomly generated forests and in a 3D  reconstruction of a real forest (Fig 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Benchmarking against  the state of the art reveals superior efficiency for the proposed  approach achieving higher overall UAV speeds and lower  exploration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Finally, aiming to set out the scaffolding  toward decentralized multi-robot exploration planning, we  show how the proposed strategy can accommodate more than  one UAV in the exploration mission, and we demonstrate that  our method performs comparably to or better than a map-  splitting centralized approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  In summary, the contributions of this work are as follows:  the design of an exploration strategy, able to strike  an effective balance between cautious exploration and  aggressive exploitation of the explored map,  the extension of the single-robot design to a multi-robot  decentralized approach, and  extensive evaluations in simulation, demonstrating bet-  ter performance than the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' RELATED WORKS  Autonomous exploration of unknown environments with  UAVs has been an active field of research over the past  few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The most popular approach to exploring an  area of interest is to use frontiers, defined as the boundary  between known and unknown space [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' These can be  utilized to identify potentially informative spatial regions  in order to drive the exploration process efficiently until  no new frontiers are found and the exploration process can  be considered complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' There are different criteria used to  decide which frontier to explore next, such as their proximity  to the current field of view, following a greedy selection  strategy, or having global planning dictate the selection [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  However, while frontier-based approaches have been proven  to yield satisfactory performances, especially in terms of  coverage [2], [3], they generally lead to inefficient motions  and sub-optimal action selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This is mostly caused by  the sensing modalities used to generate the map of the  environment to explore, as the most common sensors, such as  RGB-D and stereo cameras, have a limited detection range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Consequently, UAVs need to fly cautiously to ensure safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Cieslewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' [4] tackle this limitation, by proposing  an exploration strategy that generates velocity commands  based on newly detected frontiers, in a bid to maximize  the UAV’s speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This method is shown to outperform  classical methods [11], but focuses only on local frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Instead, FUEL [5] proposes a hierarchical planner which  generates efficient global paths, while encouraging safe and  agile local maneuvers for high-speed exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' FUEL’s  strategy performs better than [4] and [11] in scenes with  low obstacle densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, it is more computationally  demanding, as it needs to maintain a list of active frontiers,  as well as to compute accurate distances between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This  additional bookkeeping becomes prohibitive and impractical  in more cluttered and complex environments such as forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  In fact, in this type of scenery, the number of frontiers  quickly increases due to occlusions caused by tree trunks,  branches, and shrubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Another line of research focuses instead on sampling-  based path planning to generate viewpoints to explore the  space [13], [14], by guiding the robot along possible trails  of sampled configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The best path is generally found  using a greedy approach [14], bringing to complete explo-  ration or accurate surface reconstruction [2] depending on  the information gain formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Nonetheless, these sampled  routes may generate trajectories that deviate from the shortest  paths, without taking into consideration the robot’s dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Consequently, this causes the UAV to navigate in zigzag  patterns, leading to inefficient, slow motions and conservative  maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  To tackle the limitations of frontier- and sampling-based  methods, also hybrid approaches have been proposed [1],  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Such methods compute global paths towards the most  informative frontiers while generating local trajectories using  sampling-based planners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, they do not exploit the  full dynamics of the platform and generate sub-optimal  routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  To boost the efficiency in exploration, various multi-robot  cooperative frontier-based methods have also been proposed  in the literature, both in centralized [16] and decentralized  formats [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In this spirit, the work in [18] greedily assigns  view configurations, while [19] distributes the workload  between agents using a Voronoi-based partitioning of the  area to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Nevertheless, these solutions suffer from  the same limitations as in the single-robot case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Instead, the  approach in [20] is able to generate efficient trajectories for  3D reconstruction, tackling the multi-robot coordination with  a centralized architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, this method requires  a prior overhead flight over the area of interest, making  it unsuitable for the exploration of forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The approach  proposed in [7] puts more focus on the problem of navigating  forests, but the emphasis is more on state estimation rather  than on path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Motivated by these limitations, in this work, we propose  a strategy that allows a robot to explore complex forest-  like environments while flying at high speeds, thanks to  the freedom and flexibility that our planner provides to  each UAV to switch between different navigation modes  online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' While slower, cautious exploration is performed  using a frontier-based approach, we efficiently clear trails of  unexplored space caused by occlusions by employing a more  aggressive local exploration strategy, boosting the efficiency  of the mission and pushing the overall time to cover a given  area of interest down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Moreover, we demonstrate that the  proposed pipeline can also be extended to the multi-robot  setting in a decentralized fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' METHODOLOGY  The overall problem considered in this work is to ex-  plore unknown cluttered environments, such as forests, in  (a) Time-instant 1  (b) Time-instant 2  (c) Time-instant 3  (d) Time-instant 4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 2: A schematic example demonstrating the problem with greedy frontier-based exploration, at progressive time-instants, generating  islands of unknown space surrounded by free regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The field of view of the robot is depicted as a light-gray shaded area delimited by  black solid lines, while the obstacles and the unexplored space are in black and dark gray, respectively (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The robot navigates towards  the most informative frontiers (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' however, due to the limited sensor range, the space occluded by the obstacles is not cleared (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Consequently, since the exploration process is biased towards larger, more informative frontiers, the na¨ıve planner flies the UAV robot  ignoring the smaller portion of unexplored space (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  the minimum time possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We assume that the robot is  equipped with a front-looking depth camera with a limited  sensing range and that the robot’s odometry information is  available at a constant rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, forest-like scenes are  characterized by a high number of obstacles in a variety  of dimensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' trunks, leaves, and branches) that make  standard frontier-based exploration approaches inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  In fact, during the exploration process, many islands of  unknown space are usually left behind, as illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 2, necessitating subsequent passes of exploration on a  nearly completely explored map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' To tackle this limitation,  we propose an exploration pipeline that can change the ex-  ploratory behavior of the robot depending on the frontiers in  its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In particular, we propose to define two different  modes of operation for the robot, namely the Explorer and  the Collector modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In the Explorer state, the robot is driven  by frontiers and it is tasked to explore large unknown areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Consequently, it predominantly operates on the most external  boundaries between known and unknown areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Conversely,  the robot in the Collector mode clears small islands of  unknown space generated by occlusions, that are left behind  during the exploratory phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The objective of a Collector is  to clear these portions of space on the go, avoiding the need  for subsequent revisits of the map, at the expense of short  local detours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, notice that these can be performed at  high speed, since, when in Collector mode, the robot operates  in mostly explored areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' By allowing a robot to switch  between these two different modes and by finding the right  trade-off between map exploration and exploitation, we can  quickly reach full coverage of large cluttered environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  In the following, we first give an overview of the proposed  system and our exploration strategies, and then we illustrate  how it can be extended to multiple robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' System Overview  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 3, the pipeline is composed of three main  components: a mapping system, a mode selector, and a path  planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Given input depth and odometry information, a voxel  grid map M of the environment is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' At every  update, frontiers are extracted from M and clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' For  each cluster, we adopt the sampling strategy from [5] to  generate viewpoints covering the frontiers, and we use them  as possible target poses during the exploration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Moreover, each cluster undergoes a binary classification  step, where unconnected islands of frontiers, or trails, are  identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This is necessary to identify those regions that are  likely to require an additional revisiting phase towards the  end of the mission if a traditional frontier-based exploration  method is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Here, a cluster is considered a trail if its  convex hull is surrounded by free space, or when it has only  another neighboring cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This implies that most clusters  at the corners of the area to be explored are classified as  trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We motivate this design choice by arguing that corners  are generally problematic for exploration due to their low  informative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In fact, they are rarely covered in a first  sweep of the map, implying the need for a revisiting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The labeled clusters are then utilized by the Mode Selector to  choose the best exploration strategy for the robot, deciding  whether it has to persevere in its current mode, or transit  to Explorer or Collector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The mode assignment is regulated  according to the frontiers in the vicinity of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Given  that our objective is to clear trails locally to avoid large  detours on the map, we assign the role of Collector if a  minimum number of trails is close to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Instead, we  adopt a more exploratory strategy once all smaller islands of  unknown space are cleared, or when the trails are far away  from the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Once a strategy is selected, the viewpoint of  the most promising cluster is selected as the new target pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  This is fed to a path planner [21] that generates the trajectory  flying the UAV toward its destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We now describe the  different exploration modes in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Exploration Strategies  1) Explorer: Driven by frontiers, the objective of an  Explorer is to cover large areas of previously unknown space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Similarly to [4], we process the incoming clusters of frontiers  C from the most recent map update and extract the one with  the lowest cost JE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Notice that these clusters are mostly  aligned with the direction of the UAV’s motion, implying  that, if one of these is selected as the target, the robot  avoids abrupt changes in the flight direction or aggressive  Mapping  3D Map  Frontiers Classification  Frontiers Extraction  Odometry  Depth  Target Pose Selector  Trajectory Generation  Path Planning  Mode Selector  Low-level  Controller  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 3: Schematic overview of the proposed exploration pipeline for a single agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The inputs to the system are the robot’s odometry and  depth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' These are used to generate a 3D grid-based map of the environment, from which frontiers are extracted and clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The trails of frontiers are identified and used to select the adequate exploration mode for the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Then, the next target pose is chosen  and a trajectory towards the goal pose is generated using [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The cost associated to the viewpoint ξc := {xc, γc}  covering cluster c ∈ C is defined as  JE(ξc) := ωDJD(ξc) + ωV JV (ξc) + ωLJL(c),  (1)  where xc ∈ R3 is the position of the viewpoint and γc ∈  R its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The cost JD is the length of the path in  M between the current robot’s position and xc, and it is  calculated using the A* algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Instead, JV is associated  with the change in direction of travel, while JL to the label  of cluster c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The terms ωD, ωV and ωL are constant weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The cost JV (ξc) is calculated as  JV (ξc) := acos(vT  R  xc − xR  ||xc − xR||2  ),  (2)  where vR ∈ R3 and xR ∈ R3 are the robot’s current velocity  and position, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This cost is directly associated  with the angle between the velocity and the direction vector  towards the candidate position xc covering cluster c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' How-  ever, it may happen that the cluster is labeled as trails, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' in  the case of occlusions caused by thin obstacles, such as tree  trunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Since an Explorer should focus on actual frontiers,  we assign a penalty to these clusters:  JL(c) =  �  0  if c is frontier  ptrail  if c is trail  ,  (3)  where ptrail is the constant penalty associated with trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  We then select as target pose the next best viewpoint  ξc∗ := {xc∗, γc∗} covering the cluster c∗ with the lowest  cost:  ξc∗ := arg min  ξc ∀c∈C JE(ξc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  (4)  In case the UAV is trapped in a dead-end, or if no new  clusters are available in front of the robot, we ignore the cost  associated with the robot’s velocity and we employ a greedy  approach to select the new target pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We find the best  cluster in the vicinity of the robot at a maximum distance  dmax using the same cost function as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 1, with JV set  to zero:  ξc∗ := arg min  ξc ∀c∈C ωDJD(ξc) + ωLJL(c)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' ||xc − xR||2 ≤ dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  (5)  2) Collector: The objective of a Collector is to clear as  many trails as possible, in order to avoid the need for a  revisiting step in poorly explored regions of the map at the  end of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Since this task implies a detour from  the main direction of exploration, the UAV’s speed needs  to be maximized in order to go back to Explorer mode as  soon as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' To reach this objective, we sort the set of  trails Ctrails ⊆ C by associating a cost JC to each cluster  c ∈ Ctrails:  JC(ξc) := ωP JP (ξc) + ωAJA(ξc),  (6)  where JP is associated with the time to reach xc and JA with  the time to cover the angular change between the current  robot’s yaw and the viewpoint’s orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Instead, ωP and  ωA are constant weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Given the path πR  c  from xR to xc and the maximum  allowed velocity vmax, JP is computed as  JP (ξc) := length(πR  c )  vmax  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  (7)  Similarly, given the robot current’s heading γR, the view-  point’s orientation γc and the maximum allowed yaw rate  ˙γmax, JA is computed as  JA(ξc) := ∠(γR, γc)  ˙γmax  ,  (8)  where ∠(γR, γc) indicates the angular difference between γR  and γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The robot then selects the target trail in a step-by-step  greedy procedure and behaves as a Collector until all close-  by trails are cleared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Since the trails are surrounded by free  known space, we double the maximum velocity compared to  when in Explorer mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Consequently, the UAV is able to  maximize its velocity, leading to fast motions that allow it  to quickly cover all the viewpoints associated with the trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Extension to Multi-Robot  1) System Architecture: The proposed exploration strat-  egy can be easily extended to the multi-robot case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The  extended pipeline is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Assuming that the  agents can localize in a common reference frame, they  exchange local sub-maps, as well as odometry informa-  tion, current target pose, and execution mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Notice that  here we propose a decentralized architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Centralized  approaches generally assume infinite-range communication  between agents and with a ground station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, standard  Wi-Fi communications have a limited range and, when  navigating in cluttered environments such as forests, com-  munication lines can be potentially obstructed and signal  can be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In this work, we propose to use a more flexible  point-to-point strategy, assuming that there exists a maximum  range of communication between each pair of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Our  Mapping  Path Planning  Mode Selector  Agent 1  Odometry  Depth  Mapping  Path Planning  Mode Selector  Agent 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=', N  Odometry  Depth  Mode  Target Poses  Local  Sub-maps  Mode  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 4: Overview of the pipeline in the multi-robot setting, where  the UAVs are tasked to collaboratively build a complete map of the  area of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' To fulfill the objective, agents exchange odometry  information and local sub-maps, as well as current execution mode  and target poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We assume there exists a maximum communi-  cation range between robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' If the distance between two agents  exceeds this limit, communication is lost and information is not  exchanged anymore, leading to poor coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  design targets to keep the agents at a valid communication  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' If the distance between agents is higher than the  maximum range, communication is lost and information is  not exchanged anymore, leading to poor inter-agent coor-  dination and sub-optimal decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We also assume  that, when communication is lost and successively regained,  agents can synchronize their maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  2) Multi-robot Coordination: We encourage coordination  between pairs of agents only if they perform compatible  actions from an exploration point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This implies that,  if two UAVs are operating in the same mode, they can  collaborate in order to either explore more unknown spaces  (Explorers) or clear trails (Collectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' On the contrary, coor-  dination between an Explorer and a Collector should not be  encouraged, since their tasks are intrinsically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In this  situation, we propose a leader-follower paradigm, where the  Explorer (leader) explores unknown areas regardless of the  position of the second agent, while the Collector (follower)  follows the leader and clears the trails left unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This  design choice allows more flexibility during the execution  of a mission, thanks to the possibility to change execution  modes online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Notice that, if communication between all  agents is lost, their exploration strategy falls back to the  single-robot case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Our collaboration strategy within a robotic team is encour-  aged as a soft constraint by modifying the cost functions in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' If we consider robot i with position xi  R,  given the positions X i  R := {xk  R}N−1  k=0 of the other N − 1  robots in the team with k ̸= i, and their current target  positions GR = {xk  c∗}N−1  k=0 , we modify the cost functions  JE and JC for robot i as follows:  JE(ξc, X i  R, GR) := ωDJD(ξc) + ωV JV (ξc)+  ωLJL(c) + ωF JF (ξc, X i  R, GR)  (9)  and  JC(ξc, X i  R, GR) := ωP JP (ξc) + ωAJA(ξc)+  ωF JF (ξc, X i  R, GR),  (10)  where  ωF  is  a  constant  weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The  cost  function  JF (ξc, X i  R, GR) is defined as  JF (ξc, X i  R, GR) := Jatt  F (X i  R) + Jrep  F  (ξc, X i  R, GR),  (11)  where Jatt  F  aims at keeping the agents i and k in com-  munication range, while Jrep  F  ensures a minimum distance  between them to avoid collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Moreover, Jrep  F  encourages  map splitting, by assigning a high cost to candidate target  positions close to other agents’ current goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  In more details, the function Jatt  F  for agent i is defined as  follows:  Jatt  F (X i  R) :=  N−1  �  k=0,k̸=i  I(i, k) · 1  2kA||xi  R − xk  R||2,  (12)  where kA is constant factor and I(i, k) is an indicator  function that embeds our coordination strategy:  I(i, k) :=  �  0  if i Explorer and k Collector  1  otherwise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  (13)  This indicates that agent i is attracted toward agent k only if  they are in a compatible execution mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' On the contrary, in  leader-follower mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' when robot i is an Explorer and  robot k is a Collector, the leader ignores the follower, and  Jatt  F  goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Notice that instead, if i is Collector and  k an Explorer, I(i, k) = 1 and Jatt  F  ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Instead, Jrep  F  is computed as follows:  Jrep  F  (ξc, X i  R, GR) :=  N−1  �  k=0,k̸=i  Jrep  ik (xi  R, xk  R) + Jrep  ik (xi  c, xk  c∗),  (14)  where, given the Euclidean distance dAB := ||xA − xB||2,  Jrep  AB(xA, xB) :=  �  �  �  �  �  kR(dc − d0)2 dcd0  d0−dc  if dAB ≤ d0  kR(dAB − d0)2  if dc ≤ dAB ≤ d0  0  otherwise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  (15)  The parameter kR is a constant weight, while d0 represents  the minimum distance between positions A and B to have  a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The parameter dc represents the distance after  which the positions should not approach any closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This  can be selected on the basis of the safety distance required  between the UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Notice that Jrep  ik  is not influenced by  the roles of agents i and k, as safety and minimum distance  requirements need to be always met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' EXPERIMENTS  We evaluate the proposed exploration pipeline in both  single- and multi-UAV setups in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In particular,  we benchmark our method on a series of realistic, randomly  generated forests of increasing tree densities [22], as well as  on a 3D reconstruction of a real forest [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  In the single-agent setup, we compare the proposed  method against FUEL [5], while in the experiments with  multiple robots we test against a centralized strategy based  on map-splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In all tests, we use grid map resolutions of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='10 m or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='15 m depending on the map size, while we set the  dynamic limits to vmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 m/s and ˙γmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9 rad/s for  all planners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We simulate a depth camera with a fixed range  Ours  FUEL [5]  REAL FOREST  Completion Time [s]  500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9  Travelled Distance [m]  645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0  533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  SPARSE FOREST (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='05 TREES / m2)  Completion Time [s]  665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4 ± 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  1114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  Travelled Distance [m]  860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9 ± 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0  758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  AVERAGE-DENSITY FOREST (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='10 TREES / m2)  Completion Time [s]  779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 ± 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9  954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  Travelled Distance [m]  910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 ± 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  DENSE FOREST (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='15 TREES / m2)  Completion Time [s]  613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2  1130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  Travelled Distance [m]  789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  VERY DENSE FOREST (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='20 TREES / m2)  Completion Time [s]  658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2  904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  Travelled Distance [m]  802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 ± 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  TABLE I: Results of the experiments for a single agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We report  the average completion time over 3 runs, as well as the average  travelled distance and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The best performance is in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The relatively high standard deviation in the timings to complete  the missions, in particular in the cases of tree densities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='10 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='20 trees/m2, is caused by the complex nature of the map and the  large number of occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  0  200  400  600  Time [s]  0  2000  Explored Volume [m3]  Ours  FUEL  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 5: The average exploration rate during the experiments in  the model REAL FOREST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The shaded region shows the standard  deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The proposed method reaches complete coverage in less  time than FUEL [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 m using the Vulkan-based renderer of [23] and the  same physical simulator as in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We report the planners’  performance in terms of the time needed to complete the  exploration of the scene, the total travelled distance, and the  average velocity of the UAVs during each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Single-robot Experiments  In the single-robot experiments, the models of the syn-  thetic forests are of size 50 m × 50 m × 2 m, while the 3D  reconstruction of the REAL FOREST has dimensions 40 m ×  40 m×2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The map resolution is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' As reported  in Table I, the proposed planner outperforms FUEL [5] across  all scenes in terms of the time taken to reach full coverage  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 5), thanks to our adaptive exploration policy that leads  0  200  400  600  Time [s]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  Velocity [m/s]  Ours  FUEL  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 6: The average UAV velocity during the experiments in the  REAL FOREST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The shaded region indicates the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The proposed strategy is able to fly the UAV at higher velocities  than FUEL leading to improved mission efficiency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' time to  mission completion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Towards the end of the mission, the UAV  following the FUEL strategy also speeds up, as by then the map  is mostly explored, and smaller trails are cleared, resembling the  Collector mode in the proposed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 7: Top view of an exploration mission, where a team of two  UAVs is tasked to map a randomly generated forest with tree  density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='05 trees/m2, illustrated with the black dots here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The  initial positions of the UAVs are denoted as colored blobs and the  final positions as enlarged drone models for clearer visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The map is represented as a 2D occupancy grid obtained by slicing  the 3D model at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 m from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  to consistently higher UAV velocity throughout each mission,  as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' These results demonstrate the benefit  of using the proposed adaptive exploration strategy over  a fixed-mode method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, notice that the proposed  design leads to longer travelled distances, albeit guaranteeing  that there are no small unexplored areas left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' In fact, decision-  making both in Explorer and Collector modes is done on a  local-map level, and this may cause the UAV to fly longer  routes, deviating from the shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Nonetheless, in the  proposed strategy we compensate for this shortcoming by  encouraging decisions leading to higher UAV velocities, and  thus shorter mission times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Multi-robot Experiments  In the multi-robot experiments, the proposed planning  strategy is tested in a variety of models with fixed, homoge-  neous obstacle density, as well as in a randomly generated  forest with tree density varying across different regions of  the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  1) Maps with a fixed tree density: The results of the multi-  robot collaborative exploration strategy of maps with fixed  tree densities with two agents are shown in Table II, where  0  +Ours  Split Map (FUEL [5])  SPARSE FOREST (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='05 TREES / m2)  Completion Time [s]  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2  834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  Travelled Distance [m]  958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2  595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  AVERAGE-DENSITY FOREST (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='10 TREES / m2)  Completion Time [s]  838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 ± 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  Travelled Distance [m]  915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  DENSE FOREST (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='15 TREES / m2)  Completion Time [s]  786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9  Travelled Distance [m]  839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1  586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2  882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  VERY DENSE FOREST (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='20 TREES / m2)  Completion Time [s]  803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2  Travelled Distance [m]  873.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0  580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 ± 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  TABLE II: Results in randomly generated forests with fixed tree  densities explored with two UAVs, averaged over 3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' For the  proposed strategy we report the average travelled distance and  velocity per agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The best performance is highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  0  200  400  600  800  Time [s]  0  2500  Explored Volume [m3]  Agent 1  Agent 2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 8: The average exploration rate per agent with the proposed  approach using two UAVs during the experiments in a random  forest with density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='10 trees/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The shaded region shows the  standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The explored volume is shown to be consistently  balanced across the two agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  a maximum connection distance of 50 m for data exchange  between agents is assumed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Here, experiments are  performed in forest models of size 100 m×50 m×2 m with  a map resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The proposed approach is com-  pared against a centralized strategy we devised, employing  the FUEL planner [5] and assigning the UAVs to explore  maps of equal sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' using map-splitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Note that this  strategy assumes homogeneous forest maps, and knowledge  of the original map size, which renders this unsuitable for  realistic deployment unlike the proposed approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' however,  comparisons are presented for the sake of benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The proposed strategy reaches comparable results with  respect to the centralized approach using FUEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Similarly  to single-robot exploration, the proposed strategy flies the  UAVs at higher speeds, incurring longer travel distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Moreover, our strategy enables automatic load balancing of  the exploration mission, yielding similar exploration rates  per agent as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' However, in denser for-  est models, the performance of the proposed approach is  seen to degrade, as the UAVs are tasked to fly within a  connection range to each other, resulting in limited freedom  of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This is exacerbated by the increased number  of obstacles and occlusions in smaller maps, leading to  lower exploration rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Nevertheless, as aforementioned, the  proposed approach is realistically deployable in contrast to  the baseline strategy using FUEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  2) Map with non-homogeneous tree density: The results  of the experiments in a map with non-homogeneous obstacle  densities with a team of two agents are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  We utilize a model with size 100 m × 200 m × 2 m, with  different tree densities (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 trees/m2) across  distinct map regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The occupancy grip map resolution is  set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='15 m, with a maximum inter-agent communication  range of 200 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We perform the same analysis as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1, assuming a realistic maximum flight time of 1500 s  for the UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' As none of the tested planners is able to fully  explore the environment within the allowed time, we report  the total team coverage at fixed timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The proposed  approach consistently outperforms the solution based on  map-splitting using FUEL [5] as the planning back-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The  gain in performance is related to the capacity of our strategy  to fly the UAVs faster in regions with lower obstacle density  and to explore cautiously more cluttered areas while clearing  smaller frontier trails on the go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This leads to a more efficient  exploration process, able to better exploit the capacity of  UAVs to perform highly dynamic flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Finally, we report the performance of our method when  a team of three robots is deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' As shown in Table  IV, a larger team size yields higher total coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This  demonstrates that this work presents an effective strategy for  peer-to-peer sharing of the responsibility of exploration of  a large forest area and that the extension to larger teams  of multiple UAVs can be realized following this paradigm,  pushing for scalable, decentralized multi-robot planning for  exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  In this work, we propose an exploration pipeline for  autonomous UAVs operating in complex, cluttered environ-  ments, with a particular focus on forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We choose this type  of environment as one of the inherently most challenging for  effective planning due to the increased number of obstacles  and occlusions that they exhibit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The proposed strategy  allows each UAV to switch between different exploratory  behaviors, autonomously balancing cautious exploration of  unknown space and more aggressive maneuvers, exploiting  already mapped space within a mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' This leads to faster  completion times due to higher-speed flights and, conse-  quently, to more efficient and faster map coverage than the  state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Moreover, we show how the proposed method  can be extended to three, and potentially more robots in a  decentralized fashion, demonstrating automatic and effective  load balancing across the participating agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Following the  Ours  Split Map (FUEL [5])  Travelled Distance [m]  1481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9 ± 467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0  735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1  1487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  Velocity [m/s]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  Explored Volume [m3] - 300 s  542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 ± 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0  565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9 ± 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  Explored Volume [m3] - 600 s  1062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 ± 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  Explored Volume [m3] - 900 s  1400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  1145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7  1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 ± 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5  798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2  Explored Volume [m3] - 1200 s  1626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 ± 431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2  1330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1 ± 905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  1466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 ± 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4  910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4 ± 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9  Explored Volume [m3] - 1500 s  1816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  1437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8  1637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='1  1040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 ± 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3  TABLE III: Results in a randomly generated forest with non-  homogeneous tree densities when explored with two UAVs, av-  eraged over 3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We report the average travelled distance and  velocity per agent at 1500 s, as well as the total explored volume  by the team at different timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The best performance is  highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  Timestamp  Two Agents  Three Agents  300 s  1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='7 m3  1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 m3  600 s  2074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 m3  2098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='4 ± 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 m3  900 s  2702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 ± 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9 m3  2748.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='3 ± 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9 m3  1200 s  3093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='2 ± 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='6 m3  3223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='5 ± 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9 m3  1500 s  3453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 ± 849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='9 m3  3621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='8 ± 321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='0 m3  TABLE IV: Results in a randomly generated forest with varying  tree densities across different regions of the model, averaged over  3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' Here, the map is explored with teams composed of two and  three UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' We report the total volume covered by the team at  different timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content=' The best performance is highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  push for automating higher-level decision-making in robotic  missions, this work constitutes a key milestone towards  effective exploration planning for robotic teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
+page_content='  The natural next step for this work is to address the inte-  gration and deployment of the proposed pipeline onboard real  platforms, while further investigations will push advancing  coordination strategies in larger multi-robot teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQfYB2G/content/2301.08537v1.pdf'}
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diff --git a/ctAyT4oBgHgl3EQfjPgK/content/tmp_files/2301.00409v1.pdf.txt b/ctAyT4oBgHgl3EQfjPgK/content/tmp_files/2301.00409v1.pdf.txt
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@@ -0,0 +1,694 @@
+Diffusion Model based Semi-supervised Learning
+on Brain Hemorrhage Images for Efficient
+Midline Shift Quantification
+Shizhan Gong1, Cheng Chen1, Yuqi Gong1, Nga Yan Chan2, Wenao Ma1,
+Calvin Hoi-Kwan Mak3, Jill Abrigo2, Qi Dou1
+1Department of Computer Science and Engineering,
+The Chinese University of Hong Kong, Hong Kong, China
+2Department of Imaging and Interventional Radiology,
+The Chinese University of Hong Kong, Hong Kong, China
+3Queen Elizabeth Hospital, Hong Kong, China
+Abstract. Brain midline shift (MLS) is one of the most critical factors
+to be considered for clinical diagnosis and treatment decision-making for
+intracranial hemorrhage. Existing computational methods on MLS quan-
+tification not only require intensive labeling in millimeter-level measure-
+ment but also suffer from poor performance due to their dependence on
+specific landmarks or simplified anatomical assumptions. In this paper,
+we propose a novel semi-supervised framework to accurately measure the
+scale of MLS from head CT scans. We formulate the MLS measurement
+task as a deformation estimation problem and solve it using a few MLS
+slices with sparse labels. Meanwhile, with the help of diffusion models,
+we are able to use a great number of unlabeled MLS data and 2793 non-
+MLS cases for representation learning and regularization. The extracted
+representation reflects how the image is different from a non-MLS image
+and regularization serves an important role in the sparse-to-dense refine-
+ment of the deformation field. Our experiment on a real clinical brain
+hemorrhage dataset has achieved state-of-the-art performance and can
+generate interpretable deformation fields.
+Keywords: Computer-aided diagnosis · Semi-supervised learning · Dif-
+fusion models · Intracranial hemorrhage
+1
+Introduction
+Intracranial hemorrhage (ICH) refers to brain bleeding within the skull, a se-
+rious medical emergency that would cause severe disability or even death [1].
+A characteristic symptom of severe ICH is brain midline shift (MLS), which
+is the lateral displacement of midline cerebral structures (see Fig. 1). MLS is
+an important and quantifiable indicator of the severity of mass effects and the
+urgency of intervention [2,3,9]. For instance, the 5 millimeters (mm) threshold
+of MLS is frequently used to determine whether immediate intervention and
+close monitoring is required [4]. MLS quantification demands high accuracy and
+arXiv:2301.00409v1  [cs.CV]  1 Jan 2023
+
+2
+S.Z. Gong et al.
+posterior falx 
+anterior falx 
+(a) No MLS
+falx 
+mls
+(b) MLS on falx
+septum
+pellucidum
+mls
+(c) MLS on septum
+pellucidum
+The third 
+ventricle 
+mls
+(d) MLS on the
+third ventricle
+Fig. 1: Examples of head CT scans to illustrate how radiologists measure MLS.
+Dash red line connecting the anterior falx and posterior falx denotes a hypothet-
+ical normal midline. Blue circles denote the shifted landmarks. Perpendicular
+red lines from the shifted landmarks to normal midline are measured as MLS
+scale.
+efficiency, which is difficult to achieve with manual quantification, especially in
+emergencies, due to the variability in shift regions, unclear landmark boundaries,
+and non-standard scanning pose. An automated MLS quantification algorithm
+that can immediately and accurately quantify MLS is highly desirable to identify
+urgent patients for timely treatment.
+To measure MLS, clinicians usually first identify a few CT slices with large
+shifts and then measure and identify the maximum deviation of landmarks such
+as the septum pellucidum, third ventricle, or falx from their normal counterpart
+as the final MLS distance (see examples in Fig. 1). Such a clinical fashion of
+MLS quantification can be difficult to be translated into a well-defined automa-
+tion process. Currently, there are only limited studies on automated MLS quan-
+tification, using different strategies and varied labeling requirements. Nguyen
+et al. proposed a landmark-based method that relies on anatomical markers to
+determine the location of the deformed midline [9]. However, this method can
+only apply to cases where MLS appears on these specific marker regions. Liao et
+al. adopted a symmetric-based method to seek a curve connecting all deformed
+structures [10], which is difficult to generalize due to over-simplified anatom-
+ical assumptions and sensitivity to patients’ scan poses. A few recent works
+try to overcome these limitations by using stronger supervision with dense la-
+beling. Some studies formulated MLS quantification as a midline segmentation
+task [5,6,7], by delineating the intact midline as labels to supervise the train-
+ing of segmentation models. Another study designed a hemisphere segmentation
+task to quantify MLS [8], which requires pixel-wise annotation for each slice.
+However, obtaining such dense annotations is very costly and time-consuming,
+while may not be necessary for MLS quantification.
+To tackle these limitations, we propose to fit MLS quantification into a de-
+formation prediction problem by using semi-supervised learning (SSL) with only
+limited annotations. Our framework avoids the strong dependency on specific
+
+Diffusion model based ICH midline shift quantification
+3
+landmarks or over-simplified assumptions in previous methods while not increas-
+ing the labeling efforts. We aim to use only sparse and weak labels as ground
+truth supervisions, which are just one shifted landmark and its normal counter-
+part on a limited number of slices provided by radiologists, but we try to fully
+exploit the unlabeled slices and non-MLS data to impose extra regularization
+for the sparse-to-dense extension. Existing SSL methods typically use a partially
+trained model with labeled data to generate pseudo labels for unlabeled data,
+assuming that labeled and unlabeled data are generally similar. These meth-
+ods can be sub-optimal in our case as labeled slices of MLS usually present the
+largest deformation while unlabeled slices contain only minor or no deformation.
+Instead, we propose our SSL strategy by generating a corresponding non-MLS
+image for each unlabeled MLS slice with generative models and regularizing
+that the deformation field should warp the generated non-MLS images into the
+original MLS ones. However, as we only have volume-wise labels for MLS and
+non-MLS classification, it can be difficult to train a slice-wise discriminator as
+required by many generative models such as GANs [12]. Fortunately, the recently
+proposed diffusion models [15], which prove to have strong power in both dis-
+tribution learning and image generation without dependency on discriminators,
+can be a potentially good solution.
+In this work, we propose a novel semi-supervised learning framework based
+on diffusion models to quantify the brain MLS from head CT images with defor-
+mation prediction. Our method effectively exploits supervision and regulariza-
+tion from all types of available data including MLS images with sparse ground
+truth labels, MLS images without labels, and non-MLS images. We validate our
+method on a real clinical head CT dataset, showing effectiveness of each proposed
+component. Our contributions include: (1) innovating an effective deformation
+strategy for brain MLS quantification, (2) incorporating diffusion models as a
+representation learner to extract features reflecting where and how an MLS im-
+age differs from a non-MLS image, and (3) proposing a diffusion model-based
+semi-supervised framework that can effectively leverage massive unlabelled data
+to improve the model performance.
+2
+Methods
+Fig. 2 illustrates our diffusion model-based semi-supervised learning framework
+for MLS quantification via deformation prediction. In Sec. 2.1, we introduce our
+deformation prediction by using only sparse supervision. In Sec. 2.2, we propose
+to incorporate non-MLS data for representation learning. In Sec. 2.3, we describe
+how to utilize unlabeled MLS images for sparse-to-dense regularization.
+2.1
+MLS Quantification through Deformation Estimation
+Our proposed deformation strategy for brain MLS quantification aims to find
+an optimal deformation field φ so that an MLS image can be regarded as a
+
+4
+S.Z. Gong et al.
+◼ Semi-supervised framework
+𝑥𝑢
+𝑥𝑙
+Deformation 
+estimation
+Image 
+generation
+𝜙(𝑥′𝑢)
+Weak label
+Deformation network
+Conditional 
+diffusion 
+network 𝐺𝜃𝑐
+Unconditional 
+diffusion 
+network 𝐺𝜃𝑢
+Ƹ𝜖𝜃𝑐 − Ƹ𝜖𝜃𝑢
+𝑥𝑡
+𝑥0
+◼ Deformation estimation
+𝜙
+◼ Image generation
+Denoising with 
+𝜆𝐺𝜃𝑐 − (1 − 𝜆)𝐺𝜃𝑢
+𝑥0
+𝑥𝐿
+𝑥0
+,
+𝑙𝑚𝑠𝑒 + 𝑙𝑐𝑒𝑖𝑙
+𝑙ℎ𝑢𝑏𝑒𝑟
++ 𝑙𝑠𝑚𝑜𝑜𝑡ℎ
+⊕ ⊕
+𝜖
+𝜖
+warp
+𝜙𝑙
+𝜙𝑢
+𝑥′𝑢
+Fig. 2: The pipeline of our proposed semi-supervised deformation strategy for
+MLS quantification. The labeled image xl is supervised by sparse labels and the
+unlabeled image xu is self-supervised with generated negative image.
+hypothetically non-MLS image warped with this deformation field. The defor-
+mation field can be parameterized by a function with high complexity so that
+it does not explicitly rely on a single landmark or over-simplified symmetric as-
+sumptions, which naturally overcomes the limitations of existing methods. We
+apply a learning-based framework to parameterize the deformation field with a
+U-Net shape neural network. The output of the network is the stationary veloc-
+ity field v. The diffeomorphic deformation field φ is then calculated through the
+integration of the velocity field, similarly to VoxelMorph [11] for image regis-
+tration. The learning process is supervised by sparse deformation ground truth.
+For each labeled slice, we have the ground truth y = (y1, y2), which is a two-
+dimensional vector directing from shifted landmark point toward its presumably
+normal location (the red arrow in Fig. 2). The predicted deformation ˆy is bi-
+linearly interpolated at the shifted landmark point from the deformation field,
+which is also a two-dimensional vector. To alleviate the influence of a few ex-
+tremely large deformation points and increase model’s robustness, we use Huber
+loss to measure the similarity between the predicted deformation and the label:
+lhuber(yd, ˆyd) =
+�
+�
+�
+|yd − ˆyd|,
+|yd − ˆyd| ≥ c,
+(yd − ˆyd)2 + c2
+2c
+,
+|yd − ˆyd| < c.
+(1)
+where d ∈ {1, 2}. The hyperparameter c defines the range for absolute error or
+squared error. We also encourage a smooth deformation field with a diffusion
+regularizer on the spatial gradients of deformation φ to avoid a discontinuous
+
+Diffusion model based ICH midline shift quantification
+5
+deformation field:
+lsmooth =
+�
+j
+�
+k
+∥φjk − φ(j−1)k∥2 + ∥φjk − φj(k−1)∥2,
+(2)
+As the deformation can be extremely large in our case, and meanwhile to force a
+smooth transition between the deformation peak and its adjacent pixels, we use
+a coarse-to-fine manner, where velocity fields are generated through upsampling
+with skip connection to progressively aggregate features of different scales.
+2.2
+Learning Negative Patterns from Non-MLS Images
+In order to learn a deformation field to warp a non-MLS image into MLS one,
+ideally we would need a pair of non-MLS and MLS images for network train-
+ing, which however does not exist in practice. Lacking such information makes
+the network difficult to learn. A naive solution is to generate a corresponding
+non-MLS image. However, generated images entail some randomness and can
+often lack important details. Depending too much on such fake inputs can lead
+to poor robustness. Inspired by the score-matching interpretation of diffusion
+models [17], we propose to learn the non-MLS distribution from massive amount
+of negative cases. Given an MLS image, we can evaluate which parts of the im-
+age make it different from a non-MLS image. This deviation can serve as latent
+features that help the deformation network with deformation prediction.
+Diffusion models, especially DDPM [14], define a forward diffusion process as
+the Markov process progressively adding random Gaussian noise to a given image
+and then trying to approximate the reverse process by a Gaussian distribution.
+The forward process can be simplified by a one-step sampling: xt = √αtx0 +
+√1 − αtϵ, where αt := �t
+s=0 1 − βt, and βt are predefined variance schedule.
+ϵ is sampled from N(0, I). The mean µθ(xt, t) and variance Σθ(xt, t) of the
+reverse process can be parameterized by neural networks. A popular choice is
+to re-parameterize µθ(xt, t) so that ˆϵθ(xt, t) instead of µθ(xt, t) is estimated by
+neural networks to approximate the noise ϵ. Moreover, the output of the diffusion
+network ϵ(xt, t) is actually a scaled score function ∇ log p(xt) as it moves the
+corrupted image towards the opposite direction of the corruption. [18].
+As a result, through pre-training one unconditional diffusion model trained
+with all data (denoted as U) and one conditional diffusion model trained with
+only non-MLS data (denoted as C), the subtraction of two outputs
+ˆϵθU (xt, t) − ˆϵθC(xt, t) ∝ ∇ log p(xt|n) − ∇ log p(xt) = ∇ log p(n|xt),
+(3)
+can be regarded as the gradient of class prediction (n = 1 for non-MLS and 0
+otherwise) w.r.t to the input image, which reflects how the input images devi-
+ate from a non-MLS image. This latent contains information regarding how to
+transform the MLS positive image into a non-MLS one and therefore is helpful
+for training the deformation network. Moreover, this feature representation ex-
+hibits less fluctuation toward the randomness of the additive noise because the
+stochastic parts are eliminated through subtraction. It is more stable than the
+
+6
+S.Z. Gong et al.
+predicted noise or generated MSL negative images. For training, we randomly
+sample t from 0 to the diffusion steps Ttrain, while for inference we fix it to be a
+certain value. We examine the effects of this value in Section 3.4.
+2.3
+Semi-Supervised Deformation Regularization
+Deformation estimation is a dense prediction problem, while we only have sparse
+supervision. This can lead to flickering and poor generalizability if the deforma-
+tion lacks certain regularization. On the other hand, we have a significant amount
+of unlabeled data from the MLS volumes that is potentially helpful. Therefore,
+we propose to include these unlabeled data during training in a semi-supervised
+manner, so that unlabeled data can provide extra regularization for training or
+produce additional training examples based on noisy pseudo labels. Many exist-
+ing semi-supervised methods seek to use the prediction for unlabeled data given
+by the same or a twin network as pseudo-labels and then supervise the model or
+impose some regularization with these pseudo-labels. However, these methods
+hold a strong assumption that labeled and unlabeled data are drawn from the
+same distribution, which is not true in our case because most labeled data are
+with large deformation while unlabeled data are with minor or no deformation.
+Therefore, we want to find another type of pseudo-label to bypass the distribu-
+tion assumption. As the deformation field is assumed to warp a hypothetically
+normal image into an MLS one, we generate hypothetically non-MLS images x′
+0
+using pre-trained diffusion models through classifier-free guidance [16]:
+ˆϵ(xt, t) = γˆϵθC(xt, t) + (1 − γ)ˆϵθU (xt, t),
+(4)
+where γ is a hyper-parameter controlling the strength of the condition. We com-
+pare x′
+0 warped with the deformation field φ(x′
+0) and calculate its similarity with
+the original x0 through MSE loss. As it can be difficult for the generated image
+to be fully faithful to the original image because the generative process entails a
+lot of random sampling, this lmse can only serve as noisy supervision. Therefore,
+instead of generating x′
+0 ahead of deformation network training, we generate it
+in an ad-hoc way so that the noisy effects can be counteracted.
+The final MLS measurement is estimated by calculating the length of the
+maximum displacement vector from the predicted deformation field, so it is
+more sensitive to over-estimation. And our results also show most of the errors
+come from over-estimation. As for unlabelled slices, we still have the prior that
+its MLS cannot be larger than the MLS of that specific volume δ, we propose to
+incorporate an additional ceiling loss to punish the over-estimation:
+lceil =
+�
+j
+�
+k
+max(0, ||φjk|| − δ).
+(5)
+Overall, the loss term is a combination of supervised loss and unsupervised loss,
+with a weight term controlling the relative importance of each loss term:
+l = lhuber + w1lsmooth + u(i)(lmse + w2lceil),
+(6)
+
+Diffusion model based ICH midline shift quantification
+7
+where w1 and w2 are two fixed weight terms and u(i) is a time-varying weight
+term that is expected to gradually increase as the training iteration i progresses
+so that the training can converge quickly through strong supervision first and
+then refine and enhance generalizability via unsupervised loss.
+3
+Experiments and Results
+3.1
+Data Acquisition and Preprocessing
+We retrospectively collected anonymous thick-slice, non-contrast head CT of pa-
+tients who were admitted with head trauma or stroke symptoms and diagnosed
+with various subtypes of intracranial hemorrhage, including epidural hemor-
+rhage, subdural hemorrhage, subarachnoid hemorrhage, intraventricular hem-
+orrhage, and intraparenchymal hemorrhage, between July 2019 and December
+2019 in the Prince of Wales Hospital, a public hospital under the Hospital Au-
+thority of Hong Kong. The ethics approval was obtained from the Joint Chinese
+University of Hong Kong-New Territories East Cluster Clinical Research ethics
+committee. The eligible patients comprised 2793 CT volumes, among them 124
+are MLS positive cases. The MLS ranges between 2.24mm and 20.12mm, with
+mean value of 8.34mm and medium value of 8.73mm. The annotation was per-
+formed by two trained physicians and verified by one experienced radiologist
+(with over 10 years of clinical experience on ICH). The labeling process followed
+a real clinical measurement pipeline, where the shifted landmark, anterior falx
+point, and posterior falx point were pointed out, and the length of the vertical
+line from the landmark to the line connecting the anterior falx point and the
+posterior falx point was the measured MLS value. For each volume, a few slices
+with large deformation were separately measured and annotated while the shift
+of the largest one served as the case-level label. On average, 4 out of 30 slices
+of each volume were labeled. We discarded the first 8 and the last 5 slices as
+they are mainly structures irrelevant to MLS. For pre-processing, we adjusted
+the pixel size of all images to 0.86mm and then cropped or padded the resulting
+images to the resolution of 256 × 256 pixels. The HU window was set to 0 and
+80. We applied intensity clipping (0.5 and 99.5 percentiles) and min-max nor-
+malization (between -1 and 1) to each image. Random rotation between −15◦
+and 15◦ was used for data augmentation.
+3.2
+Implementation Details
+For the diffusion network, we use the network architecture designed in DDPM [15]
+and set the noise level from 10−4 to 2 × 10−2 by linearly scheduling with
+Ttrain = 1000. For non-MLS image generation, we apply the Denoising Diffu-
+sion Implicit Model (DDIM) [13] with 50 steps and set the noise scale to 15 to
+shorten the generative time. We set the hyper-parameters as α = 1, β = 1, c = 3
+and γ = 2. u(i) is set from 1 to 10 with the linear schedule. The diffusion models
+are trained by the AdamW optimizer with an initial learning rate of 1 × 10−4,
+
+8
+S.Z. Gong et al.
+Table 1: Comparison of different methods with 5-fold cross-validation.
+Methods
+Training data
+Volume-wise
+Slice-wise
+Labeled
+Unlabeled
+MAE↓
+(mm)
+RMSE↓
+(mm)
+MAE↓
+(mm)
+RMSE↓
+(mm)
+Regression
+✓
+3.91
+4.90
+3.56
+4.16
+Deformation
+✓
+3.80
+4.47
+2.51
+3.17
+Mean-Teacher [19]
+✓
+✓
+2.89
+3.67
+2.43
+3.22
+CPS [20]
+✓
+✓
+2.72
+3.42
+2.38
+3.15
+Ours
+✓
+✓
+2.43
+3.17
+2.25
+3.09
+batch size 4, for 2 × 105 iterations. We up-sample the MLS positive data by 10×
+when training the unconditional diffusion model. The deformation network is
+trained by the AdamW optimizer with an initial learning rate of 1×10−4, batch
+size 16, for 100 epochs. All models are implemented with PyTorch 1.12.1 using
+one Nvidia GeForce RTX 3090 GPU.
+3.3
+Quantification Accuracy and Deformation Quality
+We evaluate the performance of our quantification strategy through mean abso-
+lute error (MAE) and root mean square error (RMSE). For volume-wise evalua-
+tion, we measure the maximum deformation of each slice of the whole volume and
+select the largest one as the final result. We also report the slice-wise evaluation,
+which is calculated based on labeled slices. This error can reflect how the models
+perform on slices with relatively large deformation. Since existing MLS estima-
+tion methods require different types of labels from ours, it is difficult to directly
+compare with those methods. We therefore first compare our deformation-based
+strategy with a regression-based strategy, which uses DenseNet-121 [21] to di-
+rectly predict the slice-wise MLS. We also compare our proposed semi-supervised
+learning approach with two popular semi-supervised learning methods, that are
+Mean-Teacher [19] and Cross Pseudo Supervision (CPS) [20], which are imple-
+mented into our deformation framework. The results are given in Table 1, which
+are based on 5-fold cross-validations.
+From the results, we can see that when only using labeled MLS slices for
+model learning, our deformation strategy already shows better performance than
+the regression model. This may attribute to that our deformation model learns
+the knowledge of both MLS values and locations while a regression model only
+captures the MLS value information. This difference can be further enlarged if
+we consider slice-wise performance. Moreover, all three semi-supervised learn-
+ing methods, i.e., Mean-Teacher, CPS, and ours, consistently improve the per-
+formance of deformation prediction, showing the benefits and importance of
+
+Diffusion model based ICH midline shift quantification
+9
+true:
+pred:
+11.00
+11.43
+true:
+pred:
+5.43
+6.02
+true:
+pred:
+6.23
+6.78
+true:
+pred:
+7.48
+8.87
+(a) Examples of predicted deformation on MLS images
+true:
+pred:
+0.00
+1.60
+true:
+pred:
+0.00
+1.75
+true:
+pred:
+0.00
+1.87
+true:
+pred:
+0.00
+1.69
+(b) Examples of predicted deformation on non-MLS images
+Fig. 3: Predicted deformation on (a) MLS images. (b) non-MLS images. The
+regions with the largest deformation are highlighted. Slice-wise predicted MSL
+and ground truth are provided.
+incorporating unlabeled data into model learning. Our semi-supervised learn-
+ing method based on diffusion models achieves better quantification results
+than Mean-Teacher and CPS, significantly reducing the volume-wise MAE from
+3.80mm to 2.43mm. An interesting observation is that the unlabeled data con-
+tribute more to the volume-wise evaluation than the slice-wise evaluation. By
+inspecting the prediction, we find that the deformation prediction trained with
+labeled data tends to overestimate the deformation of slices with little or no
+deformation, which makes the volume-wise prediction error-prone. As most un-
+labeled data are slices with minor shifts, incorporating these data for semi-
+supervised learning can impose constraints to avoid large deformation, which
+greatly improves the model’s robustness.
+We also visualize the predicted deformation field of several sample cases.
+From Fig. 3 (a), we can see the model can well posit the location where the
+maximum shift appears and push it to its hypothetically normal counterpart.
+The largest deformation happens exactly at the site with the maximum shift. To
+validate the robustness of our model, we also select several patients diagnosed
+with no MLS and plot the predicted deformation of these samples. As can be
+seen in Fig. 3 (b), our method is able to provide a reasonable prediction for
+non-MLS images by outputting much smaller values than that for MLS images.
+Our model’s predictions for non-MLS images are not exactly zero are caused on
+one hand by that even for a completely healthy person, the midline cannot be
+perfectly aligned due to multiple factors such as scan pose, on the other hand, our
+
+O10
+S.Z. Gong et al.
+Methods
+MAE↓
+(mm)
+RMSE↓
+(mm)
+Fully-supervised
+3.61
+4.47
++ Representation
+3.22
+3.69
+Semi-supervised
+2.61
+3.24
++ Representation
+2.45
+3.05
+Table 2: Effects of the representation.
+0
+200
+400
+600
+800
+t
+2.6
+2.8
+3.0
+3.2
+3.4
+Effects of noise level t
+MAE
+RMSE
+Fig. 4: Effects of the noise level.
+models tend to overestimate the shift because we are calculating the maximum
+deformation as final measurement.
+3.4
+Ablation Study
+We conduct several ablation experiments to study the effects of several compo-
+nents in our proposed framework on the model performance. The volume-wise
+results reported are trained on four folders and tested on one folder.
+Effects for representation learning. We first conduct ablation studies to
+verify that the latent feature extracted from the two diffusion models is truly
+useful for deformation prediction. To this end, we select two deformation models,
+one trained with only labeled data and the other using semi-supervised learning,
+and compare their performance with and without the extracted representation
+as input. The results are given in Table 2. As expected, incorporating the rep-
+resentation can improve the model performance in both cases.
+The noise level is an important component of diffusion models. Only with a
+proper noise level, can the model accurately estimate the deviation of the image
+toward the negative sample space. Therefore, we do inference with multiple noise
+levels and compare its effect on model performance. The results are shown in
+Fig. 4. Our model is very robust towards this hyper-parameter. As long as t is
+not too small, the model gives very similar performances. The best performance
+appears in the middle when t = 600. This is reasonable as small noise fails to
+corrupt the original image thus degenerating the performance of score estimation
+while large noise may obscure too many details of the original image.
+Quantity of unlabeled images. To verify the usefulness of unlabeled images,
+we conduct ablation studies on the number of unlabeled images used. For each
+experiment, we randomly sample 20%, 40%, 60%, and 80% volumes, and we
+incorporate unlabeled slices of these volumes for semi-supervised training. For
+the rest volumes, we are only using the labeled slices. We also do one experiment
+that completely removes the uses of unlabeled images. For each experiment, the
+pre-trained diffusion models are the same, which uses all the data. In other
+words, these unlabeled images somehow still contribute to the model training.
+The results are shown in Fig. 5 (a). As can be seen, the model performance
+and robustness can be enhanced as we incorporate more unlabeled images. This
+
+Diffusion model based ICH midline shift quantification
+11
+(a)
+(b)
+Fig. 5: Results of our ablation experiments in terms of: (a) proportion of unla-
+beled data used, and (b) proportion of negative data used.
+provides strong evidence for our claim that our model truly learns valuable
+information from unlabeled data.
+Quantity of non-MLS images. To further measure the benefits of includ-
+ing non-MLS cases, we conduct another ablation study on the proportion of
+non-MLS cases. Non-MLS cases are used to train diffusion models. As currently,
+the amount of non-MLS cases is much higher than MLS cases, we upsample the
+MLS cases so that their quantities are approximately the same when training the
+unconditional diffusion model. For ablation, we first downsample the non-MLS
+data so that their quantity is 1×, 5×, and 10× that of the MLS cases, and then
+upsample the MLS cases to make them balanced. From the results in Fig. 5 (b),
+we find model performance improves with more non-MLS cases incorporated. In-
+creasing non-MLS cases can help train diffusion models and further improve the
+quality of generated images and extracted feature representations. However, this
+effect will soon be saturated as the amount of MLS cases is relatively small. This
+can be a bottleneck for effectively using the non-MLS cases as it is challenging
+to train unconditional diffusion models with such imbalanced datasets.
+4
+Conclusions and Future Work
+In this paper, we propose a novel framework based on deformation field esti-
+mation to automatically measure the brain MLS. The labels we are using are
+sparse which can greatly alleviate the labeling workload. We also propose a
+semi-supervised learning strategy based on diffusion models which significantly
+improves the model performance. Experiments on a clinic dataset show our meth-
+ods can achieve satisfying performance. We also verify that using unlabeled data
+and non-MLS cases can truly help improve the model’s performance.
+Our methods have several limitations. First, the model performance highly
+relies on pre-trained diffusion models. Training diffusion models with extremely
+imbalanced data requires great effort. Second, the measurement results exhibit
+randomness due to noise corruption. Finally, the measurement results are prone
+to overestimation. Our future work will figure out solutions for these limitations.
+
+Effects of unlabeled data
+4.5
+MAE
+RMSE
+4.0
+3.5
+3.0
+2.5
+2.0
+0
+20%
+40%
+60%
+80%
+100%
+proportion of unlabeled image usedEffects of negative cases
+5.0
+MAE
+4.5
+RMSE
+4.0
+3.5
+3.0
+2.5
+2.0
+0
+1X
+5x
+10x
+all
+ratio of negative cases / positive cases12
+S.Z. Gong et al.
+References
+1. Caceres, J.A., Goldstein, J.N.: Intracranial hemorrhage. Emerg. Med. Clin. North
+Am. 30(3), 771 (2012)
+2. Quattrocchi, K.B., et al.: Quantification of midline shift as a predictor of poor out-
+come following head injury. In: Surgical Neurology, vol. 35(3), pp. 183–188, (1991).
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+shift threshold to predict poor outcome in patients with supratentorial spontaneous
+intracerebral hemorrhage. In: Neurocritical care, vol. 28(3), pp.314–321, (2018).
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+tomation: A Review of Techniques and Algorithms. In: International Journal of
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+5. Pisov, M., et al.: Incorporating task-specific structural knowledge into CNNs for
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+structural connectivity prior for brain midline delineation. In: Martel, A.L., et al.
+(eds.) MICCAI 2020. LNCS, vol. 12267, pp. 208–217. Springer, Cham (2020).
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+deformed brain midline. In: Shen, D., et al. (eds.) MICCAI 2019. LNCS, vol. 11766,
+pp. 839–847. Springer, Cham (2019).
+8. Qin, C., Li, H., Liu, Y., Shang, H., Pei, H., Wang, X., Chen, Y., Chang, J., Feng,
+M., et al.: 3D Brain Midline Delineation for Hematoma Patients. In: Bruijne, M.,
+et al. (eds.) MICCAI 2021. LNCS, vol. 12905, pp. 510–518. Springer, Cham (2021).
+9. Nguyen, N.P., Yoo, Y., Chekkoury, A., Eibenberger, E., et al.: Brain midline shift
+detection and quantification by a cascaded deep network pipeline on non-contrast
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+10. Liao, C., Xiao, F., et al.: Automatic recognition of midline shift on brain CT
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+11. Balakrishnan, G., Zhao, A., Sabuncu, M.R., Guttag, J., Dalca, A.V.: Voxelmorph:
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+12. Goodfellow, I.J.,Pouget, J.A., Mirza, M., Xu, B., Farley, D.W., Ozair, S., Courville,
+A., Bengio, Y.: Generative Adversarial Networks. In: NeurIPS 2014.
+13. Song, J., Meng, C., Ermon, S.: Denoising diffusion implicit models. In: ICLR 2020.
+14. Ho, J., et al.: Denoising diffusion probabilistic models. In: NeurIPS 2020.
+15. Dhariwal, et al.: Diffusion models beat gans on image synthesis. In: NeurIPS 2021.
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+
diff --git a/ctAyT4oBgHgl3EQfjPgK/content/tmp_files/load_file.txt b/ctAyT4oBgHgl3EQfjPgK/content/tmp_files/load_file.txt
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@@ -0,0 +1,519 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf,len=518
+page_content='Diffusion Model based Semi-supervised Learning  on Brain Hemorrhage Images for Efficient  Midline Shift Quantification  Shizhan Gong1, Cheng Chen1, Yuqi Gong1, Nga Yan Chan2, Wenao Ma1,  Calvin Hoi-Kwan Mak3, Jill Abrigo2, Qi Dou1  1Department of Computer Science and Engineering,  The Chinese University of Hong Kong, Hong Kong, China  2Department of Imaging and Interventional Radiology,  The Chinese University of Hong Kong, Hong Kong, China  3Queen Elizabeth Hospital, Hong Kong, China  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Brain midline shift (MLS) is one of the most critical factors  to be considered for clinical diagnosis and treatment decision-making for  intracranial hemorrhage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Existing computational methods on MLS quan-  tification not only require intensive labeling in millimeter-level measure-  ment but also suffer from poor performance due to their dependence on  specific landmarks or simplified anatomical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In this paper,  we propose a novel semi-supervised framework to accurately measure the  scale of MLS from head CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We formulate the MLS measurement  task as a deformation estimation problem and solve it using a few MLS  slices with sparse labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Meanwhile, with the help of diffusion models,  we are able to use a great number of unlabeled MLS data and 2793 non-  MLS cases for representation learning and regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The extracted  representation reflects how the image is different from a non-MLS image  and regularization serves an important role in the sparse-to-dense refine-  ment of the deformation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Our experiment on a real clinical brain  hemorrhage dataset has achieved state-of-the-art performance and can  generate interpretable deformation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Keywords: Computer-aided diagnosis · Semi-supervised learning · Dif-  fusion models · Intracranial hemorrhage  1  Introduction  Intracranial hemorrhage (ICH) refers to brain bleeding within the skull, a se-  rious medical emergency that would cause severe disability or even death [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  A characteristic symptom of severe ICH is brain midline shift (MLS), which  is the lateral displacement of midline cerebral structures (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' MLS is  an important and quantifiable indicator of the severity of mass effects and the  urgency of intervention [2,3,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For instance, the 5 millimeters (mm) threshold  of MLS is frequently used to determine whether immediate intervention and  close monitoring is required [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' MLS quantification demands high accuracy and  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='00409v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='CV]  1 Jan 2023  2  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  posterior falx  anterior falx  (a) No MLS  falx  mls  (b) MLS on falx  septum  pellucidum  mls  (c) MLS on septum  pellucidum  The third  ventricle  mls  (d) MLS on the  third ventricle  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 1: Examples of head CT scans to illustrate how radiologists measure MLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Dash red line connecting the anterior falx and posterior falx denotes a hypothet-  ical normal midline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Blue circles denote the shifted landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Perpendicular  red lines from the shifted landmarks to normal midline are measured as MLS  scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  efficiency, which is difficult to achieve with manual quantification, especially in  emergencies, due to the variability in shift regions, unclear landmark boundaries,  and non-standard scanning pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' An automated MLS quantification algorithm  that can immediately and accurately quantify MLS is highly desirable to identify  urgent patients for timely treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  To measure MLS, clinicians usually first identify a few CT slices with large  shifts and then measure and identify the maximum deviation of landmarks such  as the septum pellucidum, third ventricle, or falx from their normal counterpart  as the final MLS distance (see examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Such a clinical fashion of  MLS quantification can be difficult to be translated into a well-defined automa-  tion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Currently, there are only limited studies on automated MLS quan-  tification, using different strategies and varied labeling requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Nguyen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' proposed a landmark-based method that relies on anatomical markers to  determine the location of the deformed midline [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' However, this method can  only apply to cases where MLS appears on these specific marker regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Liao et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' adopted a symmetric-based method to seek a curve connecting all deformed  structures [10], which is difficult to generalize due to over-simplified anatom-  ical assumptions and sensitivity to patients’ scan poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' A few recent works  try to overcome these limitations by using stronger supervision with dense la-  beling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Some studies formulated MLS quantification as a midline segmentation  task [5,6,7], by delineating the intact midline as labels to supervise the train-  ing of segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Another study designed a hemisphere segmentation  task to quantify MLS [8], which requires pixel-wise annotation for each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  However, obtaining such dense annotations is very costly and time-consuming,  while may not be necessary for MLS quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  To tackle these limitations, we propose to fit MLS quantification into a de-  formation prediction problem by using semi-supervised learning (SSL) with only  limited annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Our framework avoids the strong dependency on specific  Diffusion model based ICH midline shift quantification  3  landmarks or over-simplified assumptions in previous methods while not increas-  ing the labeling efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We aim to use only sparse and weak labels as ground  truth supervisions, which are just one shifted landmark and its normal counter-  part on a limited number of slices provided by radiologists, but we try to fully  exploit the unlabeled slices and non-MLS data to impose extra regularization  for the sparse-to-dense extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Existing SSL methods typically use a partially  trained model with labeled data to generate pseudo labels for unlabeled data,  assuming that labeled and unlabeled data are generally similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' These meth-  ods can be sub-optimal in our case as labeled slices of MLS usually present the  largest deformation while unlabeled slices contain only minor or no deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Instead, we propose our SSL strategy by generating a corresponding non-MLS  image for each unlabeled MLS slice with generative models and regularizing  that the deformation field should warp the generated non-MLS images into the  original MLS ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' However, as we only have volume-wise labels for MLS and  non-MLS classification, it can be difficult to train a slice-wise discriminator as  required by many generative models such as GANs [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Fortunately, the recently  proposed diffusion models [15], which prove to have strong power in both dis-  tribution learning and image generation without dependency on discriminators,  can be a potentially good solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  In this work, we propose a novel semi-supervised learning framework based  on diffusion models to quantify the brain MLS from head CT images with defor-  mation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Our method effectively exploits supervision and regulariza-  tion from all types of available data including MLS images with sparse ground  truth labels, MLS images without labels, and non-MLS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We validate our  method on a real clinical head CT dataset, showing effectiveness of each proposed  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Our contributions include: (1) innovating an effective deformation  strategy for brain MLS quantification, (2) incorporating diffusion models as a  representation learner to extract features reflecting where and how an MLS im-  age differs from a non-MLS image, and (3) proposing a diffusion model-based  semi-supervised framework that can effectively leverage massive unlabelled data  to improve the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  2  Methods  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 2 illustrates our diffusion model-based semi-supervised learning framework  for MLS quantification via deformation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='1, we introduce our  deformation prediction by using only sparse supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='2, we propose  to incorporate non-MLS data for representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='3, we describe  how to utilize unlabeled MLS images for sparse-to-dense regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='1  MLS Quantification through Deformation Estimation  Our proposed deformation strategy for brain MLS quantification aims to find  an optimal deformation field φ so that an MLS image can be regarded as a  4  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  ◼ Semi-supervised framework  𝑥𝑢  𝑥𝑙  Deformation  estimation  Image  generation  𝜙(𝑥′𝑢)  Weak label  Deformation network  Conditional  diffusion  network 𝐺𝜃𝑐  Unconditional  diffusion  network 𝐺𝜃𝑢  Ƹ𝜖𝜃𝑐 − Ƹ𝜖𝜃𝑢  𝑥𝑡  𝑥0  ◼ Deformation estimation  𝜙  ◼ Image generation  Denoising with  𝜆𝐺𝜃𝑐 − (1 − 𝜆)𝐺𝜃𝑢  𝑥0  𝑥𝐿  𝑥0  ,  𝑙𝑚𝑠𝑒 + 𝑙𝑐𝑒𝑖𝑙  𝑙ℎ𝑢𝑏𝑒𝑟  + 𝑙𝑠𝑚𝑜𝑜𝑡ℎ  ⊕ ⊕  𝜖  𝜖  warp  𝜙𝑙  𝜙𝑢  𝑥′𝑢  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 2: The pipeline of our proposed semi-supervised deformation strategy for  MLS quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The labeled image xl is supervised by sparse labels and the  unlabeled image xu is self-supervised with generated negative image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  hypothetically non-MLS image warped with this deformation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The defor-  mation field can be parameterized by a function with high complexity so that  it does not explicitly rely on a single landmark or over-simplified symmetric as-  sumptions, which naturally overcomes the limitations of existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We  apply a learning-based framework to parameterize the deformation field with a  U-Net shape neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The output of the network is the stationary veloc-  ity field v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The diffeomorphic deformation field φ is then calculated through the  integration of the velocity field, similarly to VoxelMorph [11] for image regis-  tration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The learning process is supervised by sparse deformation ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  For each labeled slice, we have the ground truth y = (y1, y2), which is a two-  dimensional vector directing from shifted landmark point toward its presumably  normal location (the red arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The predicted deformation ˆy is bi-  linearly interpolated at the shifted landmark point from the deformation field,  which is also a two-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' To alleviate the influence of a few ex-  tremely large deformation points and increase model’s robustness, we use Huber  loss to measure the similarity between the predicted deformation and the label:  lhuber(yd, ˆyd) =  �  �  �  |yd − ˆyd|,  |yd − ˆyd| ≥ c,  (yd − ˆyd)2 + c2  2c  ,  |yd − ˆyd| < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  (1)  where d ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The hyperparameter c defines the range for absolute error or  squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We also encourage a smooth deformation field with a diffusion  regularizer on the spatial gradients of deformation φ to avoid a discontinuous  Diffusion model based ICH midline shift quantification  5  deformation field:  lsmooth =  �  j  �  k  ∥φjk − φ(j−1)k∥2 + ∥φjk − φj(k−1)∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  (2)  As the deformation can be extremely large in our case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' and meanwhile to force a  smooth transition between the deformation peak and its adjacent pixels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' we use  a coarse-to-fine manner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' where velocity fields are generated through upsampling  with skip connection to progressively aggregate features of different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='2  Learning Negative Patterns from Non-MLS Images  In order to learn a deformation field to warp a non-MLS image into MLS one,  ideally we would need a pair of non-MLS and MLS images for network train-  ing, which however does not exist in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Lacking such information makes  the network difficult to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' A naive solution is to generate a corresponding  non-MLS image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' However, generated images entail some randomness and can  often lack important details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Depending too much on such fake inputs can lead  to poor robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Inspired by the score-matching interpretation of diffusion  models [17], we propose to learn the non-MLS distribution from massive amount  of negative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Given an MLS image, we can evaluate which parts of the im-  age make it different from a non-MLS image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This deviation can serve as latent  features that help the deformation network with deformation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Diffusion models, especially DDPM [14], define a forward diffusion process as  the Markov process progressively adding random Gaussian noise to a given image  and then trying to approximate the reverse process by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  The forward process can be simplified by a one-step sampling: xt = √αtx0 +  √1 − αtϵ, where αt := �t  s=0 1 − βt, and βt are predefined variance schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  ϵ is sampled from N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The mean µθ(xt, t) and variance Σθ(xt, t) of the  reverse process can be parameterized by neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' A popular choice is  to re-parameterize µθ(xt, t) so that ˆϵθ(xt, t) instead of µθ(xt, t) is estimated by  neural networks to approximate the noise ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Moreover, the output of the diffusion  network ϵ(xt, t) is actually a scaled score function ∇ log p(xt) as it moves the  corrupted image towards the opposite direction of the corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  As a result, through pre-training one unconditional diffusion model trained  with all data (denoted as U) and one conditional diffusion model trained with  only non-MLS data (denoted as C), the subtraction of two outputs  ˆϵθU (xt, t) − ˆϵθC(xt, t) ∝ ∇ log p(xt|n) − ∇ log p(xt) = ∇ log p(n|xt),  (3)  can be regarded as the gradient of class prediction (n = 1 for non-MLS and 0  otherwise) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='t to the input image, which reflects how the input images devi-  ate from a non-MLS image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This latent contains information regarding how to  transform the MLS positive image into a non-MLS one and therefore is helpful  for training the deformation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Moreover, this feature representation ex-  hibits less fluctuation toward the randomness of the additive noise because the  stochastic parts are eliminated through subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' It is more stable than the  6  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  predicted noise or generated MSL negative images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For training, we randomly  sample t from 0 to the diffusion steps Ttrain, while for inference we fix it to be a  certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We examine the effects of this value in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='3  Semi-Supervised Deformation Regularization  Deformation estimation is a dense prediction problem, while we only have sparse  supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This can lead to flickering and poor generalizability if the deforma-  tion lacks certain regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' On the other hand, we have a significant amount  of unlabeled data from the MLS volumes that is potentially helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Therefore,  we propose to include these unlabeled data during training in a semi-supervised  manner, so that unlabeled data can provide extra regularization for training or  produce additional training examples based on noisy pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Many exist-  ing semi-supervised methods seek to use the prediction for unlabeled data given  by the same or a twin network as pseudo-labels and then supervise the model or  impose some regularization with these pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' However, these methods  hold a strong assumption that labeled and unlabeled data are drawn from the  same distribution, which is not true in our case because most labeled data are  with large deformation while unlabeled data are with minor or no deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Therefore, we want to find another type of pseudo-label to bypass the distribu-  tion assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As the deformation field is assumed to warp a hypothetically  normal image into an MLS one, we generate hypothetically non-MLS images x′  0  using pre-trained diffusion models through classifier-free guidance [16]:  ˆϵ(xt, t) = γˆϵθC(xt, t) + (1 − γ)ˆϵθU (xt, t),  (4)  where γ is a hyper-parameter controlling the strength of the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We com-  pare x′  0 warped with the deformation field φ(x′  0) and calculate its similarity with  the original x0 through MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As it can be difficult for the generated image  to be fully faithful to the original image because the generative process entails a  lot of random sampling, this lmse can only serve as noisy supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Therefore,  instead of generating x′  0 ahead of deformation network training, we generate it  in an ad-hoc way so that the noisy effects can be counteracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  The final MLS measurement is estimated by calculating the length of the  maximum displacement vector from the predicted deformation field, so it is  more sensitive to over-estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' And our results also show most of the errors  come from over-estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As for unlabelled slices, we still have the prior that  its MLS cannot be larger than the MLS of that specific volume δ, we propose to  incorporate an additional ceiling loss to punish the over-estimation:  lceil =  �  j  �  k  max(0, ||φjk|| − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  (5)  Overall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' the loss term is a combination of supervised loss and unsupervised loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  with a weight term controlling the relative importance of each loss term:  l = lhuber + w1lsmooth + u(i)(lmse + w2lceil),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  (6)  Diffusion model based ICH midline shift quantification  7  where w1 and w2 are two fixed weight terms and u(i) is a time-varying weight  term that is expected to gradually increase as the training iteration i progresses  so that the training can converge quickly through strong supervision first and  then refine and enhance generalizability via unsupervised loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  3  Experiments and Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='1  Data Acquisition and Preprocessing  We retrospectively collected anonymous thick-slice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' non-contrast head CT of pa-  tients who were admitted with head trauma or stroke symptoms and diagnosed  with various subtypes of intracranial hemorrhage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' including epidural hemor-  rhage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' subdural hemorrhage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' subarachnoid hemorrhage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' intraventricular hem-  orrhage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' and intraparenchymal hemorrhage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' between July 2019 and December  2019 in the Prince of Wales Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' a public hospital under the Hospital Au-  thority of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The ethics approval was obtained from the Joint Chinese  University of Hong Kong-New Territories East Cluster Clinical Research ethics  committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The eligible patients comprised 2793 CT volumes, among them 124  are MLS positive cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The MLS ranges between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='24mm and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='12mm, with  mean value of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='34mm and medium value of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='73mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The annotation was per-  formed by two trained physicians and verified by one experienced radiologist  (with over 10 years of clinical experience on ICH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The labeling process followed  a real clinical measurement pipeline, where the shifted landmark, anterior falx  point, and posterior falx point were pointed out, and the length of the vertical  line from the landmark to the line connecting the anterior falx point and the  posterior falx point was the measured MLS value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For each volume, a few slices  with large deformation were separately measured and annotated while the shift  of the largest one served as the case-level label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' On average, 4 out of 30 slices  of each volume were labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We discarded the first 8 and the last 5 slices as  they are mainly structures irrelevant to MLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For pre-processing, we adjusted  the pixel size of all images to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='86mm and then cropped or padded the resulting  images to the resolution of 256 × 256 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The HU window was set to 0 and  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We applied intensity clipping (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='5 and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='5 percentiles) and min-max nor-  malization (between -1 and 1) to each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Random rotation between −15◦  and 15◦ was used for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='2  Implementation Details  For the diffusion network, we use the network architecture designed in DDPM [15]  and set the noise level from 10−4 to 2 × 10−2 by linearly scheduling with  Ttrain = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For non-MLS image generation, we apply the Denoising Diffu-  sion Implicit Model (DDIM) [13] with 50 steps and set the noise scale to 15 to  shorten the generative time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We set the hyper-parameters as α = 1, β = 1, c = 3  and γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' u(i) is set from 1 to 10 with the linear schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The diffusion models  are trained by the AdamW optimizer with an initial learning rate of 1 × 10−4,  8  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Table 1: Comparison of different methods with 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Methods  Training data  Volume-wise  Slice-wise  Labeled  Unlabeled  MAE↓  (mm)  RMSE↓  (mm)  MAE↓  (mm)  RMSE↓  (mm)  Regression  ✓  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='91  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='90  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='56  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='16  Deformation  ✓  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='80  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='47  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='51  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='17  Mean-Teacher [19]  ✓  ✓  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='89  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='67  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='43  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='22  CPS [20]  ✓  ✓  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='72  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='42  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='38  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='15  Ours  ✓  ✓  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='43  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='25  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='09  batch size 4, for 2 × 105 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We up-sample the MLS positive data by 10×  when training the unconditional diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The deformation network is  trained by the AdamW optimizer with an initial learning rate of 1×10−4, batch  size 16, for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' All models are implemented with PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='1 using  one Nvidia GeForce RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='3  Quantification Accuracy and Deformation Quality  We evaluate the performance of our quantification strategy through mean abso-  lute error (MAE) and root mean square error (RMSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For volume-wise evalua-  tion, we measure the maximum deformation of each slice of the whole volume and  select the largest one as the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We also report the slice-wise evaluation,  which is calculated based on labeled slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This error can reflect how the models  perform on slices with relatively large deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Since existing MLS estima-  tion methods require different types of labels from ours, it is difficult to directly  compare with those methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We therefore first compare our deformation-based  strategy with a regression-based strategy, which uses DenseNet-121 [21] to di-  rectly predict the slice-wise MLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We also compare our proposed semi-supervised  learning approach with two popular semi-supervised learning methods, that are  Mean-Teacher [19] and Cross Pseudo Supervision (CPS) [20], which are imple-  mented into our deformation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The results are given in Table 1, which  are based on 5-fold cross-validations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  From the results, we can see that when only using labeled MLS slices for  model learning, our deformation strategy already shows better performance than  the regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This may attribute to that our deformation model learns  the knowledge of both MLS values and locations while a regression model only  captures the MLS value information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This difference can be further enlarged if  we consider slice-wise performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Moreover, all three semi-supervised learn-  ing methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Mean-Teacher, CPS, and ours, consistently improve the per-  formance of deformation prediction, showing the benefits and importance of  Diffusion model based ICH midline shift quantification  9  true:  pred:  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='00  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='43  true:  pred:  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='43  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='02  true:  pred:  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='23  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='78  true:  pred:  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='48  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='87  (a) Examples of predicted deformation on MLS images  true:  pred:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='60  true:  pred:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='75  true:  pred:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='87  true:  pred:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='69  (b) Examples of predicted deformation on non-MLS images  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 3: Predicted deformation on (a) MLS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' (b) non-MLS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The  regions with the largest deformation are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Slice-wise predicted MSL  and ground truth are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  incorporating unlabeled data into model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Our semi-supervised learn-  ing method based on diffusion models achieves better quantification results  than Mean-Teacher and CPS, significantly reducing the volume-wise MAE from  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='80mm to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='43mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' An interesting observation is that the unlabeled data con-  tribute more to the volume-wise evaluation than the slice-wise evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' By  inspecting the prediction, we find that the deformation prediction trained with  labeled data tends to overestimate the deformation of slices with little or no  deformation, which makes the volume-wise prediction error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As most un-  labeled data are slices with minor shifts, incorporating these data for semi-  supervised learning can impose constraints to avoid large deformation, which  greatly improves the model’s robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  We also visualize the predicted deformation field of several sample cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 3 (a), we can see the model can well posit the location where the  maximum shift appears and push it to its hypothetically normal counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  The largest deformation happens exactly at the site with the maximum shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' To  validate the robustness of our model, we also select several patients diagnosed  with no MLS and plot the predicted deformation of these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As can be  seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 3 (b), our method is able to provide a reasonable prediction for  non-MLS images by outputting much smaller values than that for MLS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Our model’s predictions for non-MLS images are not exactly zero are caused on  one hand by that even for a completely healthy person, the midline cannot be  perfectly aligned due to multiple factors such as scan pose, on the other hand, our  O10  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Methods  MAE↓  (mm)  RMSE↓  (mm)  Fully-supervised  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='61  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='47  + Representation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='22  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='69  Semi-supervised  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='61  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='24  + Representation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='45  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='05  Table 2: Effects of the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  0  200  400  600  800  t  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='4  Effects of noise level t  MAE  RMSE  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 4: Effects of the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  models tend to overestimate the shift because we are calculating the maximum  deformation as final measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='4  Ablation Study  We conduct several ablation experiments to study the effects of several compo-  nents in our proposed framework on the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The volume-wise  results reported are trained on four folders and tested on one folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Effects for representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We first conduct ablation studies to  verify that the latent feature extracted from the two diffusion models is truly  useful for deformation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' To this end, we select two deformation models,  one trained with only labeled data and the other using semi-supervised learning,  and compare their performance with and without the extracted representation  as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The results are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As expected, incorporating the rep-  resentation can improve the model performance in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  The noise level is an important component of diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Only with a  proper noise level, can the model accurately estimate the deviation of the image  toward the negative sample space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Therefore, we do inference with multiple noise  levels and compare its effect on model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The results are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Our model is very robust towards this hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As long as t is  not too small, the model gives very similar performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The best performance  appears in the middle when t = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This is reasonable as small noise fails to  corrupt the original image thus degenerating the performance of score estimation  while large noise may obscure too many details of the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Quantity of unlabeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' To verify the usefulness of unlabeled images,  we conduct ablation studies on the number of unlabeled images used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For each  experiment, we randomly sample 20%, 40%, 60%, and 80% volumes, and we  incorporate unlabeled slices of these volumes for semi-supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For  the rest volumes, we are only using the labeled slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We also do one experiment  that completely removes the uses of unlabeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For each experiment, the  pre-trained diffusion models are the same, which uses all the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In other  words, these unlabeled images somehow still contribute to the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As can be seen, the model performance  and robustness can be enhanced as we incorporate more unlabeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This  Diffusion model based ICH midline shift quantification  11  (a)  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 5: Results of our ablation experiments in terms of: (a) proportion of unla-  beled data used, and (b) proportion of negative data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  provides strong evidence for our claim that our model truly learns valuable  information from unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Quantity of non-MLS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' To further measure the benefits of includ-  ing non-MLS cases, we conduct another ablation study on the proportion of  non-MLS cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Non-MLS cases are used to train diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' As currently,  the amount of non-MLS cases is much higher than MLS cases, we upsample the  MLS cases so that their quantities are approximately the same when training the  unconditional diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' For ablation, we first downsample the non-MLS  data so that their quantity is 1×, 5×, and 10× that of the MLS cases, and then  upsample the MLS cases to make them balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' From the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 5 (b),  we find model performance improves with more non-MLS cases incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In-  creasing non-MLS cases can help train diffusion models and further improve the  quality of generated images and extracted feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' However, this  effect will soon be saturated as the amount of MLS cases is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' This  can be a bottleneck for effectively using the non-MLS cases as it is challenging  to train unconditional diffusion models with such imbalanced datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  4  Conclusions and Future Work  In this paper, we propose a novel framework based on deformation field esti-  mation to automatically measure the brain MLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' The labels we are using are  sparse which can greatly alleviate the labeling workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We also propose a  semi-supervised learning strategy based on diffusion models which significantly  improves the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Experiments on a clinic dataset show our meth-  ods can achieve satisfying performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' We also verify that using unlabeled data  and non-MLS cases can truly help improve the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Our methods have several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' First, the model performance highly  relies on pre-trained diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Training diffusion models with extremely  imbalanced data requires great effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Second, the measurement results exhibit  randomness due to noise corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Finally, the measurement results are prone  to overestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Our future work will figure out solutions for these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  Effects of unlabeled data  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content='0  0  1X  5x  10x  all  ratio of negative cases / positive cases12  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Caceres, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Goldstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : Intracranial hemorrhage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Emerg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' North  Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=' Quattrocchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : Quantification of midline shift as a predictor of poor out-  come following head injury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: Surgical Neurology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 35(3), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 183–188, (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : Defining the optimal midline  shift threshold to predict poor outcome in patients with supratentorial spontaneous  intracerebral hemorrhage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: Neurocritical care, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 28(3), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='314–321, (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Liao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', and Xiao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=': Brain Midline Shift Measurement and Its Au-  tomation: A Review of Techniques and Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: International Journal of  Biomedical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 4303161, (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Pisov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : Incorporating task-specific structural knowledge into CNNs for  brain midline shift detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: Suzuki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=') ML-CDS/IMIMIC -2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  LNCS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 11797, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 30–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Springer, Cham (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Liang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : Context-aware refinement network incorporating  structural connectivity prior for brain midline delineation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: Martel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=') MICCAI 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' LNCS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 12267, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 208–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Springer, Cham (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Wei, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : Regression-based line detection network for delineation of largely  deformed brain midline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: Shen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=') MICCAI 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' LNCS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 11766,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=' Springer, Cham (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=' Qin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=', Shang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Chang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Feng,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : 3D Brain Midline Delineation for Hematoma Patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: Bruijne, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=') MICCAI 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' LNCS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 12905, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' 510–518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Springer, Cham (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=' Nguyen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=', Yoo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Chekkoury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Eibenberger, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : Brain midline shift  detection and quantification by a cascaded deep network pipeline on non-contrast  computed tomography scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=': Generative Adversarial Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=' In: NeurIPS 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=': Generative Modeling by Estimating Gradients of the  Data Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: NeurIPS 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
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+page_content=': Understanding Diffusion Models: A Unified Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' arXiv preprint  arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='11970 (2022)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Tarvainen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Valpala, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=': Mean teachers are better role models: Weight-averaged  consistency targets improve semi-supervised deep learning results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: NeurIPS 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Zeng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=': Semi-Supervised Semantic Segmentation  with Cross Pseudo Supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: CVPR 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', van der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=', Weinberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' : Densely Connected  Convolutional Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
+page_content=' In: CVPR 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfjPgK/content/2301.00409v1.pdf'}
diff --git a/dtAzT4oBgHgl3EQf3f5y/content/tmp_files/2301.01830v1.pdf.txt b/dtAzT4oBgHgl3EQf3f5y/content/tmp_files/2301.01830v1.pdf.txt
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@@ -0,0 +1,590 @@
+arXiv:2301.01830v1  [hep-ph]  4 Jan 2023
+Search for an Ultraviolet Zero in the Seven-Loop Beta Function of the λφ4
+4 Theory
+Robert Shrock
+C. N. Yang Institute for Theoretical Physics and
+Department of Physics and Astronomy
+Stony Brook University, Stony Brook, NY 11794
+We investigate whether the seven-loop beta function of the λφ4
+4 theory exhibits evidence for
+an ultraviolet zero. In addition to a direct analysis of the beta function, we calculate and study
+Pad´e approximants and discuss effects of scheme transformations on the results. Confirming and
+extending our earlier studies of the five-loop and six-loop beta functions, we find that in the range
+of λ where the perturbative calculation of the seven-loop beta function is reliable, the theory does
+not exhibit evidence for an ultraviolet zero.
+I.
+INTRODUCTION
+In this paper we consider the renormalization-group
+(RG) behavior of the λφ4 field theory in d = 4 spacetime
+dimensions, where φ is a real scalar field. This theory,
+commonly denoted φ4
+4, is described by the Lagrangian
+L = 1
+2(∂νφ)(∂νφ) − m2
+2 φ2 − λ
+4! φ4 .
+(1.1)
+The Lagrangian (1.1) is invariant under the global dis-
+crete Z2 symmetry φ → −φ. Quantum loop corrections
+lead to a dependence of the physical quartic coupling
+λ = λ(µ) on the Euclidean energy/momentum scale µ
+at which this coupling is measured. The dependence of
+λ(µ) on µ is described by the RG beta function of the
+theory, βλ = dλ/dt, or equivalently, βa = da/dt, where
+dt = d ln µ [1] and
+a ≡
+λ
+(4π)2 .
+(1.2)
+(The argument µ will often be suppressed in the nota-
+tion.) Since we will investigate the properties of the the-
+ory for large µ in the ultraviolet (UV), the value of m2
+will not play an important role in our analysis. For tech-
+nical convenience, we assume that m2 is positive. At a
+reference scale µ0, the quartic coupling λ(µ0) is taken to
+be positive for the stability of the theory. The one-loop
+term in this beta function has a positive coefficient, so
+that for small λ, βλ > 0 and hence as µ → 0, the cou-
+pling λ(µ) → 0, i.e., the theory is infrared (IR)-free. This
+perturbative result is in agreement with nonperturbative
+approaches [2]; some reviews include [3, 4].
+The beta function βa has the series expansion
+βa = a
+∞
+�
+ℓ=1
+bℓ aℓ .
+(1.3)
+The n-loop (nℓ) beta function, denoted βa,nℓ, is given by
+Eq. (1.3) with the upper limit of the loop summation in-
+dex ℓ = n instead of ℓ = ∞. The one-loop and two-loop
+terms in βa are independent of the scheme used for regu-
+larization and renormalization, while terms of loop order
+ℓ ≥ 3 are scheme-dependent [5, 6]. For the O(N) λ|⃗φ|4
+theory with an N-component field, ⃗φ = (φ1, ..., φN), the
+coefficients b1, b2, and b3 were calculated in [5]. Higher-
+loop coefficients bℓ with ℓ ≥ 3 have been computed using
+the MS minimal subtraction scheme [7, 8]. A calculation
+of b5 and discussion of earlier computations of b4 and
+b5 (e.g., [9–11]) was given in [4, 12]. The coefficient b6
+was calculated for N = 1 in [13] and for general N in
+[14]. Most recently, the seven-loop coefficient b7 was cal-
+culated in [15]. In analyzing the series expansion (1.3),
+one recalls that it is an asymptotic expansion and the
+large-order behavior has been the subject of extensive
+study [16], including [17] and references therein.
+An interesting question is whether, for the region of
+λ where a perturbative calculation of βλ is reliable, this
+beta function exhibits evidence for a zero at some (pos-
+itive) value of the quartic coupling. This would be an
+ultraviolet fixed point (UVFP) of the renormalization
+group, i.e., as µ → ∞, λ(µ) would approach this value
+(from below). In previous work we have investigated this
+question up to the five-loop order for the O(N) λ|⃗φ|4
+theory in [18] and up to the six-loop order for the real
+λφ4 theory in [19] and the O(N) λ|⃗φ|4 theory in [20],
+finding evidence against such a UVFP. In the present
+paper, using the results of [15], we extend our analysis to
+the seven-loop level. Our analysis in [20] covered a large
+range of specific N values and also included an argument
+for the absence of a UV zero in the (rescaled) n-loop beta
+function at large N (see Eqs. (3.12)-(3.13) in [20]). Thus,
+it will suffice to focus on the N = 1 theory here.
+In view of this previous evidence against a UV zero in
+βλ and associated UVFP in the O(N) λ|⃗φ|4 theory, it is
+worthwhile to mention one case where an IR-free quan-
+tum field theory is known to have a UVFP, namely, the
+nonlinear O(N) σ model in d = 2 + ǫ spacetime dimen-
+sions. In this theory, an exact solution was obtained in
+the limit N → ∞ with λ(µ)N = x(µ) a fixed function of
+µ and yielded the beta function
+βx = dx
+dt = ǫx
+�
+1 −
+x
+xUV
+�
+(1.4)
+for small ǫ, where xUV = 2πǫ is a UV fixed point of the
+renormalization group [21]. Since the leading term in βx
+is positive for ǫ > 0, this theory is IR-free. Thus, in this
+nonlinear O(N) σ model in d = 2 + ǫ dimensions, the
+
+2
+coupling x(µ) flows (monotonically) from x = 0 at µ = 0
+to x = xUV as µ → ∞. Note that by making ǫ ≪ 1 one
+can arrange that the UVFP at xUV = 2πǫ occurs at an
+arbitrarily small value of the scaled coupling x.
+This paper is organized as follows. In Section II we re-
+view some relevant background. In Section III we present
+the results of our analysis of the seven-loop beta function.
+Section IV contains a further analysis of this question of
+a UV zero using Pad´e approximants, while Section V dis-
+cusses effects of scheme transformations. Our conclusions
+are given in Section VI.
+II.
+BETA FUNCTION
+The n-loop truncation of (1.3), denoted βa,nℓ, is a poly-
+nomial in a of degree n + 1 having an overall factor of
+a2. We may extract this factor and define a reduced beta
+function
+βa,r =
+βa
+βa,1ℓ
+=
+βa
+b1a2
+= 1 + 1
+b1
+∞
+�
+ℓ=2
+bℓaℓ−1 .
+(2.1)
+The n-loop truncation of βa,r, denoted βa,r,nℓ ≡ Rn, is
+defined by taking the upper limit of the sum in (2.1) to
+be ℓ = n rather than ℓ = ∞. .
+The first two coefficients in the beta function of this
+theory are b1 = 3 and b2 = −17/3 [5]. The coefficients bℓ
+with 3 ≤ ℓ ≤ 7 and the resultant higher-loop beta func-
+tion discussed below, are calculated in the MS scheme.
+The coefficients up to the five-loop level are [4, 5, 9, 12]
+b3 = 145
+8
++ 12ζ3 = 32.5497 ,
+(2.2)
+b4 = −3499
+48
+− 78ζ3 + 18ζ4 − 120ζ5
+= −271.606 ,
+(2.3)
+and
+b5 = 764621
+2304
++ 7965
+16 ζ3 − 1189
+8
+ζ4 + 987ζ5 + 45ζ2
+3
+− 675
+2 ζ6 + 1323ζ7
+= 2848.57 ,
+(2.4)
+where the floating-point values are given to the indicated
+accuracy and
+ζs =
+∞
+�
+n=1
+1
+ns
+(2.5)
+is the Riemann zeta function. If s = 2r is even, then
+ζs can be expressed as a rational number times π2r,
+namely ζ2r = (−1)r+1B2r(2π)2r/[2(2r)!], where Bn are
+the Bernoulli numbers; however, we leave these ζ2r in
+their generic form here and below. The six-loop coeffi-
+cient is [13, 14]
+b6 = −18841427
+11520
+− 779603
+240
+ζ3 + 16989
+16
+ζ4 − 63723
+10
+ζ5 − 8678
+5
+ζ2
+3 + 6691
+2
+ζ6 + 162ζ3ζ4 − 63627
+5
+ζ7
+− 4704ζ3ζ5 + 264543
+25
+ζ8 − 51984
+25
+ζ3,5 − 768ζ3
+3 − 46112
+3
+ζ9
+= −34776.13 ,
+(2.6)
+where [22]
+ζ3,5 =
+�
+m>n≥1
+1
+n3m5 .
+(2.7)
+The seven-loop coefficient is considerably more compli-
+cated than b6, and we refer the reader to [15] for the
+analytic expression. The numerical value is
+b7 = 474651.0 .
+(2.8)
+Thus, in summary, the seven-loop beta function of the
+λφ4 theory (calculated in the MS scheme), is
+βa,7ℓ = a2�
+3 − 17
+3 a + 32.5497a2 − 271.606a3
++ 2848.57a4 − 34776.1a5 + 474651a6�
+.
+(2.9)
+III.
+ZEROS OF THE n-LOOP BETA FUNCTION
+UP TO LOOP ORDER n = 7
+In this section we investigate a possible UV zero, de-
+noted aUV,nℓ, of the n-loop beta function, βa,nℓ.
+The
+double zero of βa,nℓ at a = 0 is always present (indepen-
+dent of n); this is an infrared zero and hence will not be
+of interest here.
+
+3
+A necessary condition for there to be robust evidence
+for a UV zero in the beta function of an IR-free the-
+ory is that the values calculated at successive loop orders
+should be close to each other.
+Although the two-loop
+beta function βa,2ℓ does have a UV zero, at aUV,2ℓ =
+9/17 = 0.52941, we found that the three-loop beta func-
+tion βa,3ℓ has no UV zero and, while a UV zero is present
+in βa,4ℓ, it occurs at a considerably smaller value, namely
+aUV,4ℓ = 0.23332. At the five-loop level, βa,5ℓ has no UV
+zero, while at the six-loop level, although βa,6ℓ has a UV
+zero, it occurs at a still smaller value, aUV,6ℓ = 0.16041
+[18, 19].
+Thus, the results of this analysis show that
+the necessary condition that the beta function calculated
+to successively higher loop order should exhibit values
+of aUV,nℓ that are close to each other is not satisfied by
+this theory. At seven-loop order, using βa,7ℓ from [15],
+we find that this function has no physical UV zero. In-
+stead, the zeros are comprised of three complex-conjugate
+pairs, −0.102135±0.079848i, 0.0142348±0.136854i, and
+0.124533 ± 0.0659940i. Summarizing,
+aUV,2ℓ = 0.52941,
+aUV,4ℓ = 0.23332,
+aUV,6ℓ = 0.16041
+no aUV,nℓ for n = 3, 5, 7.
+(3.1)
+The calculations up to seven loops show a pattern,
+namely that for even n = 2, 4, 6, βa,nℓ has a zero, aUV,nℓ,
+but the values for different n are not close to each other,
+while for odd n = 1, 3, 5, 7, βa,nℓ has no UV zero.
+In Fig. 1 we plot the n-loop beta functions for 2 ≤ n ≤
+7 loops. Another way to show this information is via the
+n-loop reduced beta function, βa,r,nℓ = Rn. We plot Rn
+in Fig. 2 for 2 ≤ n ≤ 7. The results discussed above are
+evident in these figures. First, one may inquire how large
+is the interval in a over which the calculations of βa,nℓ to
+the respective n-loop orders are in mutual agreement. As
+one can see from Figs. 1 and 2, the n-loop beta functions
+βa,nℓ with 2 ≤ n ≤ 7 only agree with each other well over
+the small interval of couplings 0 ≤ a <∼ 0.05. As shown
+in Fig. 1, the βa,nℓ with even n = 2, 4, 6 reach maxima
+and then decrease, crossing the (positive) real axis at
+different values listed in Eq. (3.1), while the βa,nℓ with
+odd n increase monotonically with a.
+This seven-loop
+analysis confirms and extends our conclusions in [19, 20]
+at the six-loop level that the zero in the two-loop beta
+function of the λφ4 theory occurs at too large a value of
+a for the perturbative calculation to be reliable.
+IV.
+ANALYSIS WITH PAD´E APPROXIMANTS
+One can gain further insight into the behavior of the
+beta function by the use of Pad´e approximants (PAs).
+We carried out this analysis up to the six-loop level in
+[19, 20], finding no indication of a physical UV zero, and
+here we extend it to the seven-loop level. Since the double
+zero in βa,nℓ at a = 0 is not relevant to the question of a
+UV zero, we use the reduced beta function βa,r,nℓ for this
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+a
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+beta
+FIG. 1:
+Plot of the n-loop β function βa,nℓ as a function of a for
+(i) n = 2 (red, solid), (ii) n = 3 (green, dashed), (iii) n = 4 (blue,
+dotted), (iv) n = 5 (black, dot-dashed), (v) n = 6 (cyan, solid),
+and (vi) n = 7 (brown, solid). At a = 0.16, going from bottom to
+top, the curves are for n = 6, n = 4, n = 2, n = 3, n = 5, n = 7.
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+a
+0.0
+0.5
+1.0
+1.5
+R_n
+FIG. 2: Plot of the ratio Rn of the n-loop beta function βa,nℓ
+divided by βa,1ℓ, as a function of a for (i) n = 2 (red, solid), (ii)
+n = 3 (green, dashed), (iii) n = 4 (blue, dotted), (iv) n = 5 (black,
+dot-dashed), (v) n = 6 (cyan, solid), and (vi) n = 7 (brown, solid).
+At a = 0.16, going from bottom to top, the curves are for n = 6,
+n = 4, n = 2, n = 3, n = 5, and n = 7.
+Pad´e analysis. The [p, q] Pad´e approximant to βa,r,nℓ is
+the rational function [23]
+[p, q]βa,r,nℓ =
+1 + �p
+j=1 rjaj
+1 + �q
+k=1 sk ak
+(4.1)
+with p + q = n − 1, where the coefficients rj and sj are
+independent of a. At seven-loop order, we can calculate
+
+4
+the Pad´e approximants [p, q]βa,r,7ℓ with [p, q] taking on
+the values [6,0], [5,1], [4,2], [3,3], [2,4], [1,5], and [0,6].
+Since the loop order is understood, we write [p, q]βa,r,7ℓ ≡
+[p, q] for brevity of notation. The PA [6,0] is equivalent
+to βa,r,7ℓ itself, which we have already analyzed, and the
+PA [0,6] has no zeros, so we focus here on the remaining
+five Pad´e approximants.
+We list our results for these Pad´e approximants to
+βa,r,7ℓ below:
+[5, 1] = 1 + 11.760a − 14.931a2 + 57.552a3 − 286.17a4 + 1367.8a5
+1 + 13.649a
+,
+(4.2)
+[4, 2] = 1 + 20.541a + 75.687a2 − 49.670a3 + 81.973a4
+1 + 22.430a + 107.21a2
+,
+(4.3)
+[3, 3] = 1 + 25.073a + 152.81a2 + 155.99a3
+1 + 26.962a + 192.89a2 + 318.33a3 ,
+(4.4)
+[2, 4] =
+1 + 22.314a + 103.55a2
+1 + 24.203a + 138.42a2 + 89.390a3 − 91.252a4 ,
+(4.5)
+[1, 5] =
+1 + 14.023a
+1 + 15.912a + 19.205a2 − 45.828a3 + 196.10a4 − 910.03a5 .
+(4.6)
+We recall some necessary requirements for a zero of a
+[p, q] Pad´e approximant to be physically relevant. These
+include the requirement that this zero should occur on
+the positive real axis in the complex a plane and the
+requirement that this zero of the PA should be closer to
+the origin a = 0 than any pole on the real positive a-axis,
+since otherwise the pole would dominate the IR to UV
+flow starting at the origin. If a Pad´e approximant were
+to exhibit such a zero, then one would proceed to inquire
+how close it is to any of the aUV,nℓ in Eq. (3.1). However,
+we find that none of these Pad´e approximants (4.2)-(4.6)
+has a zero on the positive real a axis.
+Explicitly, the
+[5,1] PA has two complex-conjugate pairs of zeros at a =
+−0.12719±0.26046i and a = 0.26922±0.20930i, together
+with a real zero at a = −0.074837.
+This real zero is
+part of a nearly coincident pole-zero pair, with the pole
+of the [5,1] PA being located at a = −0.073267. The
+appearance of a nearly coincident pole-zero pair close to
+a point a0 in a [p, q] Pad´e approximant is typically an
+indication that the function that the PA is fitting has
+neither a pole nor a zero in the local neighborhood of
+a0, since as the locations of the nearly coincident pole-
+zero pair approach each other, they simply divide out in
+the ratio (4.1). Each of the Pad´e approximants that we
+calculate here has a pole-zero pair. The [4,2] PA has zeros
+at the complex-conjugate pair a = 0.42009 ± 0.96575i,
+together with the real values a = {−0.16929, −0.064970}
+and poles at a = {−0.14481, −0.064414}. The [3,3] PA
+has zeros at a = {−0.78531, −0.13282, −0.061458}, and
+poles at a = {−0.42342, −0.12140, −0.061112}. The
+[2,4] PA has zeros at a = {−0.15193, −0.063563}, and
+poles at a = {−0.69186, −0.13432, −0.063100, 1.8689}.
+Finally, the [1,5] PA has a zero at a = −0.071313 and
+poles at a = {−0.22780, −0.070185, 0.44160, 0.035937±
+0.39287i}. Thus, our analysis with Pad´e approximants of
+the seven-loop beta function yields the same conclusion
+as our analysis of the beta function itself, namely that
+there is no evidence for a stable, reliably perturbatively
+calculable UV zero up to this seven-loop level.
+V.
+EFFECTS OF SCHEME
+TRANSFORMATIONS
+Since the terms in the beta function at loop order n ≥ 3
+are scheme-dependent, it is necessary to assess the effect
+of scheme transformations in an analysis of zeros of a
+higher-loop beta function. A scheme transformation can
+be expressed as a mapping between a and a transformed
+coupling a′,
+a = a′f(a′) ,
+(5.1)
+where f(a′) is the scheme transformation function. Since
+this transformation has no effect in the free theory, one
+has f(0) = 1. We consider f(a′) functions that are ana-
+lytic about a = a′ = 0 and hence can be expanded in the
+form
+f(a′) = 1 +
+smax
+�
+s=1
+ks(a′)s ,
+(5.2)
+
+5
+where the ks are constants and smax may be finite or
+infinite. The beta function in the transformed scheme,
+βa′ = da′/d ln µ, has the expansion
+βa′ = a′
+∞
+�
+ℓ=1
+b′
+ℓ(a′)ℓ .
+(5.3)
+In [24], formulas were derived for the b′
+ℓ in terms of bℓ
+and the ks. In addition to b′
+1 = b1 and b′
+2 = b2, these are
+b′
+3 = b3 + k1b2 + (k2
+1 − k2)b1 ,
+(5.4)
+b′
+4 = b4 + 2k1b3 + k2
+1b2 + (−2k3
+1 + 4k1k2 − 2k3)b1 , (5.5)
+and so forth for higher ℓ. These results are applicable
+to the study of both an IR zero in the beta function of
+an asymptotically free theory and a possible UV zero in
+the beta function of an IR-free theory. They were exten-
+sively applied to assess scheme dependence in higher-loop
+studies of an IR fixed point in asymptotically free non-
+Abelian gauge theories [24–28].
+For the present λφ4 theory, a study of scheme depen-
+dence was carried out in [18]. It was shown that even
+when one shifts to a scheme different from the usual MS
+scheme, the beta function still does not satisfy a requi-
+site condition for a physical UV zero, namely that the
+value of this zero (in a given scheme) should not change
+strongly when it is calculated to successive loop orders.
+This result from [18] also holds in the same way in the
+present seven-loop context.
+VI.
+CONCLUSIONS
+In this paper we have investigated whether the real
+scalar field theory with a λφ4 interaction exhibits evi-
+dence of an ultraviolet zero in the beta function.
+Us-
+ing the seven-loop coefficient b7 from [15], our present
+study extends our previous six-loop study in [19, 20] to
+the seven-loop level. Our work includes a study of the
+seven-loop beta function itself, together with an analysis
+of Pad´e approximants. We conclude that, for the range
+of couplings where the perturbative calculation of this
+beta function may be reliable, it does not exhibit robust
+evidence for an ultraviolet zero.
+Acknowledgments
+I would like to thank Oliver Schnetz for valuable dis-
+cussions on [15]. This research was supported in part by
+the U.S. National Science Foundation Grant NSF-PHY-
+22-15093.
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+
diff --git a/dtAzT4oBgHgl3EQf3f5y/content/tmp_files/load_file.txt b/dtAzT4oBgHgl3EQf3f5y/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..912e72909a706d743d55b04417f116e8c2ce5438
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@@ -0,0 +1,594 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf,len=593
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='01830v1  [hep-ph]  4 Jan 2023  Search for an Ultraviolet Zero in the Seven-Loop Beta Function of the λφ4  4 Theory  Robert Shrock  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Yang Institute for Theoretical Physics and  Department of Physics and Astronomy  Stony Brook University, Stony Brook, NY 11794  We investigate whether the seven-loop beta function of the λφ4  4 theory exhibits evidence for  an ultraviolet zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In addition to a direct analysis of the beta function, we calculate and study  Pad´e approximants and discuss effects of scheme transformations on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Confirming and  extending our earlier studies of the five-loop and six-loop beta functions, we find that in the range  of λ where the perturbative calculation of the seven-loop beta function is reliable, the theory does  not exhibit evidence for an ultraviolet zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  INTRODUCTION  In this paper we consider the renormalization-group  (RG) behavior of the λφ4 field theory in d = 4 spacetime  dimensions, where φ is a real scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' This theory,  commonly denoted φ4  4, is described by the Lagrangian  L = 1  2(∂νφ)(∂νφ) − m2  2 φ2 − λ  4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' φ4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1)  The Lagrangian (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1) is invariant under the global dis-  crete Z2 symmetry φ → −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Quantum loop corrections  lead to a dependence of the physical quartic coupling  λ = λ(µ) on the Euclidean energy/momentum scale µ  at which this coupling is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The dependence of  λ(µ) on µ is described by the RG beta function of the  theory, βλ = dλ/dt, or equivalently, βa = da/dt, where  dt = d ln µ [1] and  a ≡  λ  (4π)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2)  (The argument µ will often be suppressed in the nota-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=') Since we will investigate the properties of the the-  ory for large µ in the ultraviolet (UV), the value of m2  will not play an important role in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' For tech-  nical convenience, we assume that m2 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' At a  reference scale µ0, the quartic coupling λ(µ0) is taken to  be positive for the stability of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The one-loop  term in this beta function has a positive coefficient, so  that for small λ, βλ > 0 and hence as µ → 0, the cou-  pling λ(µ) → 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=', the theory is infrared (IR)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' This  perturbative result is in agreement with nonperturbative  approaches [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' some reviews include [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  The beta function βa has the series expansion  βa = a  ∞  �  ℓ=1  bℓ aℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3)  The n-loop (nℓ) beta function, denoted βa,nℓ, is given by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3) with the upper limit of the loop summation in-  dex ℓ = n instead of ℓ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The one-loop and two-loop  terms in βa are independent of the scheme used for regu-  larization and renormalization, while terms of loop order  ℓ ≥ 3 are scheme-dependent [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' For the O(N) λ|⃗φ|4  theory with an N-component field, ⃗φ = (φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=', φN), the  coefficients b1, b2, and b3 were calculated in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Higher-  loop coefficients bℓ with ℓ ≥ 3 have been computed using  the MS minimal subtraction scheme [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' A calculation  of b5 and discussion of earlier computations of b4 and  b5 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=', [9–11]) was given in [4, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The coefficient b6  was calculated for N = 1 in [13] and for general N in  [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Most recently, the seven-loop coefficient b7 was cal-  culated in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In analyzing the series expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3),  one recalls that it is an asymptotic expansion and the  large-order behavior has been the subject of extensive  study [16], including [17] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  An interesting question is whether, for the region of  λ where a perturbative calculation of βλ is reliable, this  beta function exhibits evidence for a zero at some (pos-  itive) value of the quartic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' This would be an  ultraviolet fixed point (UVFP) of the renormalization  group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=', as µ → ∞, λ(µ) would approach this value  (from below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In previous work we have investigated this  question up to the five-loop order for the O(N) λ|⃗φ|4  theory in [18] and up to the six-loop order for the real  λφ4 theory in [19] and the O(N) λ|⃗φ|4 theory in [20],  finding evidence against such a UVFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In the present  paper, using the results of [15], we extend our analysis to  the seven-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Our analysis in [20] covered a large  range of specific N values and also included an argument  for the absence of a UV zero in the (rescaled) n-loop beta  function at large N (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='12)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='13) in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Thus,  it will suffice to focus on the N = 1 theory here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  In view of this previous evidence against a UV zero in  βλ and associated UVFP in the O(N) λ|⃗φ|4 theory, it is  worthwhile to mention one case where an IR-free quan-  tum field theory is known to have a UVFP, namely, the  nonlinear O(N) σ model in d = 2 + ǫ spacetime dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In this theory, an exact solution was obtained in  the limit N → ∞ with λ(µ)N = x(µ) a fixed function of  µ and yielded the beta function  βx = dx  dt = ǫx  �  1 −  x  xUV  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='4)  for small ǫ, where xUV = 2πǫ is a UV fixed point of the  renormalization group [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Since the leading term in βx  is positive for ǫ > 0, this theory is IR-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Thus, in this  nonlinear O(N) σ model in d = 2 + ǫ dimensions, the  2  coupling x(µ) flows (monotonically) from x = 0 at µ = 0  to x = xUV as µ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Note that by making ǫ ≪ 1 one  can arrange that the UVFP at xUV = 2πǫ occurs at an  arbitrarily small value of the scaled coupling x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In Section II we re-  view some relevant background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In Section III we present  the results of our analysis of the seven-loop beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Section IV contains a further analysis of this question of  a UV zero using Pad´e approximants, while Section V dis-  cusses effects of scheme transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Our conclusions  are given in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  BETA FUNCTION  The n-loop truncation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3), denoted βa,nℓ, is a poly-  nomial in a of degree n + 1 having an overall factor of  a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' We may extract this factor and define a reduced beta  function  βa,r =  βa  βa,1ℓ  =  βa  b1a2  = 1 + 1  b1  ∞  �  ℓ=2  bℓaℓ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1)  The n-loop truncation of βa,r, denoted βa,r,nℓ ≡ Rn, is  defined by taking the upper limit of the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1) to  be ℓ = n rather than ℓ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  The first two coefficients in the beta function of this  theory are b1 = 3 and b2 = −17/3 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The coefficients bℓ  with 3 ≤ ℓ ≤ 7 and the resultant higher-loop beta func-  tion discussed below, are calculated in the MS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  The coefficients up to the five-loop level are [4, 5, 9, 12]  b3 = 145  8  + 12ζ3 = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5497 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2)  b4 = −3499  48  − 78ζ3 + 18ζ4 − 120ζ5  = −271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='606 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3)  and  b5 = 764621  2304  + 7965  16 ζ3 − 1189  8  ζ4 + 987ζ5 + 45ζ2  3  − 675  2 ζ6 + 1323ζ7  = 2848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='57 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='4)  where the floating-point values are given to the indicated  accuracy and  ζs =  ∞  �  n=1  1  ns  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5)  is the Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' If s = 2r is even, then  ζs can be expressed as a rational number times π2r,  namely ζ2r = (−1)r+1B2r(2π)2r/[2(2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' ], where Bn are  the Bernoulli numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' however, we leave these ζ2r in  their generic form here and below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The six-loop coeffi-  cient is [13, 14]  b6 = −18841427  11520  − 779603  240  ζ3 + 16989  16  ζ4 − 63723  10  ζ5 − 8678  5  ζ2  3 + 6691  2  ζ6 + 162ζ3ζ4 − 63627  5  ζ7  − 4704ζ3ζ5 + 264543  25  ζ8 − 51984  25  ζ3,5 − 768ζ3  3 − 46112  3  ζ9  = −34776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='13 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='6)  where [22]  ζ3,5 =  �  m>n≥1  1  n3m5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='7)  The seven-loop coefficient is considerably more compli-  cated than b6, and we refer the reader to [15] for the  analytic expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The numerical value is  b7 = 474651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='8)  Thus, in summary, the seven-loop beta function of the  λφ4 theory (calculated in the MS scheme), is  βa,7ℓ = a2�  3 − 17  3 a + 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5497a2 − 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='606a3  + 2848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='57a4 − 34776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1a5 + 474651a6�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='9)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  ZEROS OF THE n-LOOP BETA FUNCTION  UP TO LOOP ORDER n = 7  In this section we investigate a possible UV zero, de-  noted aUV,nℓ, of the n-loop beta function, βa,nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  The  double zero of βa,nℓ at a = 0 is always present (indepen-  dent of n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' this is an infrared zero and hence will not be  of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  3  A necessary condition for there to be robust evidence  for a UV zero in the beta function of an IR-free the-  ory is that the values calculated at successive loop orders  should be close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Although the two-loop  beta function βa,2ℓ does have a UV zero, at aUV,2ℓ =  9/17 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='52941, we found that the three-loop beta func-  tion βa,3ℓ has no UV zero and, while a UV zero is present  in βa,4ℓ, it occurs at a considerably smaller value, namely  aUV,4ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='23332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' At the five-loop level, βa,5ℓ has no UV  zero, while at the six-loop level, although βa,6ℓ has a UV  zero, it occurs at a still smaller value, aUV,6ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='16041  [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Thus, the results of this analysis show that  the necessary condition that the beta function calculated  to successively higher loop order should exhibit values  of aUV,nℓ that are close to each other is not satisfied by  this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' At seven-loop order, using βa,7ℓ from [15],  we find that this function has no physical UV zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In-  stead, the zeros are comprised of three complex-conjugate  pairs, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='102135±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='079848i, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='0142348±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='136854i, and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='124533 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='0659940i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Summarizing,  aUV,2ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='52941,  aUV,4ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='23332,  aUV,6ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='16041  no aUV,nℓ for n = 3, 5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1)  The calculations up to seven loops show a pattern,  namely that for even n = 2, 4, 6, βa,nℓ has a zero, aUV,nℓ,  but the values for different n are not close to each other,  while for odd n = 1, 3, 5, 7, βa,nℓ has no UV zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 1 we plot the n-loop beta functions for 2 ≤ n ≤  7 loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Another way to show this information is via the  n-loop reduced beta function, βa,r,nℓ = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' We plot Rn  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 2 for 2 ≤ n ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The results discussed above are  evident in these figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' First, one may inquire how large  is the interval in a over which the calculations of βa,nℓ to  the respective n-loop orders are in mutual agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' As  one can see from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 1 and 2, the n-loop beta functions  βa,nℓ with 2 ≤ n ≤ 7 only agree with each other well over  the small interval of couplings 0 ≤ a <∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 1, the βa,nℓ with even n = 2, 4, 6 reach maxima  and then decrease, crossing the (positive) real axis at  different values listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1), while the βa,nℓ with  odd n increase monotonically with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  This seven-loop  analysis confirms and extends our conclusions in [19, 20]  at the six-loop level that the zero in the two-loop beta  function of the λφ4 theory occurs at too large a value of  a for the perturbative calculation to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  ANALYSIS WITH PAD´E APPROXIMANTS  One can gain further insight into the behavior of the  beta function by the use of Pad´e approximants (PAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  We carried out this analysis up to the six-loop level in  [19, 20], finding no indication of a physical UV zero, and  here we extend it to the seven-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Since the double  zero in βa,nℓ at a = 0 is not relevant to the question of a  UV zero, we use the reduced beta function βa,r,nℓ for this  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='6  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5  beta  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 1:  Plot of the n-loop β function βa,nℓ as a function of a for  (i) n = 2 (red, solid), (ii) n = 3 (green, dashed), (iii) n = 4 (blue,  dotted), (iv) n = 5 (black, dot-dashed), (v) n = 6 (cyan, solid),  and (vi) n = 7 (brown, solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' At a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='16, going from bottom to  top, the curves are for n = 6, n = 4, n = 2, n = 3, n = 5, n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='6  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5  R_n  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 2: Plot of the ratio Rn of the n-loop beta function βa,nℓ  divided by βa,1ℓ, as a function of a for (i) n = 2 (red, solid), (ii)  n = 3 (green, dashed), (iii) n = 4 (blue, dotted), (iv) n = 5 (black,  dot-dashed), (v) n = 6 (cyan, solid), and (vi) n = 7 (brown, solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  At a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='16, going from bottom to top, the curves are for n = 6,  n = 4, n = 2, n = 3, n = 5, and n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Pad´e analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The [p, q] Pad´e approximant to βa,r,nℓ is  the rational function [23]  [p, q]βa,r,nℓ =  1 + �p  j=1 rjaj  1 + �q  k=1 sk ak  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1)  with p + q = n − 1, where the coefficients rj and sj are  independent of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' At seven-loop order, we can calculate  4  the Pad´e approximants [p, q]βa,r,7ℓ with [p, q] taking on  the values [6,0], [5,1], [4,2], [3,3], [2,4], [1,5], and [0,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Since the loop order is understood, we write [p, q]βa,r,7ℓ ≡  [p, q] for brevity of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The PA [6,0] is equivalent  to βa,r,7ℓ itself, which we have already analyzed, and the  PA [0,6] has no zeros, so we focus here on the remaining  five Pad´e approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  We list our results for these Pad´e approximants to  βa,r,7ℓ below:  [5, 1] = 1 + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='760a − 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='931a2 + 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='552a3 − 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='17a4 + 1367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='8a5  1 + 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='649a  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2)  [4, 2] = 1 + 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='541a + 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='687a2 − 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='670a3 + 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='973a4  1 + 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='430a + 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='21a2  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3)  [3, 3] = 1 + 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='073a + 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='81a2 + 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='99a3  1 + 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='962a + 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='89a2 + 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='33a3 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='4)  [2, 4] =  1 + 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='314a + 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='55a2  1 + 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='203a + 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='42a2 + 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='390a3 − 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='252a4 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5)  [1, 5] =  1 + 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='023a  1 + 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='912a + 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='205a2 − 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='828a3 + 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='10a4 − 910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='03a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='6)  We recall some necessary requirements for a zero of a  [p, q] Pad´e approximant to be physically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' These  include the requirement that this zero should occur on  the positive real axis in the complex a plane and the  requirement that this zero of the PA should be closer to  the origin a = 0 than any pole on the real positive a-axis,  since otherwise the pole would dominate the IR to UV  flow starting at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' If a Pad´e approximant were  to exhibit such a zero, then one would proceed to inquire  how close it is to any of the aUV,nℓ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' However,  we find that none of these Pad´e approximants (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='6)  has a zero on the positive real a axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Explicitly, the  [5,1] PA has two complex-conjugate pairs of zeros at a =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='12719±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='26046i and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='26922±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='20930i, together  with a real zero at a = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='074837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  This real zero is  part of a nearly coincident pole-zero pair, with the pole  of the [5,1] PA being located at a = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='073267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The  appearance of a nearly coincident pole-zero pair close to  a point a0 in a [p, q] Pad´e approximant is typically an  indication that the function that the PA is fitting has  neither a pole nor a zero in the local neighborhood of  a0, since as the locations of the nearly coincident pole-  zero pair approach each other, they simply divide out in  the ratio (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Each of the Pad´e approximants that we  calculate here has a pole-zero pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The [4,2] PA has zeros  at the complex-conjugate pair a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='42009 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='96575i,  together with the real values a = {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='16929, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='064970}  and poles at a = {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='14481, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='064414}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The [3,3] PA  has zeros at a = {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='78531, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='13282, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='061458}, and  poles at a = {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='42342, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='12140, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='061112}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The  [2,4] PA has zeros at a = {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='15193, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='063563}, and  poles at a = {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='69186, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='13432, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='063100, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='8689}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Finally, the [1,5] PA has a zero at a = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='071313 and  poles at a = {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='22780, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='070185, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='44160, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='035937±  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='39287i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Thus, our analysis with Pad´e approximants of  the seven-loop beta function yields the same conclusion  as our analysis of the beta function itself, namely that  there is no evidence for a stable, reliably perturbatively  calculable UV zero up to this seven-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  EFFECTS OF SCHEME  TRANSFORMATIONS  Since the terms in the beta function at loop order n ≥ 3  are scheme-dependent, it is necessary to assess the effect  of scheme transformations in an analysis of zeros of a  higher-loop beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' A scheme transformation can  be expressed as a mapping between a and a transformed  coupling a′,  a = a′f(a′) ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='1)  where f(a′) is the scheme transformation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Since  this transformation has no effect in the free theory, one  has f(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' We consider f(a′) functions that are ana-  lytic about a = a′ = 0 and hence can be expanded in the  form  f(a′) = 1 +  smax  �  s=1  ks(a′)s ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='2)  5  where the ks are constants and smax may be finite or  infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' The beta function in the transformed scheme,  βa′ = da′/d ln µ, has the expansion  βa′ = a′  ∞  �  ℓ=1  b′  ℓ(a′)ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='3)  In [24], formulas were derived for the b′  ℓ in terms of bℓ  and the ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' In addition to b′  1 = b1 and b′  2 = b2, these are  b′  3 = b3 + k1b2 + (k2  1 − k2)b1 ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='4)  b′  4 = b4 + 2k1b3 + k2  1b2 + (−2k3  1 + 4k1k2 − 2k3)b1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='5)  and so forth for higher ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' These results are applicable  to the study of both an IR zero in the beta function of  an asymptotically free theory and a possible UV zero in  the beta function of an IR-free theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' They were exten-  sively applied to assess scheme dependence in higher-loop  studies of an IR fixed point in asymptotically free non-  Abelian gauge theories [24–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  For the present λφ4 theory, a study of scheme depen-  dence was carried out in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' It was shown that even  when one shifts to a scheme different from the usual MS  scheme, the beta function still does not satisfy a requi-  site condition for a physical UV zero, namely that the  value of this zero (in a given scheme) should not change  strongly when it is calculated to successive loop orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  This result from [18] also holds in the same way in the  present seven-loop context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  CONCLUSIONS  In this paper we have investigated whether the real  scalar field theory with a λφ4 interaction exhibits evi-  dence of an ultraviolet zero in the beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Us-  ing the seven-loop coefficient b7 from [15], our present  study extends our previous six-loop study in [19, 20] to  the seven-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Our work includes a study of the  seven-loop beta function itself, together with an analysis  of Pad´e approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' We conclude that, for the range  of couplings where the perturbative calculation of this  beta function may be reliable, it does not exhibit robust  evidence for an ultraviolet zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Acknowledgments  I would like to thank Oliver Schnetz for valuable dis-  cussions on [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' This research was supported in part by  the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' National Science Foundation Grant NSF-PHY-  22-15093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  [1] Some early studies on the renormalization group include  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Stueckelberg and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Peterman, Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Acta  26, 499 (1953);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Gell-Mann and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Low, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 95,  1300 (1954);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Bogolubov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Shirkov, Doklad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Nauk SSSR 103, 391 (1955);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Callan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' D 2, 1541 (1970);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Symanzik, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 18, 227 (1970);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Wilson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' D 3, 1818  (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  [2] Some early references include K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Wilson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Kogut,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Repts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 12, 75 (1974);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Aizenman, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 82, 69 (1982);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Freedman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Smolensky,  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Weingarten, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' B 113, 491 (1982);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Fr¨ohlich, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' B 200, 281 (1982);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Dashen  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Neuberger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 50, 1897 (1983);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Lang, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' B 240, 577 (1984);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Kuti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Lin,  and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Shen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 61, 678 (1988);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' L¨uscher  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Weisz, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' B 290, 25 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Zinn-Justin, Quantum Field Theory and Critical Phe-  nomena, 4th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' (Oxford Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Press, Oxford, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Kleinert and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Schulte-Frohlinde, Critical Properties  of φ4 Theories (World Scientific, Singapore, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Le Guillou, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Zinn-Justin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' D 9, 1121 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Gross, in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Balian and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Zinn-Justin, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Meth-  ods in Field Theory, Les Houches 1975 (North Holland,  Amsterdam, 1976), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' ’t Hooft, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' B 61, 455 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Bardeen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Muta,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' D 18, 3998 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Vladimirov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content='  Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' JETP 50,  521 (1979)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Chetyrkin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Larin and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Tkachov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Larin, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Tkachov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content='  Kleinert,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Neu,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Schulte-Frohlinde,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content='  Chetyrkin, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Larin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Kompaniets and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content='09210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Kompaniets and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Panzer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content='0648].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Lipatov, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content='  Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQf3f5y/content/2301.01830v1.pdf'}
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+Demystify Problem-Dependent Power of Quantum Neural Networks on Multi-Class
+Classification
+Yuxuan Du,1, ∗ Yibo Yang,1 Dacheng Tao,1 and Min-Hsiu Hsieh2
+1JD Explore Academy, Beijing 10010, China
+2Hon Hai (Foxconn) Research Institute, Taipei, Taiwan
+Quantum neural networks (QNNs) have become an important tool for understanding the physical
+world, but their advantages and limitations are not fully understood. Some QNNs with specific en-
+coding methods can be efficiently simulated by classical surrogates, while others with quantum mem-
+ory may perform better than classical classifiers. Here we systematically investigate the problem-
+dependent power of quantum neural classifiers (QCs) on multi-class classification tasks. Through
+the analysis of expected risk, a measure that weighs the training loss and the generalization error of
+a classifier jointly, we identify two key findings: first, the training loss dominates the power rather
+than the generalization ability; second, QCs undergo a U-shaped risk curve, in contrast to the
+double-descent risk curve of deep neural classifiers. We also reveal the intrinsic connection between
+optimal QCs and the Helstrom bound and the equiangular tight frame. Using these findings, we
+propose a method that uses loss dynamics to probe whether a QC may be more effective than a
+classical classifier on a particular learning task. Numerical results demonstrate the effectiveness of
+our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and
+their limitations over convolutional neural networks on image datasets. Our work sheds light on the
+problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.
+I.
+INTRODUCTION
+The advent of hardware fabrication pushes the bound-
+ary of quantum computing from verifying its superiority
+on artificial tasks [1–3] to conquering realistic problems
+with merits [4–6]. This has led to the emergence of a
+popular paradigm known as quantum neural networks
+(QNNs), which combine variational quantum Ans¨atze
+with classical optimizers [7, 8].
+So far, various QNN-
+based methods have been proposed to address difficult
+problems in areas such as quantum physics [9–12], quan-
+tum information theory [13–16], combinatorial optimiza-
+tion [17–21], and machine learning [22–26]. Among these
+applications, QNNs are often deployed as quantum clas-
+sifiers (QCs) to predict correct labels of the input data
+[27–32], e.g., categorize image objects [33–35], classify
+phases of quantum matters [36–39], and distinguish en-
+tangled states from separable states [40, 41].
+To comprehend the full potential of existing quantum
+classifiers (QCs) and to spur the development of novel
+QCs, huge efforts have been made to unveil the learnabil-
+ity of QCs [42–44]. Prior literature establishes the foun-
+dations of QCs from three primary aspects, i.e., model
+capacity [45–48], trainability [49–51], and generalization
+[52–57]. Nevertheless, the advantages and constraints of
+QCs have rarely been proven [57–62]. Meanwhile, pre-
+vious results cannot rigorously explain the empirical ob-
+servations such that QCs generally outperform classical
+classifiers (CCs) on handcraft or quantum data [44, 63]
+but are inferior to them on realistic problems [64]. As
+a result, the need for QCs to address classical issues re-
+mains highly questionable.
+∗ duyuxuan123@gmail.com
+A principal criteria in characterizing the power of a
+classifier is the expected risk [65], which weighs the em-
+pirical risk (i.e., training loss) and the generalization er-
+ror (i.e., test loss) jointly. An optimal classifier is one
+which achieves zero expected risk. As shown in Fig. 1(a),
+the success of deep neural classifiers is attributed to their
+double-descent risk curves [66, 67]. This means that as
+the hypothesis space is continually expanded, the ex-
+pected risk of a trained deep neural classifier initially
+decreases, increases, and when it overfits the train set,
+undergoes a second descent. As such, to show the supe-
+riority of QCs over CCs, it demands to distill ubiquitous
+rules that capture the risk curve of diverse QCs in addi-
+tion to conditions where the expected risk of QCs can be
+lower than CCs.
+In this study, we unify a broad class of QCs in
+the same framework and understand their problem-
+dependent ability under the expected risk (see Fig. 1(b)).
+Our analysis reveals two substantial outcomes: (i) train-
+ability dominates QCs’ ability more than generalization
+ability; (ii) QCs undergo a U-shape risk curve instead
+of the double-descent curve for CCs.
+These outcomes
+consolidate and refine previous observations. Concretely,
+the first outcome suggests that the deficiency of QCs on
+classical data stems from their limited ability to fit the
+train set, resulting in a larger training loss compared to
+CCs. The second outcome highlights the distinct learn-
+ing behavior of QCs and CCs. Despite the fact that over-
+parameterization is fundamental to enhance the perfor-
+mance of CCs, it adversely affects the power of QCs.
+In line with the diverse dynamics of the risk curves for
+QCs and CCs, we devise an efficient problem-dependent
+method to recognize potential merits of QCs, as shown
+in Fig. 1(a). Conceptually, for a given learning task, our
+method fits the loss (risk) dynamics of QC and CC under
+the prior (i.e., U-shape versus double descent) and then
+arXiv:2301.01597v1  [quant-ph]  29 Dec 2022
+
+2
+(a)
+(b)
+𝑜(") 𝑜($) 𝑜(%)
+𝜌(&)
+ℇ(𝑥(&))
+𝑥(&)
+𝑜(')
+ℛ
+𝐶-ℛ
+Hypothesis space
+𝑄-ℛ
+𝑄-ℛ!"#
+ℛ
+𝐶-ℛ
+Hypothesis space
+𝑄-ℛ
+𝑄-ℛ!"#
+(c)
+𝜌̅(%), 𝑜(%)
+𝜌̅ ' , 𝑜(')
+𝜌̅ ( , 𝑜(() 𝜌̅ ) , 𝑜())
+optimize
+𝜌̅(%), 𝑜(%)
+𝜌̅ ' , 𝑜(')
+𝜌̅ ( , 𝑜(() 𝜌̅ ) , 𝑜())
+FIG. 1. Risk curve and geometry of the unified QCs. (a) The risk curve of QCs and CCs are highlighted by the solid red
+and blue lines (labeled by ‘Q-R’ and ‘C-R’), respectively. The former yields a ‘U’ shape while the latter yields a double-descent
+tendency. Potential advantages of QCs are dominated by the empirical risk, highlighted by the dashed curve. The shaded
+region refers to the potential merits of QCs. (b) The unified QC consists of two parts, the feature state ρ and the measure
+operator o. This model covers diverse QCs. (c) Geometric relationship between {ρ(i,k)} and o of QCs with (near) zero training
+loss: (i) the feature states associated with train samples belonging to the same class concentrate around their class-feature
+mean, i.e., ¯ρ∗(k) := ρ∗(1,k) = ... = ρ∗(nc,k) for ∀k ∈ [K]; (ii) the class-feature means are maximally distant with each other, i.e.,
+Tr(¯ρ∗(k)¯ρ∗(k′)) ∼ δk,k′; (iii) the measure operator should align with class-feature means, i.e., Tr(¯ρ∗(k)o∗(k′)) ∼ δk,k′.
+identify the ‘advantage’ regime where the risk of QC is
+lower than CC. Numerical simulations are conducted to
+support our theoretical results.
+On the technical level, we approach the two outcomes
+by separately quantifying the empirical risk and gener-
+alization error of QCs. Specifically, we first prove con-
+ditions of QCs that lead to near-zero empirical risk, the
+geometric interpretation of which is depicted in Fig. 1(c).
+As a byproduct, we elucidate how such conditions are
+inherently linked to quantum state discrimination and
+quantum measurement theory.
+In addition, we prove
+that deep QCs can never reach the vanished empirical
+risk by utilizing the concentration property of quantum
+observables [68, 69]. We next analyze the generalization
+error of QCs by exploiting algorithmic robustness [70].
+The derived bound surpasses prior results because it is
+the first non-vacuous bound in the over-parameterized
+regime.
+By combining the unreachable zero empirical
+risk with the manipulatable generalization error, we ob-
+tain the first outcome. The second outcome is gained by
+integrating the fact that deep QCs are unable to reach
+the vanished empirical risk with the first outcome.
+II.
+MAIN RESULTS
+Expected risk.— Let us first introduce a K-class (K ≥
+2) classification task. Denote the input space as X, the
+label (class) space as Y = {1, · · · , K}, and the train set
+as D = �K
+k=1{(x(i,k), y(i,k))}nk
+i=1 with |D| samples drawn
+i.i.d.
+from an unknown probability distribution D on
+Z = X × Y. In standard scenarios, the number of train
+samples in each class is the same, i.e., n1 = ... = nk ≡ nc
+and |D| := n = Knc.
+The purpose of a classification
+algorithm A is using D to infer a hypothesis (a.k.a., a
+classifier) hAD : X → RK from the hypothesis space H
+to separate train examples from different classes. This
+is equivalent to identifying an optimal hypothesis in H
+minimizing the expected risk R(h) = E(x,y)∼D[ℓ(h(x), y)],
+where ℓ(·, ·) is the per-sample loss and for clarity we spec-
+ify it as the square error with ℓ(a, b) = 1
+2∥a − b∥2
+2 [71].
+Unfortunately, the inaccessible distribution D forbids us
+to assess the expected risk directly. In practice, A al-
+ternatively learns an empirical classifier ˆh ∈ H, as the
+global minimizer of the (regularized) loss function
+L(h, D) = 1
+n
+nc
+�
+i=1
+K
+�
+k=1
+ℓ(h(x(i,k)), y(i,k)) + E(h),
+(1)
+where E(h) is an optional regularizer.
+The foremost role of the risk means that quantum ad-
+vantages can be ascertained if R(ˆhQ) < R(ˆhC), where ˆhQ
+and ˆhC are the empirical QC and CC on D. Unlike con-
+ventions merely focusing on a QC on one specific task,
+the above criteria orients to unearth ubiquitous rules of
+QCs with computational advantages.
+To reconcile the
+intractable issue of R(ˆh) and proceed the following anal-
+ysis, we decomposed it into two measurable terms, i.e.,
+R(ˆh) = RERM(ˆh) + RGene(ˆh),
+(2)
+where RERM(ˆh) = 1
+n
+�n
+i=1
+�K
+k=1 ℓ(ˆh(x(i,k)), y(i,k)) is the
+empirical risk and RGene(ˆh) = R(ˆh)−RERM(ˆh) is the gen-
+eralization error. Based on Eq. (2), detecting advances
+of QCs is translated to deriving under what conditions
+do QCs commit both lower RERM and RGene than CCs.
+To better elucidate our results, let us recall that the
+general form of QC is ˆhQ = arg minhQ∈HQ L(hQ, D),
+
+Learning tasks
+Advantages
+No Advantages介83
+where L is defined in Eq. (1) and HQ is the hypothe-
+sis space. For an N-qubit QC, its hypothesis space is
+HQ =
+��
+hQ(·, U(θ), O(k))
+�
+k=1:K
+���θ ∈ Θ
+�
+,
+(3)
+where [·]k=1:K is a K-dimensional vector, its k-th entry
+hQ(x, U(θ), O(k)) = Tr(O(k)U(θ)σ(x)U(θ)†) for ∀k ∈
+[K] refers to the output (prediction) of quantum cir-
+cuits, σ(x) = UE(x)(|0⟩ ⟨0|)⊗NUE(x)† is the input state
+of x with the encoding circuit UE(·), O = {O(k)}K
+k=1
+is a set of measure operators, and U(θ) is the adopted
+Ansatz with trainable parameters θ living in the pa-
+rameter space Θ. Without loss of generality, we define
+U(θ) = �Nt
+l=1(ul(θ)ue) ∈ U(2N), where ul(θ) ∈ U(2m)
+is the l-th parameterized quantum gate operated with at
+most m qubits (m ≤ N) and ue refers to fixed quan-
+tum gates. Similarly, we define UE(x) = �Ng
+g=1 ug(x) ∈
+U(2N), where ug(x) ∈ U(2m) refers to the g-th quan-
+tum gate operated with at most m qubits, and Ng gates
+contain Nge tunable gates and Ng − Nge fixed gates.
+Due to the diverse constructions of U(θ) and UE(·), it
+is necessary to unify various QCs into the same frame-
+work to obtain the generic results. Notably, the unified
+QC should be agnostic to particular forms of these two
+terms. A feasible way is rewritten hQ(·, U(θ), O(k)) as
+hQ(ρ(i,k), o(k)) := Tr(ρ(i,k)o(k)) ∀k ∈ [K],
+(4)
+where O(k) = I2N−D ⊗o(k) with the nontrivial local oper-
+ator o(k) ∈ C2D×2D, D describes the locality, and ρ(i,k) =
+TrD(U(θ)σ(x(i,k))U(θ)†) corresponds to the state before
+measurements, named as feature state. An intuition of
+the unified QC is shown in Fig. 1(b).
+We are now ready to exploit the unified framework
+to analyze the expected risk of QCs. Let ρ = {ρ(i,k)}
+and o = {o(k)} be two sets collecting all feature states
+and measure operators. The following theorem exhibits
+conditions in which QCs allow a low expected risk, where
+the formal statement and the proof are deferred to SM A.
+Theorem 1 (informal). Following notations in Eqs. (1)-
+(4), when the train data size is nO(KNge log KNg
+ϵδ ) with ϵ
+being the tolerable error, and the optimal sets of ρ∗ and
+o∗ satisfy three conditions: (i) the feature states have
+the vanished variability in the same class; (ii) all feature
+states are equal length and are orthogonal in the varied
+classes; (iii) any feature state is alignment with the mea-
+sure operator in the same class, with probability 1−δ, the
+expected risk of QC tends to be zero, i.e., R(ˆhQ) → 0.
+Conditions (i)-(iii) visualized in Fig. 1(c) sculpt the ge-
+ometric interpretations of ρ∗ and o∗. These properties
+come across the design philosophy of CCs, e.g., linear dis-
+criminant analysis and neural collapse phenomenon ap-
+peared in most deep neural classifiers [71–73]. Moreover,
+these conditions unveil the intrinsic connection between
+optimal QCs and the quantum state discrimination [74],
+since ρ∗ and o∗ should maximize the Helstrom bound
+[75], which explains the ultimate limit of QCs observed
+in [76]. However, as will be explained later (see Corol-
+lary 1 and Lemma 1), under certain scenarios, it is hard
+for QCs to meet these conditions. A typical instance is
+applying QC to learn the image dataset, where the dif-
+ficulty stems from the limited nonlinearity of QC to fit
+the train set, thereby inducing a large empirical risk.
+Conditions (i)-(iii) also imply how the quantum mea-
+surement theory can be used to guide the design of QCs.
+Namely, the mean feature states of each class {¯ρ∗(k)}
+compose the equiangular tight frame (ETF) and Con-
+dition (iii) suggests that the optimal measure operators
+{o∗} also satisfy this ETF [77]. Due to the relation be-
+tween symmetric informationally complete (SIC) mea-
+surements and ETF [78, 79], the optimal measure op-
+erators could be estimated by various SIC construction
+strategies [80].
+Besides, the locality D of {o∗} should
+be carefully selected in QCs in which a small D pre-
+cludes the acquisition of the optimal QCs (i.e., the com-
+plex ETF does not exist when 2D = K [81, 82]), while an
+extremely large D may incur the barren plateaus [83, 84].
+Furthermore, when K is large, Pauli-based measurements
+are preferable than computational basis measurements in
+QCs, since the former allows classical shadow techniques
+to accelerate the training of QCs [85, 86].
+The scaling behavior of n indicates that it is data-
+efficient for QCs to attain a low generalization error,
+where the size of train set only linearly depends on the
+class number K and the number of encoding gates Nge
+(see Lemma 3 for the technical elaboration).
+In other
+words, the generalization error of QCs can be well con-
+trolled by the modest-size train data.
+According to Theorem 1, the challenges in satisfying
+Conditions (i)-(iii) and the well controlled generalization
+error pinpoint that the risk of a QC is mostly dominated
+by its empirical loss rather than its generalization error.
+In this view, the core in devising advanced QCs is tailor-
+ing HQ in Eq. (3) so that ˆhQ has a (near) zero empirical
+risk on D, or equivalently examining whether the em-
+ployed QCs can fulfill Conditions (i)-(iii).
+U-shape risk curve.—The risk curve concerns how
+the expected risk of a classifier behaves with the varied
+hypothesis space. It is desired that as with deep neu-
+ral classifiers, QCs undergo a double-descent risk curve
+in the sense that the over-parameterized QCs consent
+a low expected risk when the trainable parameters Nt
+is much greater than the train data n.
+If so, ‘over-
+parameterization’ could serve as a golden law in designing
+QCs. However, the following corollary refutes the exis-
+tence of the double-descent risk curve for QCs.
+Corollary
+1.
+Following
+notations
+in
+Theorem
+1,
+when {UE(x)|x
+∈
+X} follows the Haar distribu-
+tion, with probability 1 − δ, the empirical QC follows
+| Tr
+�
+σ(x(i,k))σ(x)
+�
+− 1
+2N | ≤
+�
+3
+22Nδ. When {U(θ)|θ ∈ Θ}
+follows the Haar distribution, with probability 1 − δ,
+the empirical QC follows | Tr(ρ(i,k)o(k′)) − Tr(o(k′))
+2D
+| <
+
+4
+�
+Tr(o(k′))2+2 Tr((o(k′))2)
+22Dδ
+.
+The proof is deferred to SM B. The achieved results re-
+veal the caveat of deep QCs. Specifically, when UE(x)
+is deep, two encoded states σ(x(i,k)) and σ(x(i′,k)) from
+the same class tend to be orthogonal, which contradicts
+with Conditions (i) in Theorem 1. Besides, when U(θ)
+is deep, the output of QC concentrates to zero, regard-
+less how o(k′) and ρ(i,k) are selected. This violates Con-
+dition (iii) in Theorem 1.
+Overall, over-parameterized
+QCs encounter the high empirical risk and thus the high
+expected risk, which suggests that QCs experience a U-
+shape risk curve. This observation differs the dynamics
+of QCs from variational quantum Eigensolvers, since the
+latter can benefit from over-parameterization, e.g., better
+trainability and convergence rate [87–90]. Moreover, the
+rule of thumb in QCs’ construction is slimming HQ to
+find the valley region. Intriguingly, tailoring the feature
+states echoes with quantum metric learning and quantum
+self-supervised learning [91–95].
+Probe power of QCs via loss dynamics.—The dis-
+tinct tendency of the risk curves between QCs and CCs
+provides a succinct way to recognize the potential quan-
+tum advantages. As shown in Fig. 1(a), given a specific
+data set, the U-shape risk curve of QCs indicates that its
+advantages mostly appear in the valley region. Precisely,
+if the risk values of QC around the basin are lower than
+those of CC, potential merits may exist; otherwise, QC
+is inferior to CC. The proved learning behavior of QCs,
+accompanied with the tight generalization bound, allows
+us to effectively fit its risk curve according to their loss
+dynamics. Specifically, our method contains three steps.
+First, W tuples of {n, Nt, T} are initialized based on The-
+orem 1 so that the collected risk points of QC span the
+basin area with low generalization error. Second, we ex-
+ecute QC and CC under these W hyper-parameter set-
+tings and fit their loss dynamics to attain the risk curve.
+Last, we compare two risk curves and probe potential
+advantages. See SM F for the implementation details.
+Technical analysis.—Theorem 1 is achieved by ana-
+lyzing when RERM(ˆhQ) and RGene(ˆhQ) are (near) zero.
+In the analysis of RERM(ˆhQ), we first consider the most
+general case in which both ρ and o are tunable, where
+ˆhQ ≡ hQ(ρ∗, o∗) with (ρ∗, o∗) = minρ,o L(ρ, o).
+Lemma 1 (Informal). When the regularizer E is consid-
+ered and (ρ∗, o∗) meets the three conditions in Theorem
+1, the global minimizer leads to RERM(ˆhQ) = C2
+1/2 with
+C1 depending on the hyper-parameters in E.
+The achieved properties of o∗ can be used as a priori to
+simplify QCs. To this end, the following lemma quantifies
+RERM(ˆhQ) when o is predefined and E = 0, where ˆhQ ≡
+hQ(ρ∗, o) with ρ∗ = minρ L(ρ, o).
+Lemma 2 (Informal). When the predefined {o(k)} are
+mutually orthogonal with each other and the conditions
+in Theorem 1 are satisfied, we have RERM(ˆhQ) = 0.
+The proofs of Lemmas 1 and 2 are given in SM C&D.
+We next analyze RGene(ˆhQ). Prior results cannot be
+used to prove Theorem 1, since such bounds polynomially
+scale with the trainable parameters and become vacuous
+in the over-parameterized regime. To remedy this issue,
+we utilize the concept of algorithmic robustness [70].
+Definition 1 (Robustness). A learning algorithm A is
+(R, ν(·))-robust with R ∈ N and ν(·) : Zn → R, if Z can
+be partitioned into R disjoint sets, denoted by {Cr}R
+r=1,
+such that the following holds for all D ⊂ Zn : ∀s =
+(x(i), y(i)) ∈ D, ∀z = (x, y) ∈ Z, ∀r ∈ [R],
+s, z ∈ Cr ⇒ |l(hAD(x(i)), y(i)) − l(hAD(x), y)| ≤ ν(D).
+Concisely, robustness measures how much the loss value
+can be varied with respect to the input space Z. A higher
+robustness of a classifier admits lower R, ν(·), and RGene
+[70]. The following lemma quantifies the upper bound of
+RGene(ˆhQ) whose proof is given in SM E.
+Lemma 3. Suppose the measure operator is bounded by
+C2 with maxk∈[K] ∥o(k)∥ ≤ C2. Define ϵ as the tolerable
+error. Following notations in Definition 1, the empiri-
+cal QC is (K(28Nge/ϵ)4mNge, 4L1KC2ϵ)-robust, and with
+probability 1 − δ we have
+RGene(ˆhQ) ≤ 4L1KC2ϵ + 5ξ(ˆhQ)
+�
+|TD|4mNge ln 56KNge
+ϵδ
+n
+,
+where L1 is the Lipschitz constant of ℓ with respect to
+hQ, ID
+r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)),
+and TD := {r ∈ [R] : |ID
+r | ≥ 1}.
+The achieved results convey threefold insights. First, our
+bound does not explicitly depend on the number of train-
+able parameters [96]. This unlocks a new way to under-
+stand the generalization ability of QCs, especially for the
+over-parameterized ones. Next, our bound hints that a
+carefully designed UE can enhance performance of QCs
+[53, 97]. Last, RGene(ˆhQ) → 0 requires n ≫ |TD|4mNge.
+Fortunately, a reasonable value of n is sufficient to war-
+rant this condition, because in general m ≤ 2, Nge ∝ |x|,
+and |TD| is continuously decreased from n to K with re-
+spect to the reduced empirical loss.
+III.
+NUMERICAL SIMULATIONS
+We conduct numerical simulations to exhibit that the
+advantages and limitations of QCs on different classifica-
+tion tasks can be interpreted by the derived risk curve
+and feature states. The omitted construction details and
+results are deferred to SM G.
+We first apply QC to accomplish the binary classifica-
+tion on the parity dataset [98–100]. The number of qubits
+is N = 6 and the hardware-efficient Ansatz is adopted
+to realize U(θ).
+The gradient descent method is used
+as the classical optimizer. Two measure operators are
+
+5
+(a)
+(b)
+FIG. 2. Binary classification on the parity dataset. (a)
+The learning performance of QC when the layer number is
+3.
+The x-axis denotes the epoch numbers.
+Shaded region
+represents variance. The Bloch spheres display the quantum
+feature states at different epochs. (b) The fitted risk curve
+of QC and MLP. The x-axis denotes the number of trainable
+parameters. The label ‘QC-risk’ (‘MLP-risk’) refers to the
+fitted risk curve of QC and MLP. The label ‘QC-res’ (‘MLP-
+res’) refers to the collected results used for fitting the curves.
+o(1) = |0⟩ ⟨0| and o(2) = |1⟩ ⟨1|. The simulation results of
+QC with Nt = 54 are displayed in Fig. 2(a). Particularly,
+the averaged train (test) accuracy steadily grows from
+44.1% to 100% within 22 epochs, and the corresponding
+loss decreases from 0.26 to 4 × 10−5. The dynamics of
+the feature states {ρ(i,t)} with t ∈ {0, 10, 20, 30, 40} vi-
+sualized by Bloch spheres echo with Lemma 2. Besides,
+QC becomes more robust when we continue the training.
+Although the train (test) accuracy reaches the optimum,
+the loss can be further reduced and suggests a lower risk
+warranted by Theorem 1. We further compare the risk
+curve between QC and multilayer Perceptron (MLP) on
+this dataset. We fit their risk curves following the pro-
+posed method to probe potential quantum merits.
+As
+shown in Fig. 2(b), QC clearly outperforms MLP when
+the trainable parameters ranges from 20 to 140 and the
+valley of the risk curve is around Nt = 70 [101].
+We then apply QC to learn the Fashion-MNIST im-
+age dataset with K = 9 [102]. The employed number of
+qubits is N = 10 and the Pauli-based measure operators
+are employed.
+Convolutional neural networks (CNNs)
+are exploited as the reference. For all classifiers, the num-
+ber of epochs is fixed to be T = 50 and the number of
+trainable parameters Nt ranges from 60 to 9000. Each
+setting is repeated with 3 times.
+As shown in Fig. 3,
+when the layer number is 50 with Nt = 1500, both the
+train and test accuracies of QC are about 50%.
+This
+performance is inferior to CNN under the similar setting.
+To explore whether QC has the potential to outperform
+CNN on this dataset, we compare their risk curves. As
+shown in Fig. 3(b), unlike the parity dataset, QC is evi-
+dently inferior to CNN on Fashion-MNIST dataset.
+(a)
+(b)
+FIG. 3. Multi-class classification on the image dataset
+with K = 9. (a) The learning performance of QC when the
+layer number is 50. (b) The fitted risk curve of QC and CNN.
+All labels have the same meaning with those used in Fig. 2.
+IV.
+DISCUSSIONS AND OUTLOOK
+We understand the potential of diverse QCs in terms of
+the expected risk. Our theoretical findings demonstrate
+that the efficacy of QCs is dependent on the problem at
+hand, which explains the empirical evidence of their supe-
+riority on synthetic and quantum datasets, yet inferiority
+on realistic tasks. With the clear difference between the
+risk curve of QCs and deep neural classifiers, we present a
+concise technique to investigate potential quantum ben-
+efits by fitting their loss dynamics.
+Numerical results
+validate our theoretical results and the effectiveness of
+our method.
+There are several interesting future research directions.
+The U-shape curve of QCs poses two open questions.
+First, can contemporary QCs attain quantum benefits
+on certain classical data when only limited data and re-
+stricted computing resources are available?
+Secondly,
+is it necessary to redesign QCs such as nonlinear QCs
+[103, 104] that can also exhibit a double-descent risk
+curve? Besides, the unearthed connection between the
+conditions towards optimal empirical risk and quantum
+state discrimination opens a new research avenue that
+amplifies the potential of QCs on quantum data aided
+by quantum information theory.
+Finally, it is intrigu-
+ing to extend the developed non-vacuous generalization
+error bound of QCs to other scenarios, such as out-of-
+distribution data, in order to identify potential quantum
+advantages.
+ACKNOWLEDGMENTS
+The authors thank Xinbiao Wang for valuable input
+and inspiring discussions.
+
+0.3
+1.0
+Loss
+0.2
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+-0.8
+Loss
+Test
+Acc
+0.1
+0.6
+0.0
+0.4
+0
+10
+20
+30
+4010)
+X[0]
+y
+X[0]
+X[0)
+y
+X[0)
+y
+X
+10.6
+0.4
+QC-risk
+K
+S
+QC-res
+R 
+0.2
+MLP-risk
+MLP-res
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+1601.00
+QC-risk
+0.75
+OC-res
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+0.50
+R
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+0.50 0
+9 0.6
+Test
+0.25
+0.3
+0.00
+0
+10
+20
+30
+40
+506
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+
+10
+The organization of the supplementary material (SM) is as follows. In SM A, we present the proof of Theorem 1.
+Then, we provide the proof of Corollary 1 in SM B. Subsequently, we demonstrate the proof of Lemma 1 and Lemma
+2 in SM C and SM D, respectively. Next, in SM E, we exhibit the proof of Lemma 3. In the end, we elucidate the
+details of numerical simulations in SM G.
+SM A: Proof of Theorem 1
+For convenience, let us first recall the settings and notations introduced in the main text. When QCs are applied to
+accomplish the multi-class classification task, the training dataset D contains n examples and the number of examples
+in each class is the same with n = ncK. Moreover, the per-sample loss is specified as the mean square error.
+We next introduce the formal description of Theorem 1. In particular, Theorem 1 is established on Lemma 2, where
+the regularization term is set as zero (i.e., E = 0) and the set of measure operator is predefined, i.e., o spans the
+space C2D×2D and satisfies Tr(o(k)o(k′)) = Bδk,k′ where B ≥ 1 is a constant. The requirements of o aims to preserve
+Condition (iii) in Lemma 1. Note that the focus on these specific settings adopted in Lemma 1 instead of the most
+general settings (i.e., o is tunable and E is nonzero) is motivated by Lemma 1, which promises a lower expected risk.
+Following the above elaboration, the loss function of QC to be minimized can be explicitly written as
+L(ρ) = 1
+2n
+nc
+�
+i=1
+K
+�
+k=1
+�
+[Tr(ρ(i,k)o(k))]k=1:K − y(i,k)�2
+,
+(A1)
+where y(i,k) is the unit basis whose k-th entry is 1 for ∀i ∈ [nc], ∀k ∈ [K]. Denote ρ∗ = minρ L(ρ) and the empirical
+risk of QC as RERM(ˆhQ) with ˆhQ ≡ ˆhQ(ρ∗). The formal statement of Theorem 1 is as follows.
+Theorem (Formal statement of Theorem 1). Following notations in Lemmas 2 and 3, with probability 1 − δ, the
+expected risk of QC tends to be zero, i.e., RERM(ˆhQ) = 0, when the size of train dataset satisfies n ≫ O(KNge log KNg
+ϵδ )
+and the global minimizer ρ∗ in Eq. (A1) satisfies
+(i)¯ρ∗(k) := ρ∗(1,k) = ... = ρ∗(nc,k); (ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = Bδk,k′; (iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.
+(A2)
+Proof of Theorem 1. Following Eq. (2) and the results in Lemma 3, with probability 1 − δ, the expected risk of an
+optimal empirical QC is upper bounded by
+R(ˆhQ) ≤ RERM(ˆhQ) + 4L1KC2ϵ + 3ξ(ˆh)
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))
+n
+.
+(A3)
+Then, when ρ∗ satisfies Eq. (A2), Lemma 2 warrants RERM(ˆhQ) = 0, which gives
+R(ˆhQ) ≤ 4L1KC2ϵ + 3ξ(ˆh)
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))
+n
+.
+(A4)
+This bound can be further simplified when the training of QC is perfect. Note that Condition (i) implies |TD| = K,
+since all feature states from the same class collapse to the same point. Meanwhile, since ξ(ˆh) and C2 are bounded,
+and m and ϵ are small constant, we can conclude that when n ≫ O(KNge log(KNg/(ϵδ))), the expected risk can
+approach to zero.
+SM B: Proof of Corollary 1
+The proof leverages the following two lemmas related to the Haar measure and the unitary t-design.
+Lemma 4. Let {Wy}y∈Y ⊂ U(d) form a unitary t-design with t > 1, and let A, B : Hd → Hd be arbitrary linear
+operators. Then
+1
+|Y |
+�
+y∈Y
+Tr[WyAW †
+y B] =
+�
+Haar
+dµ(W) Tr[WyAW †
+y B] = Tr[A] Tr[B]
+d
+.
+(B1)
+
+11
+Lemma 5. Let {Wy}y∈Y ⊂ U(d) form a unitary t-design with t > 1, and let A, B, C, D : Hd → Hd be arbitrary
+linear operators. Then
+1
+|Y |
+�
+y∈Y
+Tr[WyAW †
+y B] Tr[WyCW †
+y D] =
+�
+Haar
+dµ(W) Tr[WyAW †
+y B] Tr[WyCW †
+y D]
+=
+1
+d2 − 1 (Tr[A] Tr[B] Tr[C] Tr[D] + Tr[AC] Tr[BD])
+−
+1
+d(d2 − 1) (Tr[AC] Tr[B] Tr[D] + Tr[A] Tr[C] Tr[BD]) .
+(B2)
+Corollary (Restatement of Corollary 1). Following notations in Lemmas 2 and 3, when the encoding unitary
+{UE(x)|x ∈ X} follows the Haar distribution, with probability 1 − δ, the empirical QC follows | Tr
+�
+σ(x(i,k))σ(x)
+�
+−
+1
+2N | ≤
+�
+3
+22Nδ. When the adopted Ansatz {U(θ)|θ ∈ Θ} follows the Haar distribution, with probability 1 − δ, the
+empirical QC follows | Tr(ρ(i,k)o(k′)) − Tr(o(k′))
+2D
+| <
+�
+Tr(o(k′))2+2 Tr((o(k′))2)
+22Dδ
+.
+Proof of Corollary 1. We complete the proof by separately analyzing the concentration behavior of the encoding
+unitary and the Ans¨atze.
+Concentration of the encoding unitary. Recall that Condition (iii) in Lemma 2 concerns the distance between two
+feature states ρ(i,k) and ρ(i′,k′) for ∀i, i ∈ [nc] and ∀k, k′ ∈ [K]. In this regard, we quantify the distance between the
+encoded state σ(x(i,k)) and σ(x) with x ∼ X when the deep encoding Ansatz UE is employed. In particular, we have
+Ex∼X
+�
+Tr
+�
+σ(x(i,k))σ(x)
+��
+=Ex∼X
+�
+Tr
+�
+σ(x(i,k))UE(x)(|0⟩ ⟨0|)⊗NUE(x)†��
+=
+�
+Haar
+dµ(U) Tr
+�
+σ(x(i,k))U(|0⟩ ⟨0|)⊗NU
+�
+=Tr(σ(x(i,k))) Tr(|0⟩ ⟨0|)⊗N)
+2N
+= 1
+2N ,
+(B3)
+where the third equality uses Lemma 4. Moreover, the variance of the term Tr(σ(x(i,k))σ(x)) yields
+Varx∼X
+�
+Tr
+�
+σ(x(i,k))σ(x)
+��
+=Ex∼X
+�
+Tr
+�
+σ(x(i,k))σ(x)
+�2�
+− Ex∼X
+�
+Tr
+�
+σ(x(i,k))σ(x)
+��2
+=
+�
+Haar
+dµ(U) Tr
+�
+σ(x(i,k))U(|0⟩ ⟨0|)⊗NU
+�
+Tr
+�
+σ(x(i,k))U(|0⟩ ⟨0|)⊗NU
+�
+−
+1
+22N
+=
+1
+22N − 1
+�
+1 + Tr(σ(x(i,k))2)
+�
+−
+1
+22N(22N − 1)
+�
+Tr(σ(x(i,k))2) + 1
+�
+−
+1
+22N
+≤
+1
+22N−2 −
+1
+22N
+= 3
+22N ,
+(B4)
+where the second equality uses the property that the deep encoding unitary follows the Haar distribution and the
+result in Eq. (B3), the third equality comes from Lemma 4, the inequality adopts Tr(σ2) ≤ 1 and 22N − 1 > 22N−1,
+and the last equality is obtained via simplification.
+Supported by the Chebyshev’s inequality Pr(|X − E[X]| ≥ a) ≤ Var[X]/a2, Eqs. (B3) and (B4) indicate
+Pr
+���� Tr
+�
+σ(x(i,k))σ(x)
+�
+− 1
+2N
+��� ≥ τ
+�
+≤
+3
+22Nτ 2 .
+Equivalently, with probability 1 − δ, we have
+��� Tr
+�
+σ(x(i,k))σ(x)
+�
+− 1
+2N
+��� ≤
+�
+3
+22Nδ .
+(B5)
+
+12
+Concentration of the deep Ansatze. Recall Condition (ii) in Lemma 2. Given a feature state ρ(i,k), for ∀i ∈ [nc] and
+∀k ∈ [K] and a measure operator o(k), the optimal feature state should satisfy
+Tr(ρ∗(i,k)o(k′)) = δk,k′.
+In other words, we should examine the value of Tr(ρ(i,k)o(k′)) when ρ(i,k) is prepared by a deep Ansatze U(θ).
+Specifically, we have
+Eθ∼Θ
+�
+Tr(ρ(i,k)o(k′))
+�
+=Eθ∼Θ
+�
+Tr(U(θ)σ(x(i,k))U(θ)†(o(k′) ⊗ I2N−D)
+�
+=
+�
+Haar
+dµ(U) Tr
+�
+Uσ(x(i,k))U †(o(k′) ⊗ I2N−D)
+�
+=Tr(o(k′))(2N−D)
+2N
+=Tr(o(k′))
+2D
+,
+(B6)
+where the first equality comes from the explicit form of QC in Eq. (4), the second equality uses the fact that U follows
+the Haar distribution, and the last second equality comes from Lemma 4.
+We then quantify the variance of Tr(ρ(i,k)o(k′)), i.e.,
+Varθ∼Θ
+�
+Tr(ρ(i,k)o(k′))
+�
+=Eθ∼Θ
+�
+Tr(ρ(i,k)o(k′))2�
+−
+�
+Eθ∼Θ
+�
+Tr(ρ(i,k)o(k′))
+��2
+=
+�
+Haar
+dµ(U) Tr
+�
+Uσ(x(i,k))U †(o(k′) ⊗ I2N−D)
+�2
+− Tr(o(k′))2
+22D
+=
+1
+22N − 1
+�
+Tr(σ(x(i,k))) Tr(o(k′) ⊗ I2N−D) Tr(σ(x(i,k))) Tr(o(k′) ⊗ I2N−D) + Tr(σ(x(i,k))2) Tr((o(k′) ⊗ I2N−D)2)
+�
+−
+1
+2N(22N − 1)
+�
+Tr(σ(x(i,k))2) Tr(o(k′) ⊗ I2N−D)2 + Tr(σ(x(i,k)))2 Tr((o(k′) ⊗ I2N−D)2)
+�
+− Tr(o(k′))2
+22D
+≤
+1
+22N − 1
+�
+Tr(o(k′) ⊗ I2N−D)2 + Tr((o(k′) ⊗ I2N−D)2)
+�
+− Tr(o(k′))2
+22D
+=
+1
+22N − 1
+�
+Tr(o(k′))222N−2D + Tr((o(k′))2)22N−2D�
+− Tr(o(k′))2
+22D
+≤Tr(o(k′))2 + Tr((o(k′))2)
+22D−1
+− Tr(o(k′))2
+22D
+=Tr(o(k′))2 + 2 Tr((o(k′))2)
+22D
+.
+(B7)
+where the second equality uses the fact that U follows the Haar distribution and Eq. (B6), the the third equality
+comes from Lemma 5, the first inequality arises from dropping some positive terms, the last second equality employs
+Tr(A⊗B) = Tr(A) Tr(B) and (A⊗B)(C⊗D) = (AC)⊗(BD), and the last inequality exploits (22N −1)−1 > (2N−1)−1,
+and the last equalities is obtained via simplification.
+Supported by the Chebyshev’s inequality Pr(|X − E[X]| ≥ a) ≤ Var[X]/a2, Eqs. (B6) and (B7) indicate
+Pr
+���� Tr(ρ(i,k)o(k′)) − E
+�
+Tr(ρ(i,k)o(k′))
+� ��� ≥ τ
+�
+≤ Tr(o(k′))2 + 2 Tr((o(k′))2)
+22Dτ 2
+.
+Equivalently, with probability 1 − δ, we have
+��� Tr(ρ(i,k)o(k′)) − Tr(o(k′))
+2D
+��� <
+�
+Tr(o(k′))2 + 2 Tr((o(k′))2)
+22Dδ
+.
+(B8)
+
+13
+SM C: Proof of Lemma 1
+In this section, we derive the geometric properties of the global optimizer under the unconstraint loss function
+L(ρ, o) in which both ρ and o are tunable and the regularization term is considered. Mathematically, the regularizer
+in Eq. (1) is defined as E = λρ
+2
+�nc
+i=1
+�K
+k=1 ∥ρ(i,k)∥2
+F + λo
+2
+�K
+k=1 ∥o(k)∥2
+F with λρ and λo being hyper-parameters. The
+explicit form of the loss function is
+L(ρ, o) = 1
+2n
+nc
+�
+i=1
+K
+�
+k=1
+��
+Tr(ρ(i,k)o(k))
+�
+k=1:K − y(i,k)�2
++ λρ
+2
+nc
+�
+i=1
+K
+�
+k=1
+∥ρ(i,k)∥2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F .
+(C1)
+Denote the global optima as (ρ∗, o∗) = minρ,o L(ρ, o) and the empirical QC as ˆhQ ≡ hQ(ρ∗, o∗). The restatement of
+Lemma 1 is as follows.
+Lemma (Formal statement of Lemma 1). Define C1 := K
+�
+ncλoλρ. If 2D ≥ K, C1 ≤ 1, and λo ≤ ncλρ, the global
+minimizer (ρ∗, o∗) of L(ρ, o) in Eq. (C1) satisfies for ∀k, k′ ∈ [K]:
+(i)¯ρ∗(k) := ρ∗(1,k) = · · · = ρ∗(nc,k);
+(ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = (1 − C1)
+�
+λo
+nλρ
+δk,k′;
+(iii)o∗(k) =
+�
+nλρ
+λo
+¯ρ∗(k).
+(C2)
+The corresponding empirical risk is RERM(ˆhQ) = C1.
+Proof of Lemma 1. Conceptually, the global optimizer can be identified by lower bounding L(ρ, o), where the equality
+conditions of ρ amount to the properties of global minimizer. In particular, the lower bound of L(ρ, o) yields
+1
+2Knc
+nc
+�
+i=1
+K
+�
+k=1
+�
+[Tr(ρ(i,k)o(j))]j=1:K − y(i,k)�2
++ λρ
+2
+nc
+�
+i=1
+K
+�
+k=1
+∥ρ(i,k)∥2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F
+≥
+1
+2Knc
+nc
+�
+i=1
+K
+�
+k=1
+�
+Tr(ρ(i,k)o(k)) − 1
+�2
++ λρ
+2
+nc
+�
+i=1
+K
+�
+k=1
+∥ρ(i,k)∥2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F
+=
+1
+2Knc
+K
+�
+k=1
+nc
+�
+i=1
+nc
+1
+nc
+�
+Tr(ρ(i,k)o(k)) − 1
+�2
++ λρ
+2
+K
+�
+k=1
+nc
+�
+i=1
+nc
+1
+nc
+∥ρ(i,k)∥2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F
+≥ 1
+2K
+K
+�
+k=1
+�
+Tr
+� nc
+�
+i=1
+1
+nc
+ρ(i,k)o(k)
+�
+− 1
+�2
++ λρ
+2
+K
+�
+k=1
+nc
+�����
+nc
+�
+i=1
+1
+nc
+ρ(i,k)
+�����
+2
+F
++ λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F ,
+(C3)
+where the first inequality uses the fact ∥a − b∥2 = �
+i(a(i) − b(i))2 ≥ (a(k) − b(k))2 and the k-th entry of y(i,k) equals
+to 1, and the second inequality comes from the Jensen’s inequality f(E(x)) ≤ E(f(x)). The equality condition of the
+first inequality holds if and only if
+Tr
+�
+ρ(i,k)o(j)�
+= 0, (∀j ∈ [K] \ {k}) ∧ (∀i ∈ [nc]) ;
+and the equality condition of the second inequality holds if and only if
+ρ(1,k) = · · · = ρ(i,k) = · · · = ρ(nc,k), ∀k ∈ [K].
+Denote the mean of the feature state for the k-th class as ¯ρ(k) = �nc
+i=1
+1
+nc ρ(i,k) for ∀k ∈ [K]. The above two equality
+conditions suggest that the global minimizer (ρ∗, o∗) satisfies
+¯ρ∗(k) ≡ ρ∗(1,k) = · · · = ρ∗(nc,k), ∀k ∈ [K]
+Tr(¯ρ∗(k)o∗(j)) = 0, ∀j ∈ [K] \ {k}.
+(C4)
+To thins end, we obtain Conditions (i) in Lemma 1, which describe the geometric properties of ρ∗, i.e.,
+(i)¯ρ∗(k) := ρ∗(1,k) = · · · = ρ∗(nc,k).
+(C5)
+
+14
+The next part of the proof is showing that the global minimizer satisfies Condition (iii). Combining Eqs. (C3) and
+(C4), the lower bound of the loss function in Eq. (C3) follows
+L(ρ, o)
+≥ 1
+2K
+K
+�
+k=1
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ λρ
+2
+K
+�
+k=1
+nc
+���¯ρ(k)���
+2
+F + λo
+2
+K
+�
+j=1
+∥o(j)∥2
+F
+= 1
+2K
+K
+�
+k=1
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ λρ
+2 K
+K
+�
+k=1
+1
+K nc
+���¯ρ(k)���
+2
+F + λo
+2 K
+K
+�
+j=1
+1
+K ∥o(j)∥2
+F
+≥1
+2
+� K
+�
+k=1
+1
+K Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ λρ
+2 Knc
+�����
+K
+�
+k=1
+1
+K ¯ρ(k)
+�����
+2
+F
++ λo
+2 K∥
+K
+�
+j=1
+1
+K o(j)∥2
+F ,
+(C6)
+where the second inequality comes from the Jensen’s inequality and the equality condition holds if and only if for
+∀k, k′ ∈ [K],
+Tr
+�
+¯ρ(k)o(k)�
+= Tr
+�
+¯ρ(k′)o(k′)�
+, ∥¯ρ(k)∥F = ∥¯ρ(k′)∥F , ∥o(k)∥F = ∥o(k′)∥F .
+(C7)
+Then, supported by the inequality a + b ≥ 2
+√
+ab, the loss L(ρ, o) can be further lower bounded by
+1
+2
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ λρ
+2 Knc
+���¯ρ(k)���
+2
+F + λo
+2 K∥o(j)∥2
+F
+≥1
+2
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
++ K
+�
+ncλoλρ
+���¯ρ(k)���
+F ∥o(j)∥F ,
+(C8)
+where the equality condition holds if and only if
+λo∥o(j)∥2
+F = ncλρ
+���¯ρ(k)���
+2
+F , ∀k ∈ [K].
+(C9)
+Note that the requirements C1 ≤ 1 and λo ≤ ncλρ in Lemma 1 imply ∥¯ρ∗(k)∥ ≤ 1 and hence ensure that ¯ρ∗(k) is a
+meaningful quantum state for ∀k ∈ [K].
+Since Tr
+�
+¯ρ(k)o(k)�
+= ∥¯ρ(k)∥∥o(k)∥ cos(∠(ρ(k), o(k))), the lower bound of L(ρ, o) in Eq. (C8) is equivalent to
+1
+2
+�
+∥¯ρ(k)∥∥o(k)∥ cos(∠(ρ(k), o(k))) − 1
+�2
++ C1
+���¯ρ(k)���
+F ∥o(j)∥F .
+Define ∥¯ρ(k)∥∥o(k)∥ = a and ∠(ρ(k), o(k)) = α. The above equation is described by the function f(a, α) = (a cos α −
+1)2/2+C1a and its minimum is C1 −C2
+1/2 when α∗ = 0 and a∗ = 1−C1. The derivation is as follows. Since a > 0 and
+its maxima is unbounded, we first consider the case 0 < a < 1. In this case, the minimum of f(a, α) is C1 −C2
+1/2 with
+α∗ = 0 and a∗ = 1 − C1. Otherwise, when a ≥ 1, the minimum of f(a, α) is C1 with α∗ = arccos(1/a) and a∗ = 1.
+Note that the minimum value of f(a, α) in the second case is always larger than that of the first case. Therefore, the
+minimum of f(a, α) is C1 − C2
+1/2 with α∗ = 0 and a∗ = 1 − C1. Combining the observation that ¯ρ∗(k) and o(k) are in
+the same direction with Eq. (C9), we achieve Condition (iii), i.e.,
+o∗(k) =
+�
+ncλρ
+λo
+¯ρ∗(k).
+The last part is proving Condition (ii). Combining the result ∥¯ρ∗(k)∥∥o∗(k)∥ = 1 − C1 for ∀k ∈ [K] with Eq. (C4)
+and Condition (iii), we immediately obtain condition (ii), i.e.,
+(ii)
+�
+ncλρ
+λo
+∥ρ∗(k)∥∥ρ∗(k′)∥ = (1 − C1)δk,k′ ⇒ Tr(¯ρ∗(k)¯ρ∗(k′)) = (1 − C1)
+�
+λo
+ncλρ
+δk,k′.
+(C10)
+To summarize, given the global optima satisfying the above three conditions, the corresponding empirical risk is
+RERM(ˆhQ) = 1
+2n
+nc
+�
+i=1
+K
+�
+k=1
+�
+[Tr(ρ∗(i,k)o∗(k))]k=1:K − y(i,k)�2
+= C2
+1
+2
+(C11)
+
+15
+SM D: Results related to Lemma 2
+This section is composed of two parts. In SM D 1, we present the proof of Lemma 2. In SM D 2, we explain that
+the requirements in Lemma 2 are mild.
+1.
+Proof of Lemma 2
+Different from Lemma 1, here we focus the setting such that the regularization term is set as E = 0 and the operator
+o is predefined. The explicit form of the loss function L is defined in Eq. (A1). Denote the optimal feature states
+ρ∗ = minρ L(ρ), we quantify the value of RERM(ˆhQ) with ˆhQ ≡ hQ(ρ∗).
+We emphasize that the modifications of E and o allow a lower optimal empirical risk. Recall the results of Lemma
+1. In the most general case, the optimal empirical risk depends on the regularization term, i.e., RERM(ˆhQ) → C2
+1/2.
+The dependance on C1 motivates us to explore the empirical risk of QC when E = 0. Furthermore, Condition (iii)
+in Lemma 1 delivers the crucial properties of the optimal measure operator, i.e., the optimal measure operators are
+orthogonal with each other. Such properties contribute to construct a more effective QCs. Instead of optimizing,
+the measure operator o can be predefined by inheriting the properties proved in Lemma 1, that is, o are required to
+span the space C2D×2D and satisfy Tr(o(k)o(k′)) = Bδk,k′ with B ≥ 1 being a constant. Notably, these requirement
+are mild, covering frequently used measures such as computational basis and Pauli-based measures, as explained in
+SM D 2.
+Lemma (Formal statement of Lemma 2). Suppose that the adopted measure operator o spans the space C2D×2D and
+satisfies Tr(o(k)o(k′)) = Bδk,k′ where B ≥ 1 is a constant. The empirical risk of ˆhQ is RERM(ˆhQ) = 0 when the global
+minimizer ρ∗ satisfies
+(i)¯ρ∗(k) := ρ∗(1,k) = ... = ρ∗(nc,k); (ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = Bδk,k′; (iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.
+(D1)
+Proof of Lemma 2. The concept of the proof is analogous to Lemma 1, i.e., the global optimizer is identified by lower
+bounding the loss L(ρ). To this end, the lower bound of L(ρ) yields
+1
+2Knc
+nc
+�
+i=1
+K
+�
+k=1
+�
+[Tr(ρ(i,k)o(j))]j=1:K − y(i,k)�2
+≥
+1
+2Knc
+nc
+�
+i=1
+K
+�
+k=1
+�
+Tr(ρ(i,k)o(k)) − 1
+�2
+=
+1
+2Knc
+K
+�
+k=1
+nc
+�
+i=1
+nc
+1
+nc
+�
+Tr(ρ(i,k)o(k)) − 1
+�2
+≥ 1
+2K
+K
+�
+k=1
+�
+Tr
+� nc
+�
+i=1
+1
+nc
+ρ(i,k)o(k)
+�
+− 1
+�2
+,
+(D2)
+where the first inequality uses the facts n = Knc, ∥a − b∥2 = �
+i(a(i) − b(i))2 ≥ (a(k) − b(k))2, and only the k-th
+entry of y(i,k) equals to 1, and the second inequality comes from the Jensen’s inequality E(f(x)) ≥ f(E(x)) when f(·)
+is convex. Note that the equality condition of the first inequality holds if and only if
+Tr(ρ(i,k)o(j)) = 0, (∀j ∈ [K] \ {k}) ∧ (∀i ∈ [nc]) ;
+And the equality condition of the second inequality holds if and only if
+ρ(1,k) = · · · = ρ(i,k) = · · · = ρ(nc,k), ∀k ∈ [K].
+Denote the mean of the feature state for the k-th class as ¯ρ(k) = �nc
+i=1
+1
+nc ρ(i,k) for ∀k ∈ [K]. The above two equality
+conditions suggest that the global minimizer yields
+¯ρ∗(k) ≡ ρ∗(1,k) = · · · = ρ∗(nc,k), ∀k ∈ [K]
+(D3)
+Tr(¯ρ∗(k)o(j)) = 0, ∀j ∈ [K] \ {k}.
+(D4)
+
+16
+Combining Eqs. (D2)-(D4), the lower bound of the loss function L(ρ) satisfies
+1
+2K
+K
+�
+k=1
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
+≥ 1
+2
+� K
+�
+k=1
+1
+K Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
+,
+(D5)
+where the inequality comes from the Jensen’s inequality and the equality condition holds if and only if ∀k, k′ ∈ [K],
+Tr
+�
+¯ρ(k)o(k)�
+= Tr
+�
+¯ρ(k′)o(k′)�
+.
+(D6)
+Supported by Eq. (D6), we can further lower bound L(ρ) with
+1
+2
+�
+Tr
+�
+¯ρ(k)o(k)�
+− 1
+�2
+≥ 0,
+(D7)
+where the equality condition is achieved when Tr(¯ρ(k)o(k)) = 1 for ∀k ∈ [K].
+Taken together, the global optimizer ρ∗ should satisfy Condition (i)&(iii) in Lemma 2, where
+(i)¯ρ∗(k) := ρ∗(1,k) = ... = ρ∗(nc,k);
+(iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.
+(D8)
+We last prove that Condition (iii) and the requirements of o lead to Condition (ii).
+In particular, denote the
+vectorization of ρ∗(k) and o(k) as |ρ∗(k)⟩⟩ and |o(k)⟩⟩, respectively. Condition (iii) can be rewritten as
+��
+¯ρ∗(k), o(k′)��
+= δk,k′.
+(D9)
+Moreover, since the set of measure operators {o(k)} is required to be complete in the space of C2D and Tr(o(k)o(k′)) =
+Bδk,k′ with B ≥ 1 for ∀k, k′ ∈ [K], we have
+�
+k
+���o(k)����
+o(k)��� = BI2D.
+Then, Condition (ii) can be derived as follows, i.e.,
+Tr(ρ∗(k)ρ∗(k′))
+=⟨⟨¯ρ∗(k)|I2D|ρ∗(k′)⟩⟩
+= 1
+B
+��
+¯ρ∗(k)���
+�
+k′′
+|o(k′′)⟩⟩⟨⟨o(k′′)|
+���ρ∗(k′)
+��
+= 1
+B
+��
+¯ρ∗(k)���|o(k)⟩⟩⟨⟨o(k)|
+���ρ∗(k′)��
++
+��
+¯ρ∗(k)���
+�
+k′′̸=k
+|o(k′′)⟩ ⟨o(k′′)|
+���ρ∗(k′)
+��
+= 1
+B δk,k′.
+(D10)
+2.
+Requirement of o used in Lemma 2
+Here we elucidate that the requirements adopted in Lemma 2, i.e., o spans the complex space 2D × 2D and satisfies
+Tr(o(k)o(k′)) = Bδk,k′ with B ≥ 1, are mild. Specifically, the employed measurements in most QNN-based classifiers
+satisfy these requirements, including the computational basis measurements and Pauli measurements.
+Computational basis measurements. In this setting, the local measurement o(k) is set as |k⟩ ⟨k| with |k⟩ being the
+k-th computational basis for ∀k ∈ [K].
+When 2D = K, {|k⟩} spans the whole space of C2D×2D and we have
+Tr(o(k)o(k′)) = (⟨k|k′⟩)2 = δk,k′ with B = 1. The assumptions are satisfied.
+Pauli measurements. Denote the Pauli operation applied to the i-th qubit as P (i)
+a
+with a ∈ {X, Y, Z, I} for ∀i ∈ [D].
+Then, there are in total 4D Pauli strings P = ⊗D
+i=1P (i)
+a
+that form a orthogonal basis for the space C2D×2D. With
+setting 2D = K, each o(k) corresponds to one Pauli string with Tr(o(k)o(k′)) = Kδk,k′ with B = K.
+
+17
+SM E: Proof of Lemma 3
+For elucidating, let us restate Lemma 3 below and introduce the proof sketch before moving on to present the proof
+details.
+Lemma (Formal statement of Lemma 3). Denote L1 as the Lipschitz constant of ℓ in Eq. (1) with respect to h. Given
+a QC defined in Eq. (3), let E be a quantum channel with
+hQ(x, U(θ), O(k)) ≡ Tr(o(k)E(σ(x))), ∀k ∈ [K].
+(E1)
+Suppose the measure operator follows maxk∈[K] ∥o(k)∥ ≤ C2. The explicit form of the encoding unitary follows UE(x) =
+�Ng
+g=1 ug(x) ∈ U(2N) with the g-th quantum gate ug(x) ∈ U(2m) operating with at most m qubits with m ≤ N and Ng
+gates consisting of Nge variational gates and Ng − Nge fixed gates,
+Following above notations and Definition 1, the empirical QC is (K( 28Nge
+ϵ
+)4mNge, 4L1KC2ϵ)-robust and with prob-
+ability 1 − δ, its generalization error yields
+RGene(ˆh) ≤ 4L1KC2ϵ + 3ξ(ˆh)
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))
+n
+,
+where L1 is the Lipschitz constant of ℓ with respect to h, ID
+r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z ℓ(ˆh, z), and
+TD := {r ∈ [R] : |ID
+r | ≥ 1}.
+The proof of Lemma 3 is established on the following lemma, which leverages the algorithmic robustness to quantify
+the upper bound of the generalization error.
+Lemma 6 (Theorem 1, [105]). If the learning algorithm A is (R, ν(·))-robust with {Cr}R
+r=1, then for any δ > 0, with
+probability at least 1−δ over an i.i.d drawn of n samples D = {z(i)}n
+i=1 with z(i) = (x(i), y(i)), the returned hypothesis
+ˆh by A on D satisfies
+RGene(ˆh) ≤ ν(D) + ξ(ˆh)
+�
+(
+√
+2 + 1)
+�
+|TD| ln(2R/δ)
+n
++ 2|TD| ln(2R/δ)
+n
+�
+,
+(E2)
+where ID
+r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)), and TD := {r ∈ [R] : |ID
+r | ≥ 1}.
+The above result hints that given a hypothesis ˆh, its generalization error is upper bounded by the disjoint sets {Cr}R
+r=1,
+where a lower cardinality R allows a lower generalization error. A natural approach to realize these disjoint partitions
+is covering number [70].
+Definition 2 (Covering number, [65]). Given a metric space (U, ∥ · ∥), the covering number N(U, ϵ, ∥ · ∥) denotes the
+least cardinality of any subset V ⊂ U that covers U at scale ϵ with a norm ∥ · ∥, i.e., supA∈U minB∈V ∥A − B∥ ≤ ϵ.
+In conjunction with Lemma 6 and Definition 2, the analysis of RGene(ˆh) of an N-qubit QC amounts to quantifying
+the covering number of the space of the input quantum states, i.e.,
+XQ =
+�
+UE(x)(|0⟩ ⟨0|)⊗NUE(x)†��x ∈ X
+�
+.
+(E3)
+The following lemma connects the robustness and covering number of XQ of QCs whose proof is provided in Sec. E 1.
+Lemma 7. Following the settings in Eqs. (E1)-(E3), the corresponding QC is (K( 28Nge
+ϵ
+)4mNge, 4L1KC2∥E∥⋄ϵ)-robust.
+We are now ready to prove Lemma 3.
+Proof of Lemma 3. The generalization error bound can be acquired by combining Lemmas 6 and 7, i.e.,
+RGene(ˆh) ≤4L1KC2∥E∥⋄ϵ + ξ(ˆh)
+�
+�(
+√
+2 + 1)
+�
+|TD| ln(2K( 28Nge
+ϵ
+)4mNge/δ)
+n
++ 2|TD| ln(2K( 28Nge
+ϵ
+)4mNge/δ)
+n
+�
+�
+≤4L1KC2∥E∥⋄ϵ + ξ(ˆh)
+�
+3
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ 2|TD|4mNge ln(56KNge/(ϵδ))
+n
+�
+≤4L1KC2ϵ + ξ(ˆh)
+�
+3
+�
+|TD|4mNge ln(56KNge/(ϵδ))
+n
++ 2|TD|4mNge ln(56KNge/(ϵδ))
+n
+�
+,
+(E4)
+where ID
+r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)), and TD := {r ∈ [R] : |ID
+r | ≥ 1}.
+
+18
+1.
+Proof of Lemma 7
+The proof uses the following lemma to quantify the covering number of XQ whose proof is given in SM E 2.
+Lemma 8. Following the settings in Eq. (E1), the covering number of XQ in Eq. (E3) is
+N(XQ, ϵ, ∥ · ∥F ) ≤
+�28Nge
+ϵ
+�4mNge
+.
+(E5)
+Proof of Lemma 7. When QC is applied to accomplish the K-class classification task, the sample space is Z = XQ ×Y
+with Y = {1, 2, ..., K}. Denote ˜
+XQ as the ϵ-cover set of XQ with the covering number N(XQ, ϵ, ∥ · ∥F ) in Definition 2.
+Supported by the ϵ-cover set ˜
+XQ, the space XQ × {i} can be divided into N(XQ, ϵ, ∥ · ∥F ) sets for ∀i ∈ [K]. In other
+words, we can divide Z into KN(XQ, ϵ, ∥ · ∥F ) sets denoted by {Zi}KN (XQ,ϵ,∥·∥F )
+i=1
+.
+We then utilize the divided sets of Z to connect the robustness with covering number according to Definition 1.
+Given a training example (x(i), y(i)) and a test example (x, y), suppose that the corresponding quantum examples
+(σ(x(i)), y(i)) and (σ(x), y) are in the same set of {Zi}KN (XQ,ϵ,∥·∥F )
+i=1
+. For convenience, we abbreviate σ(x(i)) and σ(x)
+as σ(i) and σ, respectively. Following the definition of covering number, we have
+y(i) = y and ∥σ(i) − σ∥F ≤ 2ϵ.
+(E6)
+Since the encoded state takes the form σ = UE(x)(|0⟩ ⟨0|)⊗NUE(x)†, we have
+rank(σ(i) − σ) ≤ 2.
+(E7)
+Then, in accordance with the definition of robustness, we bound the discrepancy of the loss values for σ(i) and σ, i.e.,
+���l(hQ(σ(i)), y(i)) − l(hQ(σ), y)
+���
+≤L1
+���[Tr(E(σ(i))o(k))]k=1:K − [Tr(E(σ))o(k))]k=1:K
+���
+2
+≤L1K max
+k∈K | Tr(E(σ(i)))o(k)) − Tr(E(σ)o(k))|
+≤L1K max
+k
+���o(k)���
+2 Tr(|E(σ(i) − σ)|)
+≤2L1KC2∥E∥⋄∥σ(i) − σ∥F
+≤4L1KC2∥E∥⋄ϵ,
+(E8)
+where the first inequality uses the Lipschitz property of the loss function with ℓ(a, b) − ℓ(c, d) ≤ L1∥a − c∥2 and the
+form of E in Lemma 7, the second inequality comes from the definition of l2 norm, the third inequality exploits von
+Neumann’s trace inequality | Tr(AB)| ≤ ∥A∥p∥B∥q with 1/p + 1/q = 1 and the linear property of CPTP map with
+E(ρ)−E(σ) = E(ρ−σ), the last second inequality employs maxk
+��o(k)��
+2 ≤ C2, the relation ∥E(ρ−σ)∥1 ≤ ∥E∥⋄∥ρ−σ∥1
+and ∥A∥1 ≤ rank(A)∥A∥F , and the last inequality adopts the result in Eq. (E6).
+The above result exhibits that the learned QC is (KN(XQ, ϵ, ∥ · ∥), 4L1KC2∥E∥⋄ϵ)-robust. In this regard, the proof
+can be completed when the upper bound of the covering number N(XQ, ϵ, ∥ · ∥F ) is known. Supported by Lemma 8,
+we obtain N(XQ, ϵ, ∥ · ∥F ) ≤ ( 28Nge
+ϵ
+)4mNge. Taken together, the learned QC is
+�
+K
+�28Nge
+ϵ
+�4mNge
+, 4L1KC2∥E∥⋄ϵ
+�
+− robust.
+2.
+Proof of Lemma 8
+The derivation of the covering number of XQ in Eq. (E3) uses the following lemma.
+Lemma 9 (Lemma 1, [106]). For 0 < ϵ < 1/10, the ϵ-covering number for the unitary group U(2m) with respect to
+the Frobenius-norm distance in Definition 2 obeys
+� 3
+4ϵ
+�4m
+≤ N(U(2m), ϵ, ∥ · ∥F ) ≤
+�7
+ϵ
+�4m
+.
+(E9)
+
+19
+Proof of of Lemma 8. Recall the input state space is XQ = {UE(x)(|0⟩ ⟨0|)⊗NUE(x)†|x ∈ X}, where the encoding
+unitary UE(x) = �Ng
+g=1 ug(x) ∈ U(2N) consists of Nge variational gates and Ng − Nge fixed gates. To quantify the
+covering number N(XQ, ϵ, ∥·∥F ), we define ˜S as the ϵ-covering set for the unitary group U(2m), ˜
+XQ as the ϵ′-covering
+set of XQ, and define a set
+˜UE :=
+�
+�
+�
+�
+i∈{Nge}
+ui(x)
+�
+j∈{Ng−Nge}
+uj(x)
+���ui(x) ∈ ˜S
+�
+�
+� ,
+(E10)
+where ui(θi) and uj specify to the variational and fixed quantum gates, respectively. Note that for any encoding circuit
+UE(x), we can always find a unitary UE,ϵ(x) ∈ ˜UE where each ug(x) is replaced by the nearest element in the covering
+set ˜S. To this end, following the definition of covering number, the discrepancy between UE(x)(|0⟩ ⟨0|)⊗NUE(x)† ∈ XQ
+and UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)† ∈ ˜
+XQ under the Frobenius norm satisfies
+��UE(x)(|0⟩ ⟨0|)⊗NUE(x)† − UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)†��
+F
+≤2
+��UE(x)(|0⟩ ⟨0|)⊗NUE(x)† − UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)†��
+≤2∥UE(x) − UEϵ(x)∥∥(|0⟩ ⟨0|)⊗N∥
+≤4Ngeϵ,
+(E11)
+where the first inequality uses ∥X∥F ≤ rank(X)∥X∥ and the relation in Eq. (E7), the second inequality comes from
+the Cauchy–Schwarz inequality, and the last inequality follows ∥UE(x) − UE,ϵ(x)∥ ≤ Ngeϵ and ∥(|0⟩ ⟨0|)⊗N∥ = 1. In
+other words, ϵ′ = 2Ngeϵ and ˜
+XQ is a (4Ngeϵ)-covering set for XQ. In conjunction with the observation that there are
+| ˜S|Nge combinations for the gates in ˜UE and the results in Lemma 9, we obtain the cardinality of the set ˜UE is upper
+bounded by | ˜UE| ≤
+� 7
+ϵ
+�4mNge. Accordingly, supported by Eq. (E11), the covering number of XQ satisfies
+N(XQ, 4Ngeϵ, ∥ · ∥F ) ≤
+�7
+ϵ
+�4mNge
+.
+(E12)
+After simplification, we have
+N(XQ, ϵ, ∥ · ∥F ) ≤
+�28Nge
+ϵ
+�4mNge
+.
+(E13)
+SM F: Implementation of the algorithm to probe potential advantages of QCs
+The expected risk is the most principal criteria to quantify the power of a classifier. As a result, to probe whether
+a QC holds potential advantages over a CC on a specific learning task, the simplest way is comparing their risk
+curves. Nevertheless, capturing these two risk curves are difficult, because of many flexible hyper-parameter settings
+to initiate a classifier.
+The developed theories in Theorem 1 and Lemmas 1-3 deliver concrete rules to set up these hyper-parameters and
+thus allow an efficient way to estimate these risk curves. In particular, the derived U-shape curve of QCs indicates
+that the minimum risk of QC locates at the modest size of the hypothesis space HQ. In other words, the number
+of trainable parameters NT should be lower than O(poly(N)), with N being the number of qubits in QC. Moreover,
+Lemma 3 hints that the generalization error of QC can be well suppressed by using the modest number of train
+examples. As such, if the available number of training examples in D is tremendous, we can distill a subset from D
+to better recognize quantum advantages.
+The Pseudo code of the proposed method is presented in Alg. 1. To make a fair comparison, the hyper-parameter
+settings applied to QC and CC, especially for those relating to the computational resources, are required to keep
+to be the same. Specifically, in each comparison, the employed loss function, the train examples n, the number of
+trainable parameters Nt, and the number of epochs T applied to QC and CC should be identical. Note that the
+learning rate, the adopted optimizer, and the batch size can be varied of different classifiers to better estimate the
+empirical hypothesis. To ensure that the collected results of QC span its basin of the risk curve, the employed W
+settings of Nt can be acquired by uniformly interpolating from O(1) to O(poly(N)). The iteration T should ensure
+the convergence of QC. Once the loss values of QC and CC under {n(w), N (w)
+t
+, T (w)}W
+w=1 are obtained, we can apply
+certain fitting algorithms to attain their risk curves.
+
+20
+Algorithm 1: Estimate risk curves of quantum and classical classifiers
+Data: The train dataset D, the test dataset DT est, QC hQ associated with the hypothesis space HQ, CC hC associated
+with the hypothesis space HQ, the loss function L(·, ·).
+Result: The estimated risk curves of QC and CC.
+Initialization: W tuples of hyper-parameter settings {n(w), N (w)
+t
+, T (w)}W
+w=1 with n being train examples, Nt being the
+number of trainable parameters, and T being the number of epochs;
+for w = 1, w ≤ W, w + + do
+Initialize train data as D(w) by distilling n(w) examples from D;
+# Collect loss dynamics of QC ;
+Minimize the loss function L(·, ·) via gradient descent methods to obtain the empirical quantum classifier ¯h(w)
+Q
+∈ HQ
+using D(w) within T (w) epochs and NT trainable parameters;
+Record the loss value L(¯h(w)
+Q , DT est) ;
+# Collect loss dynamics of CC ;
+Minimize the loss function L(·, ·) via gradient descent methods to obtain the empirical classical classifier ¯h(w)
+C
+∈ HC
+using D(w) within T (w) epochs and NT trainable parameters;
+Record the loss value L(¯h(w)
+C , DT est) ;
+end
+Fitting the loss dynamics of {L(¯h(w)
+Q , DT est)}W
+w=1 to obtain the estimated risk curve of QC ;
+Fitting the loss dynamics of {L(¯h(w)
+C , DT est)}W
+w=1 to obtain the estimated risk curve of CC.
+Ul(✓)
+RZ(✓(l,1,1))
+RY (✓(l,1,2))
+RZ(✓(l,1,3))
+RZ(✓(l,2,1))
+RY (✓(l,2,2))
+RZ(✓(l,2,3))
+RZ(✓(l,3,1))
+RY (✓(l,3,2))
+RZ(✓(l,3,3))
+RZ(✓(l,4,1))
+RY (✓(l,4,2))
+RZ(✓(l,4,3))
+×𝐿
+(a)
+(b)
+Class 1:
+Class 2:
+Class 3:
+FIG. G.4.
+Visualization of image dataset and hardware-efficient Ansatz.
+(a) Image instances sampled from the
+Fashion-MNIST dataset. (b) The circuit architecture of the employed Hardware-efficient Ansatz. The label ‘×L’ denotes the
+layer number, which means repeating the gates in the dashed box with L times.
+SM G: Numerical simulation details
+Dataset. The construction of the parity dataset mainly follows from Ref. [98]. Note that this task has also been
+broadly studied in the field of deep learning to show the limits of deep neural classifiers [107, 108]. The constructed
+dataset contains in total 64 examples. Each example corresponds to a bit-string with the length 6, i.e., x ∈ {0, 1}6.
+The label of x is assigned to be 1 if the number of ‘0’ in x is even; otherwise, the label is 0. We split it into train dataset
+and test dataset with the train-test-split ratio being 0.75. The number of train examples in each class is controlled to
+be the same. For each example, its feature dimension is 10. The image dataset is adapted from Ref. [102]. Specifically,
+the data from the first nine classes are preserved and the total number of examples is 180. The train-test-split ratio is
+set as 0.5 to construct the train and test dataset. Each example corresponds to an image with 28 × 28 pixels. In the
+preprocessing stage, we flatten all examples followed by padding and normalization. The processed example yields an
+10-qubit state with x ∈ R210 and ∥x∥2
+2 = 1. Some examples after preprocessing are illustrated in Fig. G.4(a).
+Construction of QCs. The quantum subroutine of QC consists of the encoding circuit UE and the Ansatz U(θ).
+For all learning tasks, the hardware-efficient Ansatz is employed whose mathematical expression is U(θ) = �L
+l Ul(θ).
+The layout of the hardware-efficient Ansatz follows the layer-wise structure and the gate arrangement at each layer
+is the same. For ∀l ∈ [L], Ul(θ) = �N
+i=1(RZ(θ(l,i,1)) RY(θ(l,i,2)) RZ(θ(l,i,3)))Uent with Uent being the entanglement
+layer formed by CNOT gates. Fig. G.4(b) depicts the adopted hardware-efficient Ansatz with L layers.
+The encoding methods for the parity dataset classification and the digit images classification are different. The
+former uses the basis encoding method. Specifically, for a classical example x ∈ Rd, the employed encoding unitary
+
+21
+(a)
+(b)
+FIG. G.5. Geometric properties of the quantum feature states on parity dataset. (a) The averaged performance of
+QC evaluated by M1 defined in Eq. (G1). The label ‘Init-C-k’ with k = 1, 2 refers that the value of M(k)
+1
+at the initialization.
+Similarly, the label ‘Final-C-k’ with k = 1, 2 refers that the value of M(k)
+1
+when the training of QC is completed. (b) The
+averaged performance of QC evaluated by M2 defined in Eq. (G2). The label ‘Init-C-1-2’ (‘Final-C-1-2’) refers that the value
+of M2 before and after training of QC. The label ‘L = a’ in the x-axis stands for that the layer number of hardware-efficient
+Ansatz is a.
+is UE(x) |0⟩⊗d = |x⟩, which maps x to a 2d dimensional quantum state UE(x) |0⟩⊗d. The latter uses the amplitude
+encoding method. Given a normalized image x ∈ R64 with ∥x∥2
+2 = 1, the corresponding unitary encodes it into a
+6-qubit state with UE(x) |0⟩⊗6 = �64
+j=1 xj |j⟩.
+The Pauli-based measure operators are used in learning Fashion-MNIST dataset. Since the preprocessed dataset
+contains 9 classes, there are in total 9 measure operators, i.e., o(1) = X⊗X⊗I⊗8, o(2) = X⊗Y ⊗I⊗8, o(3) = X⊗Z⊗I⊗8,
+o(4) = Y ⊗X ⊗I⊗8, o(5) = Y ⊗Y ⊗I⊗8, o(6) = Y ⊗Z ⊗I⊗8, o(7) = Z ⊗X ⊗I⊗8, o(8) = Z ⊗Y ⊗I⊗8, o(9) = Z ⊗Z ⊗I⊗8.
+Multilayer Perceptron.
+To better justify the capability and performance of QCs, we apply the multilayer
+perceptron (MLP) as the reference [109]. MLP is composed of an input layer, L hidden layers with L ≥ 1, and an
+output layer. The dimension of the input layer is equivalent to the feature dimension of the input. ReLU activations
+are added in the hidden layer to perform nonlinear transformation. In the output layer, the activation function,
+Softmax, is employed. The number of layers L depends on the assigned tuples {n, Nt, T}.
+Convolutional neural network. In the task of image classification, convolutional neural networks (CNNs) is
+employed as the reference [109]. The employed CNN is formed by two convolutional layers and one fully-connected
+layer. ReLU activations and the pooling operation are added in the hidden layer to perform nonlinear transformation.
+The number of channels for the first convolutional layer is fixed to be 8 and the corresponding kernel size is 9 × 9.
+The kernel size of the pooling operation applied to the two convolutional layers is 2×2. The kernel size for the second
+convolutional layer is fixed to be 5×5 but the number of output channels is varied depending on the settings in Alg. 1.
+For the sake of fair comparison, the number of output channels is set as 2, 6, 15, 30, 50, 75, where the corresponding
+number of parameters is 860, 1284, 2238, 3828, 5948, and 8598, respectively.
+Optimizer and other hyper-parameters. The adaptive gradient descent method, named AdaGrad optimizer
+[110], is used to optimize QCs and MLPs. Compared to the vanilla gradient descent method, AdaGrad permits better
+performance, since it adapts the learning rate for each feature depending on the estimated geometry of the problem.
+In the task of parity learning, the initial learning rate is set as η = 0.5 for QC and η = 0.01 for MLP, respectively.
+For both classifiers, the batch size is fixed to be 4. In the task of image classification, the initial learning rate is set
+as η = 0.05 for QC and η = 0.01 for CNN, respectively. The batch size for both classifiers is set as 1.
+Curve fitting method. To capture the risk curve, Alg. 1 requests a curve fitting method. For all experiments,
+we adopt the polynomial fitting to derive the risk curve by using the collected results. The least squares method in
+determining the best fitting functions.
+Source code.
+The source code used in numerical simulations will be available at Github repository https:
+//github.com/yuxuan-du/Problem-dependent-power-of-QNNs.
+
+0.8
+Init-C-1-2
+Fin-C-1-2
+0.6
+0.4
+0.2
+0.0
+L=1L=2L=3L=4L=5L=6L=74.8
+Init-C-1
+Init-C-2
+4.0
+Fin-C-1
+Fin-C-2
+3.2
+Distance
+2.4
+1.6
+0.8
+0.0
+L=1L=2L=3L=4L=5L=6L=722
+(a)
+(b)
+FIG. G.6. Train (test) accuracy versus epoch on parity dataset. (a) Train accuracy and test accuracy of QC with the
+varied layer number. The label ‘L = a’ refers that the layer number used in hardware-efficient Ansatz is a. The solid line and
+the dashed line separately correspond to the train and test accuracies of QC. (b) Train accuracy and test accuracy of MLP
+with the varied number of hidden neurons. The label ‘h = a’ refers that the number of neurons is a. The solid and dashed
+lines have the same meaning with those in QC.
+1.
+Simulation results of the binary classification for the parity dataset
+The feature states before and after training. We explore the geometric properties of feature states when the
+layer number of hardware-efficient Ansatz varies from L = 1 to L = 7. Other settings are identical to those introduced
+in the main text. Condition (i) in Lemma 2 is evaluated by the metric
+M(k)
+1
+=
+nc
+�
+i=1
+∥ρ(i,k) − ¯ρ(k)∥,
+(G1)
+where the number of train examples {ρ(i,k)}nc
+i=1 belonging to the k-th class is nc and ¯ρ(k) refers to their class-feature
+mean. Since parity learning is a binary classification task, Condition (ii) in Lemma 2 is evaluated by
+M2 = Tr(¯ρ(0)¯ρ(1)).
+(G2)
+The geometric properties of the feature states in the measure of M(k)
+1
+and M2 are visualized in Fig. G.5. The left
+panel shows that when L ∈ {2, 3, 4, 5}, both the value of M(1)
+1
+(highlighted by the green color) and M(2)
+1
+(highlighted
+by the pink color) decrease from ∼ 3.2 (epoch t = 0) to ∼ 0.5 (epoch t = 40). These results comply with Condition
+(i) in the sense that the feature states in the same class concentrates to the class-feature mean and leads to the low
+empirical risk. By contrast, when L is too small or too large, the value of M(1)
+1
+changes subtly before and after
+optimization, which is above 3.2. The large deviation of feature states incurs the degrade performance of QC. The
+right panel depicts that when L ∈ {2, 3, 4, 5}, the value of M(2)
+1
+decreases from 0.5 (epoch t = 0) to 0.05 (epoch
+t = 40). This reduction means that the class-feature means are maximally separated and thus ensure a good learning
+performance. On the contrary, when L ∈ {1, 6, 7}, the the value of M(2)
+1
+oscillates around 0.5, which implies that the
+class-feature means ¯ρ(1) and ¯ρ(2) are highly overlapped.
+The learning dynamics of QC and MLP. Fig. G.6 visualizes the learning dynamics of QC and MLP with
+respect to the varied trainable parameters. The left panel indicates that when the layer number is L = 2, 3, 4, both
+train and test accuracies of QC fast converge to 100% with 25 epochs. When L = 1, both train and test accuracies
+oscillate to 50%. When L = 7, the number of train data becomes insufficient and the overfitting phenomenon appears.
+These results accord with the U-shape risk curve of QCs. The right panel shows that when the number of hidden
+neurons ranges from h = 1 to h = 18, the test accuracy of MLP is no higher that 55%. These results reflect the
+incapability of MLP in learning parity dataset compared with QCs.
+2.
+Simulation results of multi-class classification for the Fashion-MNIST images dataset
+The feature states before and after training. Here we discuss the geometric properties of feature states when
+the layer number of hardware-efficient Ansatz varies from L = 2 to L = 150. The metrics M(k)
+1
+and M2 defined in
+
+1.00
+h=1
+h=2
+0.85-
+h=6
+h=10
+0.70
+h = 14
+Acc
+h= 18
+0.55
+0.40
+0
+10
+20
+30
+401.00
+0.85
+0.70
+0.55
+0.40
+0
+10
+20
+30
+4023
+(a)
+(b)
+FIG. G.7. Geometric properties of the quantum feature states on Fashion-MNIST dataset. (a) The averaged
+performance of QC evaluated by M1 defined in Eq. (G1). (b) The averaged performance of QC evaluated by M2 defined in
+Eq. (G2). All labels have the same meaning with those introduced in Fig. G.5.
+(a)
+(b)
+FIG. G.8. Train (test) accuracy versus epoch on Fashion-MNIST dataset. (a) Train accuracy and test accuracy of
+QC with the varied layer number. The labels have the same meaning with those presented in Fig. G.6. (b) Train accuracy
+and test accuracy of CNN with the varied number of trainable parameters. The label ‘h = a’ refers that the number of output
+channels at the second layer is a. The solid and dashed lines have the same meaning with those in QC.
+Eqs. (G1) and (G2) are employed. In the measure of M2, since the performance of QC for any two classes is similar,
+we only study the first two classes for ease of visualization.
+Fig. G.7 depicts the geometric properties of the feature states in the measure of M(k)
+1
+and M2. The left panel
+shows that for all settings with L ∈ {2, 5, 25, 50, 100, 150}, the value M(k)
+1
+at the initial step and the final step is very
+similar and M(k)
+1
+is larger than 0.2 for ∀k ∈ {1, 2, ..., 9}. These results indicate that QC cannot satisfy Condition
+(i) when learning Fashion-MNIST dataset, where the feature states from the same class cannot collapse to a unique
+point. Moreover, when we examine the performance of intra-class, the right panel implies that after training, the
+class-feature means of QC are still highly overlapping. The distance for all settings of L is above 0.3. The inability
+to achieve the optimal training loss shows the the limited power of QC on learning Fashion-MNIST dataset.
+The learning dynamics of QC and CNN. Fig. G.8 depicts the learning dynamics of QC and CNN with the
+varied number of trainable parameters. The left panel indicates that QC achieves the best performance when the layer
+number is L ∈ [25, 100], where the corresponding number of parameters ranges from 750 to 3000. In these settings,
+both train and test accuracies of QC are around 30% after 50 epochs. When L < 25 or L > 100, both train and test
+accuracies oscillate at 15%. These results accord with the U-shape risk curve of QCs. The right panel shows that
+the train and test accuracies of CNN are steadily growing with the increased number of channels. That is, when the
+number of channels at the second layer is not less than 6, both the train and test accuracies are higher than 60%.
+These results indicate that the employed QC does not have potential advantages in learning image dataset compared
+with CNN.
+
+Init-C-1-2
+Fin-C-1-2
+0.8
+Distance
+e
+0.6
+0.4
+0.2
+0.0
+L = 2
+L= 5
+5L=25L=50L=100L=150Init-C-1
+Fin-C-5
+Fin-C-1
+Init-C-6
+1.6
+Init-C-2
+Fin-C-6
+Fin-C-2
+Init-C-7
+Init-C-3
+Fin-C-7
+e 1.2
+Fin-C-3
+Init-C-8
+Distance
+Init-C-4
+Fin-C-8
+Fin-C-4
+Init-C-9
+0.8
+Init-C-5
+Fin-C-9
+0.4
+0.0
+L = 2
+L= 5
+L=25L=50L=100L=1501.00
+0.75
+h=2
+00.50
+h=6
+h= 15
+h = 30
+0.25
+h= 50
+h= 75
+0.00
+0
+10
+20
+30
+40
+500.45
+L=2
+L= 5
+L= 25
+0.30
+L = 50
+L= 100
+Acc
+_= 150
+0.15
+0.00
+0
+10
+20
+30
+40
+50
\ No newline at end of file
diff --git a/ftAzT4oBgHgl3EQfof1Y/content/tmp_files/load_file.txt b/ftAzT4oBgHgl3EQfof1Y/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf,len=1247
+page_content='Demystify Problem-Dependent Power of Quantum Neural Networks on Multi-Class  Classification  Yuxuan Du,1, ∗ Yibo Yang,1 Dacheng Tao,1 and Min-Hsiu Hsieh2  1JD Explore Academy, Beijing 10010, China  2Hon Hai (Foxconn) Research Institute, Taipei, Taiwan  Quantum neural networks (QNNs) have become an important tool for understanding the physical  world, but their advantages and limitations are not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Some QNNs with specific en-  coding methods can be efficiently simulated by classical surrogates, while others with quantum mem-  ory may perform better than classical classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Here we systematically investigate the problem-  dependent power of quantum neural classifiers (QCs) on multi-class classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Through  the analysis of expected risk, a measure that weighs the training loss and the generalization error of  a classifier jointly, we identify two key findings: first, the training loss dominates the power rather  than the generalization ability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' second, QCs undergo a U-shaped risk curve, in contrast to the  double-descent risk curve of deep neural classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' We also reveal the intrinsic connection between  optimal QCs and the Helstrom bound and the equiangular tight frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Using these findings, we  propose a method that uses loss dynamics to probe whether a QC may be more effective than a  classical classifier on a particular learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Numerical results demonstrate the effectiveness of  our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and  their limitations over convolutional neural networks on image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Our work sheds light on the  problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  INTRODUCTION  The advent of hardware fabrication pushes the bound-  ary of quantum computing from verifying its superiority  on artificial tasks [1–3] to conquering realistic problems  with merits [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' This has led to the emergence of a  popular paradigm known as quantum neural networks  (QNNs), which combine variational quantum Ans¨atze  with classical optimizers [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  So far, various QNN-  based methods have been proposed to address difficult  problems in areas such as quantum physics [9–12], quan-  tum information theory [13–16], combinatorial optimiza-  tion [17–21], and machine learning [22–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Among these  applications, QNNs are often deployed as quantum clas-  sifiers (QCs) to predict correct labels of the input data  [27–32], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', categorize image objects [33–35], classify  phases of quantum matters [36–39], and distinguish en-  tangled states from separable states [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  To comprehend the full potential of existing quantum  classifiers (QCs) and to spur the development of novel  QCs, huge efforts have been made to unveil the learnabil-  ity of QCs [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Prior literature establishes the foun-  dations of QCs from three primary aspects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', model  capacity [45–48], trainability [49–51], and generalization  [52–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Nevertheless, the advantages and constraints of  QCs have rarely been proven [57–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Meanwhile, pre-  vious results cannot rigorously explain the empirical ob-  servations such that QCs generally outperform classical  classifiers (CCs) on handcraft or quantum data [44, 63]  but are inferior to them on realistic problems [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' As  a result, the need for QCs to address classical issues re-  mains highly questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  ∗ duyuxuan123@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='com  A principal criteria in characterizing the power of a  classifier is the expected risk [65], which weighs the em-  pirical risk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', training loss) and the generalization er-  ror (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', test loss) jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' An optimal classifier is one  which achieves zero expected risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1(a),  the success of deep neural classifiers is attributed to their  double-descent risk curves [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' This means that as  the hypothesis space is continually expanded, the ex-  pected risk of a trained deep neural classifier initially  decreases, increases, and when it overfits the train set,  undergoes a second descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' As such, to show the supe-  riority of QCs over CCs, it demands to distill ubiquitous  rules that capture the risk curve of diverse QCs in addi-  tion to conditions where the expected risk of QCs can be  lower than CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In this study, we unify a broad class of QCs in  the same framework and understand their problem-  dependent ability under the expected risk (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Our analysis reveals two substantial outcomes: (i) train-  ability dominates QCs’ ability more than generalization  ability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (ii) QCs undergo a U-shape risk curve instead  of the double-descent curve for CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  These outcomes  consolidate and refine previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Concretely,  the first outcome suggests that the deficiency of QCs on  classical data stems from their limited ability to fit the  train set, resulting in a larger training loss compared to  CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The second outcome highlights the distinct learn-  ing behavior of QCs and CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Despite the fact that over-  parameterization is fundamental to enhance the perfor-  mance of CCs, it adversely affects the power of QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In line with the diverse dynamics of the risk curves for  QCs and CCs, we devise an efficient problem-dependent  method to recognize potential merits of QCs, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Conceptually, for a given learning task, our  method fits the loss (risk) dynamics of QC and CC under  the prior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', U-shape versus double descent) and then  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='01597v1  [quant-ph]  29 Dec 2022  2  (a)  (b)  𝑜(") 𝑜($) 𝑜(%)  𝜌(&)  ℇ(𝑥(&))  𝑥(&)  𝑜(\')  ℛ  𝐶-ℛ  Hypothesis space  𝑄-ℛ  𝑄-ℛ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' "#  ℛ  𝐶-ℛ  Hypothesis space  𝑄-ℛ  𝑄-ℛ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' "#  (c)  𝜌̅(%), 𝑜(%)  𝜌̅ \' , 𝑜(\')  𝜌̅ ( , 𝑜(() 𝜌̅ ) , 𝑜())  optimize  𝜌̅(%), 𝑜(%)  𝜌̅ \' , 𝑜(\')  𝜌̅ ( , 𝑜(() 𝜌̅ ) , 𝑜())  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Risk curve and geometry of the unified QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (a) The risk curve of QCs and CCs are highlighted by the solid red  and blue lines (labeled by ‘Q-R’ and ‘C-R’), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The former yields a ‘U’ shape while the latter yields a double-descent  tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Potential advantages of QCs are dominated by the empirical risk, highlighted by the dashed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The shaded  region refers to the potential merits of QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (b) The unified QC consists of two parts, the feature state ρ and the measure  operator o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' This model covers diverse QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (c) Geometric relationship between {ρ(i,k)} and o of QCs with (near) zero training  loss: (i) the feature states associated with train samples belonging to the same class concentrate around their class-feature  mean, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', ¯ρ∗(k) := ρ∗(1,k) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' = ρ∗(nc,k) for ∀k ∈ [K];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (ii) the class-feature means are maximally distant with each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',  Tr(¯ρ∗(k)¯ρ∗(k′)) ∼ δk,k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (iii) the measure operator should align with class-feature means, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', Tr(¯ρ∗(k)o∗(k′)) ∼ δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  identify the ‘advantage’ regime where the risk of QC is  lower than CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Numerical simulations are conducted to  support our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  On the technical level, we approach the two outcomes  by separately quantifying the empirical risk and gener-  alization error of QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Specifically, we first prove con-  ditions of QCs that lead to near-zero empirical risk, the  geometric interpretation of which is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  As a byproduct, we elucidate how such conditions are  inherently linked to quantum state discrimination and  quantum measurement theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In addition, we prove  that deep QCs can never reach the vanished empirical  risk by utilizing the concentration property of quantum  observables [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' We next analyze the generalization  error of QCs by exploiting algorithmic robustness [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The derived bound surpasses prior results because it is  the first non-vacuous bound in the over-parameterized  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  By combining the unreachable zero empirical  risk with the manipulatable generalization error, we ob-  tain the first outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The second outcome is gained by  integrating the fact that deep QCs are unable to reach  the vanished empirical risk with the first outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  MAIN RESULTS  Expected risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='— Let us first introduce a K-class (K ≥  2) classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Denote the input space as X, the  label (class) space as Y = {1, · · · , K}, and the train set  as D = �K  k=1{(x(i,k), y(i,k))}nk  i=1 with |D| samples drawn  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  from an unknown probability distribution D on  Z = X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In standard scenarios, the number of train  samples in each class is the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', n1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' = nk ≡ nc  and |D| := n = Knc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The purpose of a classification  algorithm A is using D to infer a hypothesis (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', a  classifier) hAD : X → RK from the hypothesis space H  to separate train examples from different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' This  is equivalent to identifying an optimal hypothesis in H  minimizing the expected risk R(h) = E(x,y)∼D[ℓ(h(x), y)],  where ℓ(·, ·) is the per-sample loss and for clarity we spec-  ify it as the square error with ℓ(a, b) = 1  2∥a − b∥2  2 [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Unfortunately, the inaccessible distribution D forbids us  to assess the expected risk directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In practice, A al-  ternatively learns an empirical classifier ˆh ∈ H, as the  global minimizer of the (regularized) loss function  L(h, D) = 1  n  nc  �  i=1  K  �  k=1  ℓ(h(x(i,k)), y(i,k)) + E(h),  (1)  where E(h) is an optional regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The foremost role of the risk means that quantum ad-  vantages can be ascertained if R(ˆhQ) < R(ˆhC), where ˆhQ  and ˆhC are the empirical QC and CC on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Unlike con-  ventions merely focusing on a QC on one specific task,  the above criteria orients to unearth ubiquitous rules of  QCs with computational advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  To reconcile the  intractable issue of R(ˆh) and proceed the following anal-  ysis, we decomposed it into two measurable terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',  R(ˆh) = RERM(ˆh) + RGene(ˆh),  (2)  where RERM(ˆh) = 1  n  �n  i=1  �K  k=1 ℓ(ˆh(x(i,k)), y(i,k)) is the  empirical risk and RGene(ˆh) = R(ˆh)−RERM(ˆh) is the gen-  eralization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (2), detecting advances  of QCs is translated to deriving under what conditions  do QCs commit both lower RERM and RGene than CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  To better elucidate our results, let us recall that the  general form of QC is ˆhQ = arg minhQ∈HQ L(hQ, D),  Learning tasks  Advantages  No Advantages介83  where L is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (1) and HQ is the hypothe-  sis space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' For an N-qubit QC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' its hypothesis space is  HQ =  ��  hQ(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' U(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' O(k))  �  k=1:K  ���θ ∈ Θ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (3)  where [·]k=1:K is a K-dimensional vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' its k-th entry  hQ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' U(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' O(k)) = Tr(O(k)U(θ)σ(x)U(θ)†) for ∀k ∈  [K] refers to the output (prediction) of quantum cir-  cuits,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' σ(x) = UE(x)(|0⟩ ⟨0|)⊗NUE(x)† is the input state  of x with the encoding circuit UE(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' O = {O(k)}K  k=1  is a set of measure operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' and U(θ) is the adopted  Ansatz with trainable parameters θ living in the pa-  rameter space Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Without loss of generality, we define  U(θ) = �Nt  l=1(ul(θ)ue) ∈ U(2N), where ul(θ) ∈ U(2m)  is the l-th parameterized quantum gate operated with at  most m qubits (m ≤ N) and ue refers to fixed quan-  tum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Similarly, we define UE(x) = �Ng  g=1 ug(x) ∈  U(2N), where ug(x) ∈ U(2m) refers to the g-th quan-  tum gate operated with at most m qubits, and Ng gates  contain Nge tunable gates and Ng − Nge fixed gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Due to the diverse constructions of U(θ) and UE(·), it  is necessary to unify various QCs into the same frame-  work to obtain the generic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Notably, the unified  QC should be agnostic to particular forms of these two  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' A feasible way is rewritten hQ(·, U(θ), O(k)) as  hQ(ρ(i,k), o(k)) := Tr(ρ(i,k)o(k)) ∀k ∈ [K],  (4)  where O(k) = I2N−D ⊗o(k) with the nontrivial local oper-  ator o(k) ∈ C2D×2D, D describes the locality, and ρ(i,k) =  TrD(U(θ)σ(x(i,k))U(θ)†) corresponds to the state before  measurements, named as feature state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' An intuition of  the unified QC is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We are now ready to exploit the unified framework  to analyze the expected risk of QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Let ρ = {ρ(i,k)}  and o = {o(k)} be two sets collecting all feature states  and measure operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The following theorem exhibits  conditions in which QCs allow a low expected risk, where  the formal statement and the proof are deferred to SM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Theorem 1 (informal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Following notations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (1)-  (4), when the train data size is nO(KNge log KNg  ϵδ ) with ϵ  being the tolerable error, and the optimal sets of ρ∗ and  o∗ satisfy three conditions: (i) the feature states have  the vanished variability in the same class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (ii) all feature  states are equal length and are orthogonal in the varied  classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (iii) any feature state is alignment with the mea-  sure operator in the same class, with probability 1−δ, the  expected risk of QC tends to be zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', R(ˆhQ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Conditions (i)-(iii) visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1(c) sculpt the ge-  ometric interpretations of ρ∗ and o∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' These properties  come across the design philosophy of CCs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', linear dis-  criminant analysis and neural collapse phenomenon ap-  peared in most deep neural classifiers [71–73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Moreover,  these conditions unveil the intrinsic connection between  optimal QCs and the quantum state discrimination [74],  since ρ∗ and o∗ should maximize the Helstrom bound  [75], which explains the ultimate limit of QCs observed  in [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' However, as will be explained later (see Corol-  lary 1 and Lemma 1), under certain scenarios, it is hard  for QCs to meet these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' A typical instance is  applying QC to learn the image dataset, where the dif-  ficulty stems from the limited nonlinearity of QC to fit  the train set, thereby inducing a large empirical risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Conditions (i)-(iii) also imply how the quantum mea-  surement theory can be used to guide the design of QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Namely, the mean feature states of each class {¯ρ∗(k)}  compose the equiangular tight frame (ETF) and Con-  dition (iii) suggests that the optimal measure operators  {o∗} also satisfy this ETF [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Due to the relation be-  tween symmetric informationally complete (SIC) mea-  surements and ETF [78, 79], the optimal measure op-  erators could be estimated by various SIC construction  strategies [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Besides, the locality D of {o∗} should  be carefully selected in QCs in which a small D pre-  cludes the acquisition of the optimal QCs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', the com-  plex ETF does not exist when 2D = K [81, 82]), while an  extremely large D may incur the barren plateaus [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Furthermore, when K is large, Pauli-based measurements  are preferable than computational basis measurements in  QCs, since the former allows classical shadow techniques  to accelerate the training of QCs [85, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The scaling behavior of n indicates that it is data-  efficient for QCs to attain a low generalization error,  where the size of train set only linearly depends on the  class number K and the number of encoding gates Nge  (see Lemma 3 for the technical elaboration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In other  words, the generalization error of QCs can be well con-  trolled by the modest-size train data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  According to Theorem 1, the challenges in satisfying  Conditions (i)-(iii) and the well controlled generalization  error pinpoint that the risk of a QC is mostly dominated  by its empirical loss rather than its generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In this view, the core in devising advanced QCs is tailor-  ing HQ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (3) so that ˆhQ has a (near) zero empirical  risk on D, or equivalently examining whether the em-  ployed QCs can fulfill Conditions (i)-(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  U-shape risk curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='—The risk curve concerns how  the expected risk of a classifier behaves with the varied  hypothesis space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' It is desired that as with deep neu-  ral classifiers, QCs undergo a double-descent risk curve  in the sense that the over-parameterized QCs consent  a low expected risk when the trainable parameters Nt  is much greater than the train data n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  If so, ‘over-  parameterization’ could serve as a golden law in designing  QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' However, the following corollary refutes the exis-  tence of the double-descent risk curve for QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Corollary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Following  notations  in  Theorem  1,  when {UE(x)|x  ∈  X} follows the Haar distribu-  tion, with probability 1 − δ, the empirical QC follows  | Tr  �  σ(x(i,k))σ(x)  �  − 1  2N | ≤  �  3  22Nδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When {U(θ)|θ ∈ Θ}  follows the Haar distribution, with probability 1 − δ,  the empirical QC follows | Tr(ρ(i,k)o(k′)) − Tr(o(k′))  2D  | <  4  �  Tr(o(k′))2+2 Tr((o(k′))2)  22Dδ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The proof is deferred to SM B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The achieved results re-  veal the caveat of deep QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Specifically, when UE(x)  is deep, two encoded states σ(x(i,k)) and σ(x(i′,k)) from  the same class tend to be orthogonal, which contradicts  with Conditions (i) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Besides, when U(θ)  is deep, the output of QC concentrates to zero, regard-  less how o(k′) and ρ(i,k) are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' This violates Con-  dition (iii) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Overall, over-parameterized  QCs encounter the high empirical risk and thus the high  expected risk, which suggests that QCs experience a U-  shape risk curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' This observation differs the dynamics  of QCs from variational quantum Eigensolvers, since the  latter can benefit from over-parameterization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', better  trainability and convergence rate [87–90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Moreover, the  rule of thumb in QCs’ construction is slimming HQ to  find the valley region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Intriguingly, tailoring the feature  states echoes with quantum metric learning and quantum  self-supervised learning [91–95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Probe power of QCs via loss dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='—The dis-  tinct tendency of the risk curves between QCs and CCs  provides a succinct way to recognize the potential quan-  tum advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1(a), given a specific  data set, the U-shape risk curve of QCs indicates that its  advantages mostly appear in the valley region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Precisely,  if the risk values of QC around the basin are lower than  those of CC, potential merits may exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' otherwise, QC  is inferior to CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The proved learning behavior of QCs,  accompanied with the tight generalization bound, allows  us to effectively fit its risk curve according to their loss  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Specifically, our method contains three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  First, W tuples of {n, Nt, T} are initialized based on The-  orem 1 so that the collected risk points of QC span the  basin area with low generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Second, we ex-  ecute QC and CC under these W hyper-parameter set-  tings and fit their loss dynamics to attain the risk curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Last, we compare two risk curves and probe potential  advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' See SM F for the implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Technical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='—Theorem 1 is achieved by ana-  lyzing when RERM(ˆhQ) and RGene(ˆhQ) are (near) zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In the analysis of RERM(ˆhQ), we first consider the most  general case in which both ρ and o are tunable, where  ˆhQ ≡ hQ(ρ∗, o∗) with (ρ∗, o∗) = minρ,o L(ρ, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma 1 (Informal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When the regularizer E is consid-  ered and (ρ∗, o∗) meets the three conditions in Theorem  1, the global minimizer leads to RERM(ˆhQ) = C2  1/2 with  C1 depending on the hyper-parameters in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The achieved properties of o∗ can be used as a priori to  simplify QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' To this end, the following lemma quantifies  RERM(ˆhQ) when o is predefined and E = 0, where ˆhQ ≡  hQ(ρ∗, o) with ρ∗ = minρ L(ρ, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma 2 (Informal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When the predefined {o(k)} are  mutually orthogonal with each other and the conditions  in Theorem 1 are satisfied, we have RERM(ˆhQ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The proofs of Lemmas 1 and 2 are given in SM C&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We next analyze RGene(ˆhQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Prior results cannot be  used to prove Theorem 1, since such bounds polynomially  scale with the trainable parameters and become vacuous  in the over-parameterized regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' To remedy this issue,  we utilize the concept of algorithmic robustness [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Definition 1 (Robustness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' A learning algorithm A is  (R, ν(·))-robust with R ∈ N and ν(·) : Zn → R, if Z can  be partitioned into R disjoint sets, denoted by {Cr}R  r=1,  such that the following holds for all D ⊂ Zn : ∀s =  (x(i), y(i)) ∈ D, ∀z = (x, y) ∈ Z, ∀r ∈ [R],  s, z ∈ Cr ⇒ |l(hAD(x(i)), y(i)) − l(hAD(x), y)| ≤ ν(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Concisely, robustness measures how much the loss value  can be varied with respect to the input space Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' A higher  robustness of a classifier admits lower R, ν(·), and RGene  [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The following lemma quantifies the upper bound of  RGene(ˆhQ) whose proof is given in SM E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Suppose the measure operator is bounded by  C2 with maxk∈[K] ∥o(k)∥ ≤ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Define ϵ as the tolerable  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Following notations in Definition 1, the empiri-  cal QC is (K(28Nge/ϵ)4mNge, 4L1KC2ϵ)-robust, and with  probability 1 − δ we have  RGene(ˆhQ) ≤ 4L1KC2ϵ + 5ξ(ˆhQ)  �  |TD|4mNge ln 56KNge  ϵδ  n  ,  where L1 is the Lipschitz constant of ℓ with respect to  hQ, ID  r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)),  and TD := {r ∈ [R] : |ID  r | ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The achieved results convey threefold insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' First, our  bound does not explicitly depend on the number of train-  able parameters [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' This unlocks a new way to under-  stand the generalization ability of QCs, especially for the  over-parameterized ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Next, our bound hints that a  carefully designed UE can enhance performance of QCs  [53, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Last, RGene(ˆhQ) → 0 requires n ≫ |TD|4mNge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Fortunately, a reasonable value of n is sufficient to war-  rant this condition, because in general m ≤ 2, Nge ∝ |x|,  and |TD| is continuously decreased from n to K with re-  spect to the reduced empirical loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  NUMERICAL SIMULATIONS  We conduct numerical simulations to exhibit that the  advantages and limitations of QCs on different classifica-  tion tasks can be interpreted by the derived risk curve  and feature states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The omitted construction details and  results are deferred to SM G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We first apply QC to accomplish the binary classifica-  tion on the parity dataset [98–100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The number of qubits  is N = 6 and the hardware-efficient Ansatz is adopted  to realize U(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The gradient descent method is used  as the classical optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Two measure operators are  5  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Binary classification on the parity dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (a)  The learning performance of QC when the layer number is  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The x-axis denotes the epoch numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Shaded region  represents variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The Bloch spheres display the quantum  feature states at different epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (b) The fitted risk curve  of QC and MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The x-axis denotes the number of trainable  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘QC-risk’ (‘MLP-risk’) refers to the  fitted risk curve of QC and MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘QC-res’ (‘MLP-  res’) refers to the collected results used for fitting the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  o(1) = |0⟩ ⟨0| and o(2) = |1⟩ ⟨1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The simulation results of  QC with Nt = 54 are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Particularly,  the averaged train (test) accuracy steadily grows from  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='1% to 100% within 22 epochs, and the corresponding  loss decreases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='26 to 4 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The dynamics of  the feature states {ρ(i,t)} with t ∈ {0, 10, 20, 30, 40} vi-  sualized by Bloch spheres echo with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Besides,  QC becomes more robust when we continue the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Although the train (test) accuracy reaches the optimum,  the loss can be further reduced and suggests a lower risk  warranted by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' We further compare the risk  curve between QC and multilayer Perceptron (MLP) on  this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' We fit their risk curves following the pro-  posed method to probe potential quantum merits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 2(b), QC clearly outperforms MLP when  the trainable parameters ranges from 20 to 140 and the  valley of the risk curve is around Nt = 70 [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We then apply QC to learn the Fashion-MNIST im-  age dataset with K = 9 [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The employed number of  qubits is N = 10 and the Pauli-based measure operators  are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Convolutional neural networks (CNNs)  are exploited as the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' For all classifiers, the num-  ber of epochs is fixed to be T = 50 and the number of  trainable parameters Nt ranges from 60 to 9000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Each  setting is repeated with 3 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 3,  when the layer number is 50 with Nt = 1500, both the  train and test accuracies of QC are about 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  This  performance is inferior to CNN under the similar setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  To explore whether QC has the potential to outperform  CNN on this dataset, we compare their risk curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 3(b), unlike the parity dataset, QC is evi-  dently inferior to CNN on Fashion-MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Multi-class classification on the image dataset  with K = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (a) The learning performance of QC when the  layer number is 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (b) The fitted risk curve of QC and CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  All labels have the same meaning with those used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  DISCUSSIONS AND OUTLOOK  We understand the potential of diverse QCs in terms of  the expected risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Our theoretical findings demonstrate  that the efficacy of QCs is dependent on the problem at  hand, which explains the empirical evidence of their supe-  riority on synthetic and quantum datasets, yet inferiority  on realistic tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' With the clear difference between the  risk curve of QCs and deep neural classifiers, we present a  concise technique to investigate potential quantum ben-  efits by fitting their loss dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Numerical results  validate our theoretical results and the effectiveness of  our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  There are several interesting future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The U-shape curve of QCs poses two open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  First, can contemporary QCs attain quantum benefits  on certain classical data when only limited data and re-  stricted computing resources are available?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Secondly,  is it necessary to redesign QCs such as nonlinear QCs  [103, 104] that can also exhibit a double-descent risk  curve?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Besides, the unearthed connection between the  conditions towards optimal empirical risk and quantum  state discrimination opens a new research avenue that  amplifies the potential of QCs on quantum data aided  by quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Finally, it is intrigu-  ing to extend the developed non-vacuous generalization  error bound of QCs to other scenarios, such as out-of-  distribution data, in order to identify potential quantum  advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors thank Xinbiao Wang for valuable input  and inspiring discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='0  Loss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2  Train  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='8  Loss  Test  Acc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4  0  10  20  30  4010)  X[0]  y  X[0]  X[0)  y  X[0)  y  X  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4  QC-risk  K  S  QC-res  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2  MLP-risk  MLP-res  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='0  0  20  40  60  80  100  120  140  1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='00  QC-risk  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='75  OC-res  K  CNN-risk  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='50  R  CNN-res  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='00  0  1000  2000  3000  4000  5000  6000  7000  8000  90001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='75  Loss  SS  Train  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='50 0  9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6  Test  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content=' An analog quantum variational em-  9  bedding classifier, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='02748v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content=' Therefore, for simplicity, we discard the  term f(U(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content='  arXiv  preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content='  Quantum learning robust against noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content=' Demonstration of quantum advantage in machine  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' npj Quantum Information, 3(1):1–5, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content='  Variational quantum  classifiers through the lens of the hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content=' In other words,  over-  parameterization and sufficient train data are two nec-  essary conditions to induce the double-descent curve,  while parity dataset can only provide limited train data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content=' Fashion-  mnist: a novel image dataset for benchmarking machine  learning algorithms, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content=' PRX  Quantum, 3:030101, Jul 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  [104] Zo¨e Holmes, Nolan Coble, Andrew T Sornborger, and  Yi˘git Suba¸sı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' On nonlinear transformations in quantum  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='12307, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  [105] Kenji Kawaguchi, Zhun Deng, Kyle Luh, and Jiaoyang  Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Robustness implies generalization via data-  dependent generalization bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In International  Conference on Machine Learning, pages 10866–10894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  [106] Thomas Barthel and Jianfeng Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Fundamental limita-  tions for measurements in quantum many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+page_content=' Learning parities with  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Advances in Neural Information Pro-  cessing Systems, 33:20356–20365, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  [108] Boaz Barak, Benjamin L Edelman, Surbhi Goel, Sham  Kakade, Eran Malach, and Cyril Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Hidden  progress in deep learning: Sgd learns parities near the  computational limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='08799,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  [109] Ian Goodfellow, Yoshua Bengio, and Aaron Courville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' MIT press, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  [110] John Duchi, Elad Hazan, and Yoram Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Adaptive  subgradient methods for online learning and stochas-  tic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Journal of machine learning research,  12(7), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  10  The organization of the supplementary material (SM) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In SM A, we present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Then, we provide the proof of Corollary 1 in SM B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Subsequently, we demonstrate the proof of Lemma 1 and Lemma  2 in SM C and SM D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Next, in SM E, we exhibit the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In the end, we elucidate the  details of numerical simulations in SM G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  SM A: Proof of Theorem 1  For convenience, let us first recall the settings and notations introduced in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When QCs are applied to  accomplish the multi-class classification task, the training dataset D contains n examples and the number of examples  in each class is the same with n = ncK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Moreover, the per-sample loss is specified as the mean square error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We next introduce the formal description of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In particular, Theorem 1 is established on Lemma 2, where  the regularization term is set as zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', E = 0) and the set of measure operator is predefined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', o spans the  space C2D×2D and satisfies Tr(o(k)o(k′)) = Bδk,k′ where B ≥ 1 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The requirements of o aims to preserve  Condition (iii) in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Note that the focus on these specific settings adopted in Lemma 1 instead of the most  general settings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', o is tunable and E is nonzero) is motivated by Lemma 1, which promises a lower expected risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Following the above elaboration, the loss function of QC to be minimized can be explicitly written as  L(ρ) = 1  2n  nc  �  i=1  K  �  k=1  �  [Tr(ρ(i,k)o(k))]k=1:K − y(i,k)�2  ,  (A1)  where y(i,k) is the unit basis whose k-th entry is 1 for ∀i ∈ [nc], ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Denote ρ∗ = minρ L(ρ) and the empirical  risk of QC as RERM(ˆhQ) with ˆhQ ≡ ˆhQ(ρ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The formal statement of Theorem 1 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Theorem (Formal statement of Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Following notations in Lemmas 2 and 3, with probability 1 − δ, the  expected risk of QC tends to be zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', RERM(ˆhQ) = 0, when the size of train dataset satisfies n ≫ O(KNge log KNg  ϵδ )  and the global minimizer ρ∗ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (A1) satisfies  (i)¯ρ∗(k) := ρ∗(1,k) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' = ρ∗(nc,k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = Bδk,k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (A2)  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (2) and the results in Lemma 3, with probability 1 − δ, the expected risk of an  optimal empirical QC is upper bounded by  R(ˆhQ) ≤ RERM(ˆhQ) + 4L1KC2ϵ + 3ξ(ˆh)  �  |TD|4mNge ln(56KNge/(ϵδ))  n  + ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (A3)  Then, when ρ∗ satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (A2), Lemma 2 warrants RERM(ˆhQ) = 0, which gives  R(ˆhQ) ≤ 4L1KC2ϵ + 3ξ(ˆh)  �  |TD|4mNge ln(56KNge/(ϵδ))  n  + ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (A4)  This bound can be further simplified when the training of QC is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Note that Condition (i) implies |TD| = K,  since all feature states from the same class collapse to the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Meanwhile, since ξ(ˆh) and C2 are bounded,  and m and ϵ are small constant, we can conclude that when n ≫ O(KNge log(KNg/(ϵδ))), the expected risk can  approach to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  SM B: Proof of Corollary 1  The proof leverages the following two lemmas related to the Haar measure and the unitary t-design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Let {Wy}y∈Y ⊂ U(d) form a unitary t-design with t > 1, and let A, B : Hd → Hd be arbitrary linear  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Then  1  |Y |  �  y∈Y  Tr[WyAW †  y B] =  �  Haar  dµ(W) Tr[WyAW †  y B] = Tr[A] Tr[B]  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (B1)  11  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Let {Wy}y∈Y ⊂ U(d) form a unitary t-design with t > 1, and let A, B, C, D : Hd → Hd be arbitrary  linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Then  1  |Y |  �  y∈Y  Tr[WyAW †  y B] Tr[WyCW †  y D] =  �  Haar  dµ(W) Tr[WyAW †  y B] Tr[WyCW †  y D]  =  1  d2 − 1 (Tr[A] Tr[B] Tr[C] Tr[D] + Tr[AC] Tr[BD])  −  1  d(d2 − 1) (Tr[AC] Tr[B] Tr[D] + Tr[A] Tr[C] Tr[BD]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (B2)  Corollary (Restatement of Corollary 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Following notations in Lemmas 2 and 3, when the encoding unitary  {UE(x)|x ∈ X} follows the Haar distribution, with probability 1 − δ, the empirical QC follows | Tr  �  σ(x(i,k))σ(x)  �  −  1  2N | ≤  �  3  22Nδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When the adopted Ansatz {U(θ)|θ ∈ Θ} follows the Haar distribution, with probability 1 − δ, the  empirical QC follows | Tr(ρ(i,k)o(k′)) − Tr(o(k′))  2D  | <  �  Tr(o(k′))2+2 Tr((o(k′))2)  22Dδ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' We complete the proof by separately analyzing the concentration behavior of the encoding  unitary and the Ans¨atze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Concentration of the encoding unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Recall that Condition (iii) in Lemma 2 concerns the distance between two  feature states ρ(i,k) and ρ(i′,k′) for ∀i, i ∈ [nc] and ∀k, k′ ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In this regard, we quantify the distance between the  encoded state σ(x(i,k)) and σ(x) with x ∼ X when the deep encoding Ansatz UE is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In particular, we have  Ex∼X  �  Tr  �  σ(x(i,k))σ(x)  ��  =Ex∼X  �  Tr  �  σ(x(i,k))UE(x)(|0⟩ ⟨0|)⊗NUE(x)†��  =  �  Haar  dµ(U) Tr  �  σ(x(i,k))U(|0⟩ ⟨0|)⊗NU  �  =Tr(σ(x(i,k))) Tr(|0⟩ ⟨0|)⊗N)  2N  = 1  2N ,  (B3)  where the third equality uses Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' the variance of the term Tr(σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))σ(x)) yields  Varx∼X  �  Tr  �  σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))σ(x)  ��  =Ex∼X  �  Tr  �  σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))σ(x)  �2�  − Ex∼X  �  Tr  �  σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))σ(x)  ��2  =  �  Haar  dµ(U) Tr  �  σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))U(|0⟩ ⟨0|)⊗NU  �  Tr  �  σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))U(|0⟩ ⟨0|)⊗NU  �  −  1  22N  =  1  22N − 1  �  1 + Tr(σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))2)  �  −  1  22N(22N − 1)  �  Tr(σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))2) + 1  �  −  1  22N  ≤  1  22N−2 −  1  22N  = 3  22N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (B4)  where the second equality uses the property that the deep encoding unitary follows the Haar distribution and the  result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (B3), the third equality comes from Lemma 4, the inequality adopts Tr(σ2) ≤ 1 and 22N − 1 > 22N−1,  and the last equality is obtained via simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Supported by the Chebyshev’s inequality Pr(|X − E[X]| ≥ a) ≤ Var[X]/a2, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (B3) and (B4) indicate  Pr  ���� Tr  �  σ(x(i,k))σ(x)  �  − 1  2N  ��� ≥ τ  �  ≤  3  22Nτ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Equivalently, with probability 1 − δ, we have  ��� Tr  �  σ(x(i,k))σ(x)  �  − 1  2N  ��� ≤  �  3  22Nδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (B5)  12  Concentration of the deep Ansatze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Recall Condition (ii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Given a feature state ρ(i,k), for ∀i ∈ [nc] and  ∀k ∈ [K] and a measure operator o(k), the optimal feature state should satisfy  Tr(ρ∗(i,k)o(k′)) = δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In other words, we should examine the value of Tr(ρ(i,k)o(k′)) when ρ(i,k) is prepared by a deep Ansatze U(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Specifically, we have  Eθ∼Θ  �  Tr(ρ(i,k)o(k′))  �  =Eθ∼Θ  �  Tr(U(θ)σ(x(i,k))U(θ)†(o(k′) ⊗ I2N−D)  �  =  �  Haar  dµ(U) Tr  �  Uσ(x(i,k))U †(o(k′) ⊗ I2N−D)  �  =Tr(o(k′))(2N−D)  2N  =Tr(o(k′))  2D  ,  (B6)  where the first equality comes from the explicit form of QC in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (4), the second equality uses the fact that U follows  the Haar distribution, and the last second equality comes from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We then quantify the variance of Tr(ρ(i,k)o(k′)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Varθ∼Θ  �  Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k′))  �  =Eθ∼Θ  �  Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k′))2�  −  �  Eθ∼Θ  �  Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k′))  ��2  =  �  Haar  dµ(U) Tr  �  Uσ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))U †(o(k′) ⊗ I2N−D)  �2  − Tr(o(k′))2  22D  =  1  22N − 1  �  Tr(σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))) Tr(o(k′) ⊗ I2N−D) Tr(σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))) Tr(o(k′) ⊗ I2N−D) + Tr(σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))2) Tr((o(k′) ⊗ I2N−D)2)  �  −  1  2N(22N − 1)  �  Tr(σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k))2) Tr(o(k′) ⊗ I2N−D)2 + Tr(σ(x(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)))2 Tr((o(k′) ⊗ I2N−D)2)  �  − Tr(o(k′))2  22D  ≤  1  22N − 1  �  Tr(o(k′) ⊗ I2N−D)2 + Tr((o(k′) ⊗ I2N−D)2)  �  − Tr(o(k′))2  22D  =  1  22N − 1  �  Tr(o(k′))222N−2D + Tr((o(k′))2)22N−2D�  − Tr(o(k′))2  22D  ≤Tr(o(k′))2 + Tr((o(k′))2)  22D−1  − Tr(o(k′))2  22D  =Tr(o(k′))2 + 2 Tr((o(k′))2)  22D  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (B7)  where the second equality uses the fact that U follows the Haar distribution and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (B6), the the third equality  comes from Lemma 5, the first inequality arises from dropping some positive terms, the last second equality employs  Tr(A⊗B) = Tr(A) Tr(B) and (A⊗B)(C⊗D) = (AC)⊗(BD), and the last inequality exploits (22N −1)−1 > (2N−1)−1,  and the last equalities is obtained via simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Supported by the Chebyshev’s inequality Pr(|X − E[X]| ≥ a) ≤ Var[X]/a2, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (B6) and (B7) indicate  Pr  ���� Tr(ρ(i,k)o(k′)) − E  �  Tr(ρ(i,k)o(k′))  � ��� ≥ τ  �  ≤ Tr(o(k′))2 + 2 Tr((o(k′))2)  22Dτ 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Equivalently, with probability 1 − δ, we have  ��� Tr(ρ(i,k)o(k′)) − Tr(o(k′))  2D  ��� <  �  Tr(o(k′))2 + 2 Tr((o(k′))2)  22Dδ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (B8)  13  SM C: Proof of Lemma 1  In this section, we derive the geometric properties of the global optimizer under the unconstraint loss function  L(ρ, o) in which both ρ and o are tunable and the regularization term is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Mathematically, the regularizer  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (1) is defined as E = λρ  2  �nc  i=1  �K  k=1 ∥ρ(i,k)∥2  F + λo  2  �K  k=1 ∥o(k)∥2  F with λρ and λo being hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The  explicit form of the loss function is  L(ρ, o) = 1  2n  nc  �  i=1  K  �  k=1  ��  Tr(ρ(i,k)o(k))  �  k=1:K − y(i,k)�2  + λρ  2  nc  �  i=1  K  �  k=1  ∥ρ(i,k)∥2  F + λo  2  K  �  j=1  ∥o(j)∥2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C1)  Denote the global optima as (ρ∗, o∗) = minρ,o L(ρ, o) and the empirical QC as ˆhQ ≡ hQ(ρ∗, o∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The restatement of  Lemma 1 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma (Formal statement of Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Define C1 := K  �  ncλoλρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' If 2D ≥ K, C1 ≤ 1, and λo ≤ ncλρ, the global  minimizer (ρ∗, o∗) of L(ρ, o) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (C1) satisfies for ∀k, k′ ∈ [K]:  (i)¯ρ∗(k) := ρ∗(1,k) = · · · = ρ∗(nc,k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = (1 − C1)  �  λo  nλρ  δk,k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (iii)o∗(k) =  �  nλρ  λo  ¯ρ∗(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C2)  The corresponding empirical risk is RERM(ˆhQ) = C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Conceptually, the global optimizer can be identified by lower bounding L(ρ, o), where the equality  conditions of ρ amount to the properties of global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' the lower bound of L(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' o) yields  1  2Knc  nc  �  i=1  K  �  k=1  �  [Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(j))]j=1:K − y(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)�2  + λρ  2  nc  �  i=1  K  �  k=1  ∥ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)∥2  F + λo  2  K  �  j=1  ∥o(j)∥2  F  ≥  1  2Knc  nc  �  i=1  K  �  k=1  �  Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k)) − 1  �2  + λρ  2  nc  �  i=1  K  �  k=1  ∥ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)∥2  F + λo  2  K  �  j=1  ∥o(j)∥2  F  =  1  2Knc  K  �  k=1  nc  �  i=1  nc  1  nc  �  Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k)) − 1  �2  + λρ  2  K  �  k=1  nc  �  i=1  nc  1  nc  ∥ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)∥2  F + λo  2  K  �  j=1  ∥o(j)∥2  F  ≥ 1  2K  K  �  k=1  �  Tr  � nc  �  i=1  1  nc  ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k)  �  − 1  �2  + λρ  2  K  �  k=1  nc  �����  nc  �  i=1  1  nc  ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)  �����  2  F  + λo  2  K  �  j=1  ∥o(j)∥2  F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C3)  where the first inequality uses the fact ∥a − b∥2 = �  i(a(i) − b(i))2 ≥ (a(k) − b(k))2 and the k-th entry of y(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k) equals  to 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' and the second inequality comes from the Jensen’s inequality f(E(x)) ≤ E(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The equality condition of the  first inequality holds if and only if  Tr  �  ρ(i,k)o(j)�  = 0, (∀j ∈ [K] \\ {k}) ∧ (∀i ∈ [nc]) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  and the equality condition of the second inequality holds if and only if  ρ(1,k) = · · · = ρ(i,k) = · · · = ρ(nc,k), ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Denote the mean of the feature state for the k-th class as ¯ρ(k) = �nc  i=1  1  nc ρ(i,k) for ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The above two equality  conditions suggest that the global minimizer (ρ∗, o∗) satisfies  ¯ρ∗(k) ≡ ρ∗(1,k) = · · · = ρ∗(nc,k), ∀k ∈ [K]  Tr(¯ρ∗(k)o∗(j)) = 0, ∀j ∈ [K] \\ {k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C4)  To thins end, we obtain Conditions (i) in Lemma 1, which describe the geometric properties of ρ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',  (i)¯ρ∗(k) := ρ∗(1,k) = · · · = ρ∗(nc,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C5)  14  The next part of the proof is showing that the global minimizer satisfies Condition (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (C3) and  (C4), the lower bound of the loss function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (C3) follows  L(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' o)  ≥ 1  2K  K  �  k=1  �  Tr  �  ¯ρ(k)o(k)�  − 1  �2  + λρ  2  K  �  k=1  nc  ���¯ρ(k)���  2  F + λo  2  K  �  j=1  ∥o(j)∥2  F  = 1  2K  K  �  k=1  �  Tr  �  ¯ρ(k)o(k)�  − 1  �2  + λρ  2 K  K  �  k=1  1  K nc  ���¯ρ(k)���  2  F + λo  2 K  K  �  j=1  1  K ∥o(j)∥2  F  ≥1  2  � K  �  k=1  1  K Tr  �  ¯ρ(k)o(k)�  − 1  �2  + λρ  2 Knc  �����  K  �  k=1  1  K ¯ρ(k)  �����  2  F  + λo  2 K∥  K  �  j=1  1  K o(j)∥2  F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C6)  where the second inequality comes from the Jensen’s inequality and the equality condition holds if and only if for  ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' k′ ∈ [K],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Tr  �  ¯ρ(k)o(k)�  = Tr  �  ¯ρ(k′)o(k′)�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' ∥¯ρ(k)∥F = ∥¯ρ(k′)∥F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' ∥o(k)∥F = ∥o(k′)∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C7)  Then, supported by the inequality a + b ≥ 2  √  ab, the loss L(ρ, o) can be further lower bounded by  1  2  �  Tr  �  ¯ρ(k)o(k)�  − 1  �2  + λρ  2 Knc  ���¯ρ(k)���  2  F + λo  2 K∥o(j)∥2  F  ≥1  2  �  Tr  �  ¯ρ(k)o(k)�  − 1  �2  + K  �  ncλoλρ  ���¯ρ(k)���  F ∥o(j)∥F ,  (C8)  where the equality condition holds if and only if  λo∥o(j)∥2  F = ncλρ  ���¯ρ(k)���  2  F , ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C9)  Note that the requirements C1 ≤ 1 and λo ≤ ncλρ in Lemma 1 imply ∥¯ρ∗(k)∥ ≤ 1 and hence ensure that ¯ρ∗(k) is a  meaningful quantum state for ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Since Tr  �  ¯ρ(k)o(k)�  = ∥¯ρ(k)∥∥o(k)∥ cos(∠(ρ(k), o(k))), the lower bound of L(ρ, o) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (C8) is equivalent to  1  2  �  ∥¯ρ(k)∥∥o(k)∥ cos(∠(ρ(k), o(k))) − 1  �2  + C1  ���¯ρ(k)���  F ∥o(j)∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Define ∥¯ρ(k)∥∥o(k)∥ = a and ∠(ρ(k), o(k)) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The above equation is described by the function f(a, α) = (a cos α −  1)2/2+C1a and its minimum is C1 −C2  1/2 when α∗ = 0 and a∗ = 1−C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The derivation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Since a > 0 and  its maxima is unbounded, we first consider the case 0 < a < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In this case, the minimum of f(a, α) is C1 −C2  1/2 with  α∗ = 0 and a∗ = 1 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Otherwise, when a ≥ 1, the minimum of f(a, α) is C1 with α∗ = arccos(1/a) and a∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Note that the minimum value of f(a, α) in the second case is always larger than that of the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Therefore, the  minimum of f(a, α) is C1 − C2  1/2 with α∗ = 0 and a∗ = 1 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Combining the observation that ¯ρ∗(k) and o(k) are in  the same direction with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (C9), we achieve Condition (iii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',  o∗(k) =  �  ncλρ  λo  ¯ρ∗(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The last part is proving Condition (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Combining the result ∥¯ρ∗(k)∥∥o∗(k)∥ = 1 − C1 for ∀k ∈ [K] with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (C4)  and Condition (iii), we immediately obtain condition (ii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',  (ii)  �  ncλρ  λo  ∥ρ∗(k)∥∥ρ∗(k′)∥ = (1 − C1)δk,k′ ⇒ Tr(¯ρ∗(k)¯ρ∗(k′)) = (1 − C1)  �  λo  ncλρ  δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (C10)  To summarize, given the global optima satisfying the above three conditions, the corresponding empirical risk is  RERM(ˆhQ) = 1  2n  nc  �  i=1  K  �  k=1  �  [Tr(ρ∗(i,k)o∗(k))]k=1:K − y(i,k)�2  = C2  1  2  (C11)  15  SM D: Results related to Lemma 2  This section is composed of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In SM D 1, we present the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In SM D 2, we explain that  the requirements in Lemma 2 are mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Proof of Lemma 2  Different from Lemma 1, here we focus the setting such that the regularization term is set as E = 0 and the operator  o is predefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The explicit form of the loss function L is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Denote the optimal feature states  ρ∗ = minρ L(ρ), we quantify the value of RERM(ˆhQ) with ˆhQ ≡ hQ(ρ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We emphasize that the modifications of E and o allow a lower optimal empirical risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Recall the results of Lemma  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In the most general case, the optimal empirical risk depends on the regularization term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', RERM(ˆhQ) → C2  1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The dependance on C1 motivates us to explore the empirical risk of QC when E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Furthermore, Condition (iii)  in Lemma 1 delivers the crucial properties of the optimal measure operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', the optimal measure operators are  orthogonal with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Such properties contribute to construct a more effective QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Instead of optimizing,  the measure operator o can be predefined by inheriting the properties proved in Lemma 1, that is, o are required to  span the space C2D×2D and satisfy Tr(o(k)o(k′)) = Bδk,k′ with B ≥ 1 being a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Notably, these requirement  are mild, covering frequently used measures such as computational basis and Pauli-based measures, as explained in  SM D 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma (Formal statement of Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Suppose that the adopted measure operator o spans the space C2D×2D and  satisfies Tr(o(k)o(k′)) = Bδk,k′ where B ≥ 1 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The empirical risk of ˆhQ is RERM(ˆhQ) = 0 when the global  minimizer ρ∗ satisfies  (i)¯ρ∗(k) := ρ∗(1,k) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' = ρ∗(nc,k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (ii) Tr(¯ρ∗(k)¯ρ∗(k′)) = Bδk,k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (D1)  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The concept of the proof is analogous to Lemma 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', the global optimizer is identified by lower  bounding the loss L(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' To this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' the lower bound of L(ρ) yields  1  2Knc  nc  �  i=1  K  �  k=1  �  [Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(j))]j=1:K − y(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)�2  ≥  1  2Knc  nc  �  i=1  K  �  k=1  �  Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k)) − 1  �2  =  1  2Knc  K  �  k=1  nc  �  i=1  nc  1  nc  �  Tr(ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k)) − 1  �2  ≥ 1  2K  K  �  k=1  �  Tr  � nc  �  i=1  1  nc  ρ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k)o(k)  �  − 1  �2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (D2)  where the first inequality uses the facts n = Knc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' ∥a − b∥2 = �  i(a(i) − b(i))2 ≥ (a(k) − b(k))2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' and only the k-th  entry of y(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='k) equals to 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' and the second inequality comes from the Jensen’s inequality E(f(x)) ≥ f(E(x)) when f(·)  is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Note that the equality condition of the first inequality holds if and only if  Tr(ρ(i,k)o(j)) = 0, (∀j ∈ [K] \\ {k}) ∧ (∀i ∈ [nc]) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  And the equality condition of the second inequality holds if and only if  ρ(1,k) = · · · = ρ(i,k) = · · · = ρ(nc,k), ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Denote the mean of the feature state for the k-th class as ¯ρ(k) = �nc  i=1  1  nc ρ(i,k) for ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The above two equality  conditions suggest that the global minimizer yields  ¯ρ∗(k) ≡ ρ∗(1,k) = · · · = ρ∗(nc,k), ∀k ∈ [K]  (D3)  Tr(¯ρ∗(k)o(j)) = 0, ∀j ∈ [K] \\ {k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (D4)  16  Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (D2)-(D4), the lower bound of the loss function L(ρ) satisfies  1  2K  K  �  k=1  �  Tr  �  ¯ρ(k)o(k)�  − 1  �2  ≥ 1  2  � K  �  k=1  1  K Tr  �  ¯ρ(k)o(k)�  − 1  �2  ,  (D5)  where the inequality comes from the Jensen’s inequality and the equality condition holds if and only if ∀k, k′ ∈ [K],  Tr  �  ¯ρ(k)o(k)�  = Tr  �  ¯ρ(k′)o(k′)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (D6)  Supported by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (D6), we can further lower bound L(ρ) with  1  2  �  Tr  �  ¯ρ(k)o(k)�  − 1  �2  ≥ 0,  (D7)  where the equality condition is achieved when Tr(¯ρ(k)o(k)) = 1 for ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Taken together, the global optimizer ρ∗ should satisfy Condition (i)&(iii) in Lemma 2, where  (i)¯ρ∗(k) := ρ∗(1,k) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' = ρ∗(nc,k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (iii) Tr(¯ρ∗(k)o(k′)) = δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (D8)  We last prove that Condition (iii) and the requirements of o lead to Condition (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In particular, denote the  vectorization of ρ∗(k) and o(k) as |ρ∗(k)⟩⟩ and |o(k)⟩⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Condition (iii) can be rewritten as  ��  ¯ρ∗(k), o(k′)��  = δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (D9)  Moreover, since the set of measure operators {o(k)} is required to be complete in the space of C2D and Tr(o(k)o(k′)) =  Bδk,k′ with B ≥ 1 for ∀k, k′ ∈ [K], we have  �  k  ���o(k)����  o(k)��� = BI2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Then, Condition (ii) can be derived as follows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',  Tr(ρ∗(k)ρ∗(k′))  =⟨⟨¯ρ∗(k)|I2D|ρ∗(k′)⟩⟩  = 1  B  ��  ¯ρ∗(k)���  �  k′′  |o(k′′)⟩⟩⟨⟨o(k′′)|  ���ρ∗(k′)  ��  = 1  B  ��  ¯ρ∗(k)���|o(k)⟩⟩⟨⟨o(k)|  ���ρ∗(k′)��  +  ��  ¯ρ∗(k)���  �  k′′̸=k  |o(k′′)⟩ ⟨o(k′′)|  ���ρ∗(k′)  ��  = 1  B δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (D10)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Requirement of o used in Lemma 2  Here we elucidate that the requirements adopted in Lemma 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', o spans the complex space 2D × 2D and satisfies  Tr(o(k)o(k′)) = Bδk,k′ with B ≥ 1, are mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Specifically, the employed measurements in most QNN-based classifiers  satisfy these requirements, including the computational basis measurements and Pauli measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Computational basis measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In this setting, the local measurement o(k) is set as |k⟩ ⟨k| with |k⟩ being the  k-th computational basis for ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  When 2D = K, {|k⟩} spans the whole space of C2D×2D and we have  Tr(o(k)o(k′)) = (⟨k|k′⟩)2 = δk,k′ with B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The assumptions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Pauli measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Denote the Pauli operation applied to the i-th qubit as P (i)  a  with a ∈ {X, Y, Z, I} for ∀i ∈ [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Then, there are in total 4D Pauli strings P = ⊗D  i=1P (i)  a  that form a orthogonal basis for the space C2D×2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' With  setting 2D = K, each o(k) corresponds to one Pauli string with Tr(o(k)o(k′)) = Kδk,k′ with B = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  17  SM E: Proof of Lemma 3  For elucidating, let us restate Lemma 3 below and introduce the proof sketch before moving on to present the proof  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma (Formal statement of Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Denote L1 as the Lipschitz constant of ℓ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (1) with respect to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Given  a QC defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (3), let E be a quantum channel with  hQ(x, U(θ), O(k)) ≡ Tr(o(k)E(σ(x))), ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E1)  Suppose the measure operator follows maxk∈[K] ∥o(k)∥ ≤ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The explicit form of the encoding unitary follows UE(x) =  �Ng  g=1 ug(x) ∈ U(2N) with the g-th quantum gate ug(x) ∈ U(2m) operating with at most m qubits with m ≤ N and Ng  gates consisting of Nge variational gates and Ng − Nge fixed gates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Following above notations and Definition 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' the empirical QC is (K( 28Nge  ϵ  )4mNge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 4L1KC2ϵ)-robust and with prob-  ability 1 − δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' its generalization error yields  RGene(ˆh) ≤ 4L1KC2ϵ + 3ξ(ˆh)  �  |TD|4mNge ln(56KNge/(ϵδ))  n  + ξ(ˆh)2|TD|4mNge ln(56KNge/(ϵδ))  n  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  where L1 is the Lipschitz constant of ℓ with respect to h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' ID  r = {i ∈ [n] : z(i) ∈ Cr},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' ξ(ˆh) := maxz∈Z ℓ(ˆh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' and  TD := {r ∈ [R] : |ID  r | ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The proof of Lemma 3 is established on the following lemma, which leverages the algorithmic robustness to quantify  the upper bound of the generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma 6 (Theorem 1, [105]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' If the learning algorithm A is (R, ν(·))-robust with {Cr}R  r=1, then for any δ > 0, with  probability at least 1−δ over an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='d drawn of n samples D = {z(i)}n  i=1 with z(i) = (x(i), y(i)), the returned hypothesis  ˆh by A on D satisfies  RGene(ˆh) ≤ ν(D) + ξ(ˆh)  �  (  √  2 + 1)  �  |TD| ln(2R/δ)  n  + 2|TD| ln(2R/δ)  n  �  ,  (E2)  where ID  r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)), and TD := {r ∈ [R] : |ID  r | ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The above result hints that given a hypothesis ˆh, its generalization error is upper bounded by the disjoint sets {Cr}R  r=1,  where a lower cardinality R allows a lower generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' A natural approach to realize these disjoint partitions  is covering number [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Definition 2 (Covering number, [65]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Given a metric space (U, ∥ · ∥), the covering number N(U, ϵ, ∥ · ∥) denotes the  least cardinality of any subset V ⊂ U that covers U at scale ϵ with a norm ∥ · ∥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', supA∈U minB∈V ∥A − B∥ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In conjunction with Lemma 6 and Definition 2, the analysis of RGene(ˆh) of an N-qubit QC amounts to quantifying  the covering number of the space of the input quantum states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',  XQ =  �  UE(x)(|0⟩ ⟨0|)⊗NUE(x)†��x ∈ X  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E3)  The following lemma connects the robustness and covering number of XQ of QCs whose proof is provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Following the settings in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (E1)-(E3), the corresponding QC is (K( 28Nge  ϵ  )4mNge, 4L1KC2∥E∥⋄ϵ)-robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We are now ready to prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The generalization error bound can be acquired by combining Lemmas 6 and 7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',  RGene(ˆh) ≤4L1KC2∥E∥⋄ϵ + ξ(ˆh)  �  �(  √  2 + 1)  �  |TD| ln(2K( 28Nge  ϵ  )4mNge/δ)  n  + 2|TD| ln(2K( 28Nge  ϵ  )4mNge/δ)  n  �  �  ≤4L1KC2∥E∥⋄ϵ + ξ(ˆh)  �  3  �  |TD|4mNge ln(56KNge/(ϵδ))  n  + 2|TD|4mNge ln(56KNge/(ϵδ))  n  �  ≤4L1KC2ϵ + ξ(ˆh)  �  3  �  |TD|4mNge ln(56KNge/(ϵδ))  n  + 2|TD|4mNge ln(56KNge/(ϵδ))  n  �  ,  (E4)  where ID  r = {i ∈ [n] : z(i) ∈ Cr}, ξ(ˆh) := maxz∈Z(ℓ(ˆh, z)), and TD := {r ∈ [R] : |ID  r | ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Proof of Lemma 7  The proof uses the following lemma to quantify the covering number of XQ whose proof is given in SM E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Following the settings in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (E1), the covering number of XQ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (E3) is  N(XQ, ϵ, ∥ · ∥F ) ≤  �28Nge  ϵ  �4mNge  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E5)  Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When QC is applied to accomplish the K-class classification task, the sample space is Z = XQ ×Y  with Y = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Denote ˜  XQ as the ϵ-cover set of XQ with the covering number N(XQ, ϵ, ∥ · ∥F ) in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Supported by the ϵ-cover set ˜  XQ, the space XQ × {i} can be divided into N(XQ, ϵ, ∥ · ∥F ) sets for ∀i ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In other  words, we can divide Z into KN(XQ, ϵ, ∥ · ∥F ) sets denoted by {Zi}KN (XQ,ϵ,∥·∥F )  i=1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  We then utilize the divided sets of Z to connect the robustness with covering number according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Given a training example (x(i), y(i)) and a test example (x, y), suppose that the corresponding quantum examples  (σ(x(i)), y(i)) and (σ(x), y) are in the same set of {Zi}KN (XQ,ϵ,∥·∥F )  i=1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' For convenience, we abbreviate σ(x(i)) and σ(x)  as σ(i) and σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Following the definition of covering number, we have  y(i) = y and ∥σ(i) − σ∥F ≤ 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E6)  Since the encoded state takes the form σ = UE(x)(|0⟩ ⟨0|)⊗NUE(x)†, we have  rank(σ(i) − σ) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E7)  Then, in accordance with the definition of robustness, we bound the discrepancy of the loss values for σ(i) and σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  ���l(hQ(σ(i)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' y(i)) − l(hQ(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' y)  ���  ≤L1  ���[Tr(E(σ(i))o(k))]k=1:K − [Tr(E(σ))o(k))]k=1:K  ���  2  ≤L1K max  k∈K | Tr(E(σ(i)))o(k)) − Tr(E(σ)o(k))|  ≤L1K max  k  ���o(k)���  2 Tr(|E(σ(i) − σ)|)  ≤2L1KC2∥E∥⋄∥σ(i) − σ∥F  ≤4L1KC2∥E∥⋄ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E8)  where the first inequality uses the Lipschitz property of the loss function with ℓ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' b) − ℓ(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' d) ≤ L1∥a − c∥2 and the  form of E in Lemma 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' the second inequality comes from the definition of l2 norm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' the third inequality exploits von  Neumann’s trace inequality | Tr(AB)| ≤ ∥A∥p∥B∥q with 1/p + 1/q = 1 and the linear property of CPTP map with  E(ρ)−E(σ) = E(ρ−σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' the last second inequality employs maxk  ��o(k)��  2 ≤ C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' the relation ∥E(ρ−σ)∥1 ≤ ∥E∥⋄∥ρ−σ∥1  and ∥A∥1 ≤ rank(A)∥A∥F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' and the last inequality adopts the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (E6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The above result exhibits that the learned QC is (KN(XQ, ϵ, ∥ · ∥), 4L1KC2∥E∥⋄ϵ)-robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In this regard, the proof  can be completed when the upper bound of the covering number N(XQ, ϵ, ∥ · ∥F ) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Supported by Lemma 8,  we obtain N(XQ, ϵ, ∥ · ∥F ) ≤ ( 28Nge  ϵ  )4mNge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Taken together, the learned QC is  �  K  �28Nge  ϵ  �4mNge  , 4L1KC2∥E∥⋄ϵ  �  − robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Proof of Lemma 8  The derivation of the covering number of XQ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (E3) uses the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Lemma 9 (Lemma 1, [106]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' For 0 < ϵ < 1/10, the ϵ-covering number for the unitary group U(2m) with respect to  the Frobenius-norm distance in Definition 2 obeys  � 3  4ϵ  �4m  ≤ N(U(2m), ϵ, ∥ · ∥F ) ≤  �7  ϵ  �4m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E9)  19  Proof of of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Recall the input state space is XQ = {UE(x)(|0⟩ ⟨0|)⊗NUE(x)†|x ∈ X}, where the encoding  unitary UE(x) = �Ng  g=1 ug(x) ∈ U(2N) consists of Nge variational gates and Ng − Nge fixed gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' To quantify the  covering number N(XQ, ϵ, ∥·∥F ), we define ˜S as the ϵ-covering set for the unitary group U(2m), ˜  XQ as the ϵ′-covering  set of XQ, and define a set  ˜UE :=  �  �  �  �  i∈{Nge}  ui(x)  �  j∈{Ng−Nge}  uj(x)  ���ui(x) ∈ ˜S  �  �  � ,  (E10)  where ui(θi) and uj specify to the variational and fixed quantum gates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Note that for any encoding circuit  UE(x), we can always find a unitary UE,ϵ(x) ∈ ˜UE where each ug(x) is replaced by the nearest element in the covering  set ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' To this end, following the definition of covering number, the discrepancy between UE(x)(|0⟩ ⟨0|)⊗NUE(x)† ∈ XQ  and UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)† ∈ ˜  XQ under the Frobenius norm satisfies  ��UE(x)(|0⟩ ⟨0|)⊗NUE(x)† − UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)†��  F  ≤2  ��UE(x)(|0⟩ ⟨0|)⊗NUE(x)† − UE,ϵ(x)(|0⟩ ⟨0|)⊗NUE,ϵ(x)†��  ≤2∥UE(x) − UEϵ(x)∥∥(|0⟩ ⟨0|)⊗N∥  ≤4Ngeϵ,  (E11)  where the first inequality uses ∥X∥F ≤ rank(X)∥X∥ and the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (E7), the second inequality comes from  the Cauchy–Schwarz inequality, and the last inequality follows ∥UE(x) − UE,ϵ(x)∥ ≤ Ngeϵ and ∥(|0⟩ ⟨0|)⊗N∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In  other words, ϵ′ = 2Ngeϵ and ˜  XQ is a (4Ngeϵ)-covering set for XQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In conjunction with the observation that there are  | ˜S|Nge combinations for the gates in ˜UE and the results in Lemma 9, we obtain the cardinality of the set ˜UE is upper  bounded by | ˜UE| ≤  � 7  ϵ  �4mNge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Accordingly, supported by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (E11), the covering number of XQ satisfies  N(XQ, 4Ngeϵ, ∥ · ∥F ) ≤  �7  ϵ  �4mNge  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E12)  After simplification, we have  N(XQ, ϵ, ∥ · ∥F ) ≤  �28Nge  ϵ  �4mNge  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (E13)  SM F: Implementation of the algorithm to probe potential advantages of QCs  The expected risk is the most principal criteria to quantify the power of a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' As a result, to probe whether  a QC holds potential advantages over a CC on a specific learning task, the simplest way is comparing their risk  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Nevertheless, capturing these two risk curves are difficult, because of many flexible hyper-parameter settings  to initiate a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The developed theories in Theorem 1 and Lemmas 1-3 deliver concrete rules to set up these hyper-parameters and  thus allow an efficient way to estimate these risk curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In particular, the derived U-shape curve of QCs indicates  that the minimum risk of QC locates at the modest size of the hypothesis space HQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In other words, the number  of trainable parameters NT should be lower than O(poly(N)), with N being the number of qubits in QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Moreover,  Lemma 3 hints that the generalization error of QC can be well suppressed by using the modest number of train  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' As such, if the available number of training examples in D is tremendous, we can distill a subset from D  to better recognize quantum advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The Pseudo code of the proposed method is presented in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' To make a fair comparison, the hyper-parameter  settings applied to QC and CC, especially for those relating to the computational resources, are required to keep  to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Specifically, in each comparison, the employed loss function, the train examples n, the number of  trainable parameters Nt, and the number of epochs T applied to QC and CC should be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Note that the  learning rate, the adopted optimizer, and the batch size can be varied of different classifiers to better estimate the  empirical hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' To ensure that the collected results of QC span its basin of the risk curve, the employed W  settings of Nt can be acquired by uniformly interpolating from O(1) to O(poly(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The iteration T should ensure  the convergence of QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Once the loss values of QC and CC under {n(w), N (w)  t  , T (w)}W  w=1 are obtained, we can apply  certain fitting algorithms to attain their risk curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  20  Algorithm 1: Estimate risk curves of quantum and classical classifiers  Data: The train dataset D, the test dataset DT est, QC hQ associated with the hypothesis space HQ, CC hC associated  with the hypothesis space HQ, the loss function L(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Result: The estimated risk curves of QC and CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Initialization: W tuples of hyper-parameter settings {n(w), N (w)  t  , T (w)}W  w=1 with n being train examples, Nt being the  number of trainable parameters, and T being the number of epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  for w = 1, w ≤ W, w + + do  Initialize train data as D(w) by distilling n(w) examples from D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  # Collect loss dynamics of QC ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Minimize the loss function L(·, ·) via gradient descent methods to obtain the empirical quantum classifier ¯h(w)  Q  ∈ HQ  using D(w) within T (w) epochs and NT trainable parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Record the loss value L(¯h(w)  Q , DT est) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  # Collect loss dynamics of CC ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Minimize the loss function L(·, ·) via gradient descent methods to obtain the empirical classical classifier ¯h(w)  C  ∈ HC  using D(w) within T (w) epochs and NT trainable parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Record the loss value L(¯h(w)  C , DT est) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  end  Fitting the loss dynamics of {L(¯h(w)  Q , DT est)}W  w=1 to obtain the estimated risk curve of QC ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Fitting the loss dynamics of {L(¯h(w)  C , DT est)}W  w=1 to obtain the estimated risk curve of CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Ul(✓)  RZ(✓(l,1,1))  RY (✓(l,1,2))  RZ(✓(l,1,3))  RZ(✓(l,2,1))  RY (✓(l,2,2))  RZ(✓(l,2,3))  RZ(✓(l,3,1))  RY (✓(l,3,2))  RZ(✓(l,3,3))  RZ(✓(l,4,1))  RY (✓(l,4,2))  RZ(✓(l,4,3))  ×𝐿  (a)  (b)  Class 1:  Class 2:  Class 3:  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Visualization of image dataset and hardware-efficient Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (a) Image instances sampled from the  Fashion-MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (b) The circuit architecture of the employed Hardware-efficient Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘×L’ denotes the  layer number, which means repeating the gates in the dashed box with L times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  SM G: Numerical simulation details  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The construction of the parity dataset mainly follows from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Note that this task has also been  broadly studied in the field of deep learning to show the limits of deep neural classifiers [107, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The constructed  dataset contains in total 64 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Each example corresponds to a bit-string with the length 6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', x ∈ {0, 1}6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The label of x is assigned to be 1 if the number of ‘0’ in x is even;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' otherwise, the label is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' We split it into train dataset  and test dataset with the train-test-split ratio being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The number of train examples in each class is controlled to  be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' For each example, its feature dimension is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The image dataset is adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Specifically,  the data from the first nine classes are preserved and the total number of examples is 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The train-test-split ratio is  set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='5 to construct the train and test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Each example corresponds to an image with 28 × 28 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In the  preprocessing stage, we flatten all examples followed by padding and normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The processed example yields an  10-qubit state with x ∈ R210 and ∥x∥2  2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Some examples after preprocessing are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Construction of QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The quantum subroutine of QC consists of the encoding circuit UE and the Ansatz U(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  For all learning tasks, the hardware-efficient Ansatz is employed whose mathematical expression is U(θ) = �L  l Ul(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The layout of the hardware-efficient Ansatz follows the layer-wise structure and the gate arrangement at each layer  is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' For ∀l ∈ [L], Ul(θ) = �N  i=1(RZ(θ(l,i,1)) RY(θ(l,i,2)) RZ(θ(l,i,3)))Uent with Uent being the entanglement  layer formed by CNOT gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4(b) depicts the adopted hardware-efficient Ansatz with L layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The encoding methods for the parity dataset classification and the digit images classification are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The  former uses the basis encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Specifically, for a classical example x ∈ Rd, the employed encoding unitary  21  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Geometric properties of the quantum feature states on parity dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (a) The averaged performance of  QC evaluated by M1 defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘Init-C-k’ with k = 1, 2 refers that the value of M(k)  1  at the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Similarly, the label ‘Final-C-k’ with k = 1, 2 refers that the value of M(k)  1  when the training of QC is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (b) The  averaged performance of QC evaluated by M2 defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘Init-C-1-2’ (‘Final-C-1-2’) refers that the value  of M2 before and after training of QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘L = a’ in the x-axis stands for that the layer number of hardware-efficient  Ansatz is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  is UE(x) |0⟩⊗d = |x⟩, which maps x to a 2d dimensional quantum state UE(x) |0⟩⊗d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The latter uses the amplitude  encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Given a normalized image x ∈ R64 with ∥x∥2  2 = 1, the corresponding unitary encodes it into a  6-qubit state with UE(x) |0⟩⊗6 = �64  j=1 xj |j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The Pauli-based measure operators are used in learning Fashion-MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Since the preprocessed dataset  contains 9 classes, there are in total 9 measure operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', o(1) = X⊗X⊗I⊗8, o(2) = X⊗Y ⊗I⊗8, o(3) = X⊗Z⊗I⊗8,  o(4) = Y ⊗X ⊗I⊗8, o(5) = Y ⊗Y ⊗I⊗8, o(6) = Y ⊗Z ⊗I⊗8, o(7) = Z ⊗X ⊗I⊗8, o(8) = Z ⊗Y ⊗I⊗8, o(9) = Z ⊗Z ⊗I⊗8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Multilayer Perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  To better justify the capability and performance of QCs, we apply the multilayer  perceptron (MLP) as the reference [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' MLP is composed of an input layer, L hidden layers with L ≥ 1, and an  output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The dimension of the input layer is equivalent to the feature dimension of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' ReLU activations  are added in the hidden layer to perform nonlinear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In the output layer, the activation function,  Softmax, is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The number of layers L depends on the assigned tuples {n, Nt, T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In the task of image classification, convolutional neural networks (CNNs) is  employed as the reference [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The employed CNN is formed by two convolutional layers and one fully-connected  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' ReLU activations and the pooling operation are added in the hidden layer to perform nonlinear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The number of channels for the first convolutional layer is fixed to be 8 and the corresponding kernel size is 9 × 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The kernel size of the pooling operation applied to the two convolutional layers is 2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The kernel size for the second  convolutional layer is fixed to be 5×5 but the number of output channels is varied depending on the settings in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  For the sake of fair comparison, the number of output channels is set as 2, 6, 15, 30, 50, 75, where the corresponding  number of parameters is 860, 1284, 2238, 3828, 5948, and 8598, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Optimizer and other hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The adaptive gradient descent method, named AdaGrad optimizer  [110], is used to optimize QCs and MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Compared to the vanilla gradient descent method, AdaGrad permits better  performance, since it adapts the learning rate for each feature depending on the estimated geometry of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  In the task of parity learning, the initial learning rate is set as η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='5 for QC and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='01 for MLP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  For both classifiers, the batch size is fixed to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In the task of image classification, the initial learning rate is set  as η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='05 for QC and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='01 for CNN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The batch size for both classifiers is set as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Curve fitting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' To capture the risk curve, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' 1 requests a curve fitting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' For all experiments,  we adopt the polynomial fitting to derive the risk curve by using the collected results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The least squares method in  determining the best fitting functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The source code used in numerical simulations will be available at Github repository https:  //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='com/yuxuan-du/Problem-dependent-power-of-QNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='8  Init-C-1-2  Fin-C-1-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='0  L=1L=2L=3L=4L=5L=6L=74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='8  Init-C-1  Init-C-2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='0  Fin-C-1  Fin-C-2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2  Distance  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='0  L=1L=2L=3L=4L=5L=6L=722  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Train (test) accuracy versus epoch on parity dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (a) Train accuracy and test accuracy of QC with the  varied layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘L = a’ refers that the layer number used in hardware-efficient Ansatz is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The solid line and  the dashed line separately correspond to the train and test accuracies of QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (b) Train accuracy and test accuracy of MLP  with the varied number of hidden neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘h = a’ refers that the number of neurons is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The solid and dashed  lines have the same meaning with those in QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Simulation results of the binary classification for the parity dataset  The feature states before and after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' We explore the geometric properties of feature states when the  layer number of hardware-efficient Ansatz varies from L = 1 to L = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Other settings are identical to those introduced  in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Condition (i) in Lemma 2 is evaluated by the metric  M(k)  1  =  nc  �  i=1  ∥ρ(i,k) − ¯ρ(k)∥,  (G1)  where the number of train examples {ρ(i,k)}nc  i=1 belonging to the k-th class is nc and ¯ρ(k) refers to their class-feature  mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Since parity learning is a binary classification task, Condition (ii) in Lemma 2 is evaluated by  M2 = Tr(¯ρ(0)¯ρ(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (G2)  The geometric properties of the feature states in the measure of M(k)  1  and M2 are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The left  panel shows that when L ∈ {2, 3, 4, 5}, both the value of M(1)  1  (highlighted by the green color) and M(2)  1  (highlighted  by the pink color) decrease from ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2 (epoch t = 0) to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='5 (epoch t = 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' These results comply with Condition  (i) in the sense that the feature states in the same class concentrates to the class-feature mean and leads to the low  empirical risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' By contrast, when L is too small or too large, the value of M(1)  1  changes subtly before and after  optimization, which is above 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The large deviation of feature states incurs the degrade performance of QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The  right panel depicts that when L ∈ {2, 3, 4, 5}, the value of M(2)  1  decreases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='5 (epoch t = 0) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='05 (epoch  t = 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' This reduction means that the class-feature means are maximally separated and thus ensure a good learning  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' On the contrary, when L ∈ {1, 6, 7}, the the value of M(2)  1  oscillates around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='5, which implies that the  class-feature means ¯ρ(1) and ¯ρ(2) are highly overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The learning dynamics of QC and MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6 visualizes the learning dynamics of QC and MLP with  respect to the varied trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The left panel indicates that when the layer number is L = 2, 3, 4, both  train and test accuracies of QC fast converge to 100% with 25 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When L = 1, both train and test accuracies  oscillate to 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When L = 7, the number of train data becomes insufficient and the overfitting phenomenon appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  These results accord with the U-shape risk curve of QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The right panel shows that when the number of hidden  neurons ranges from h = 1 to h = 18, the test accuracy of MLP is no higher that 55%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' These results reflect the  incapability of MLP in learning parity dataset compared with QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Simulation results of multi-class classification for the Fashion-MNIST images dataset  The feature states before and after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Here we discuss the geometric properties of feature states when  the layer number of hardware-efficient Ansatz varies from L = 2 to L = 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The metrics M(k)  1  and M2 defined in  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='00  h=1  h=2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='85-  h=6  h=10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='70  h = 14  Acc  h= 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='40  0  10  20  30  401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='40  0  10  20  30  4023  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Geometric properties of the quantum feature states on Fashion-MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (a) The averaged  performance of QC evaluated by M1 defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (b) The averaged performance of QC evaluated by M2 defined in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' All labels have the same meaning with those introduced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Train (test) accuracy versus epoch on Fashion-MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (a) Train accuracy and test accuracy of  QC with the varied layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The labels have the same meaning with those presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (b) Train accuracy  and test accuracy of CNN with the varied number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The label ‘h = a’ refers that the number of output  channels at the second layer is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The solid and dashed lines have the same meaning with those in QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' (G1) and (G2) are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In the measure of M2, since the performance of QC for any two classes is similar,  we only study the first two classes for ease of visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='7 depicts the geometric properties of the feature states in the measure of M(k)  1  and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The left panel  shows that for all settings with L ∈ {2, 5, 25, 50, 100, 150}, the value M(k)  1  at the initial step and the final step is very  similar and M(k)  1  is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2 for ∀k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=', 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' These results indicate that QC cannot satisfy Condition  (i) when learning Fashion-MNIST dataset, where the feature states from the same class cannot collapse to a unique  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Moreover, when we examine the performance of intra-class, the right panel implies that after training, the  class-feature means of QC are still highly overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The distance for all settings of L is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The inability  to achieve the optimal training loss shows the the limited power of QC on learning Fashion-MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  The learning dynamics of QC and CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='8 depicts the learning dynamics of QC and CNN with the  varied number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The left panel indicates that QC achieves the best performance when the layer  number is L ∈ [25, 100], where the corresponding number of parameters ranges from 750 to 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' In these settings,  both train and test accuracies of QC are around 30% after 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' When L < 25 or L > 100, both train and test  accuracies oscillate at 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' These results accord with the U-shape risk curve of QCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' The right panel shows that  the train and test accuracies of CNN are steadily growing with the increased number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content=' That is, when the  number of channels at the second layer is not less than 6, both the train and test accuracies are higher than 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  These results indicate that the employed QC does not have potential advantages in learning image dataset compared  with CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='  Init-C-1-2  Fin-C-1-2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='8  Distance  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='0  L = 2  L= 5  5L=25L=50L=100L=150Init-C-1  Fin-C-5  Fin-C-1  Init-C-6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='6  Init-C-2  Fin-C-6  Fin-C-2  Init-C-7  Init-C-3  Fin-C-7  e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='2  Fin-C-3  Init-C-8  Distance  Init-C-4  Fin-C-8  Fin-C-4  Init-C-9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='8  Init-C-5  Fin-C-9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='0  L = 2  L= 5  L=25L=50L=100L=1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='75  h=2  00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='50  h=6  h= 15  h = 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
+page_content='25  h= 50  h= 75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfof1Y/content/2301.01597v1.pdf'}
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+Published as a conference paper at EMNLP 2022
+Towards Table-to-Text Generation with Pretrained Language Model: A
+Table Structure Understanding and Text Deliberating Approach
+Miao Chen§ ∗, Xinjiang Lu¶ �, Tong Xu§, Yanyan Li ¶, Jingbo Zhou¶, Dejing Dou¶, Hui Xiong† ‡
+¶BIL, Baidu Research,
+§University of Science and Technology of China
+†Hong Kong University of Science and Technology (Guangzhou)
+‡Guangzhou HKUST Fok Ying Tung Research Institute
+cmer@mail.ustc.edu.cn, {luxinjiang,liyanyanliyanyan,zhoujingbo}@baidu.com,
+tongxu@ustc.edu.cn, doudejing@baidu.com, xionghui@ust.hk
+Abstract
+Although remarkable progress on the neural
+table-to-text methods has been made, the gen-
+eralization issues hinder the applicability of
+these models due to the limited source tables.
+Large-scale pretrained language models sound
+like a promising solution to tackle such issues.
+However, how to effectively bridge the gap
+between the structured table and the text in-
+put by fully leveraging table information to
+fuel the pretrained model is still not well ex-
+plored. Besides, another challenge of integrat-
+ing the deliberation mechanism into the text-
+to-text pretrained model for solving the table-
+to-text task remains seldom studied. In this
+paper, to implement the table-to-text genera-
+tion with pretrained language model, we pro-
+pose a table structure understanding and text
+deliberating approach, namely TASD. To be
+specific, we devise a three-layered multi-head
+attention network to realize the table-structure-
+aware text generation model with the help of
+the pretrained language model. Furthermore,
+a multi-pass decoder framework is adopted to
+enhance the capability of polishing generated
+text for table descriptions. The empirical stud-
+ies, as well as human evaluation, on two public
+datasets, validate that our approach can gener-
+ate faithful and fluent descriptive texts for dif-
+ferent types of tables.
+1
+Introduction
+The task of learning to generate natural language
+descriptions from non-linguistic input, which is
+referred to as data-to-text, is important for many
+applications, such as weather forecast genera-
+tion (Mei et al., 2016), sports news writing (Wise-
+man et al., 2017), biography writing (Lebret et al.,
+2016), market comments writing (Murakami et al.,
+2017) and automatic question-answering (Li et al.,
+2021b). The input data can be in various forms
+∗ This work was done when the first author was an intern
+at Baidu Research under the supervision of the second author.
+for data-to-text though, here we focus on the text
+generation task that takes the table as input.
+Inspired by neural machine translation models,
+previous studies on table-to-text tasks mainly adopt
+traditional seq2seq methods to generate table de-
+scriptions (Lebret et al., 2016; Wiseman et al.,
+2017; Liu et al., 2018; Gong et al., 2019b; Wang
+et al., 2020; Li et al., 2021a). Despite generating
+text with high fluency, lacking numerous source
+tables leads to lower generalizability of the table-
+to-text model. Recent progress in the pretrained
+language model (Devlin et al., 2019; Radford et al.,
+2019) shows remarkable performance in solving
+natural language processing tasks. The model pre-
+trained on large-scale data possesses rich knowl-
+edge, which inspires us with the potential for solv-
+ing generalization issues of the text generation task.
+To exploit the expressive power of the pretrained
+model for the table-to-text task, it is necessary to
+serialize the input table effectively. Several works
+have put efforts to bridge this gap, such as serial-
+izing the table into a token sequence (Zhang et al.,
+2020; Suadaa et al., 2021; Xing and Wan, 2021),
+or introducing an extra task to control the table
+representation (Gong et al., 2020). However, none
+of these leveraged the table structure information
+effectively. Furthermore, the text-to-text pretrained
+model decodes and generates a sequence in a one-
+pass forward process, which means it cannot per-
+ceive the future words in advance on the target side.
+Recently, the deliberation mechanism (Niehues
+et al., 2016; Geng et al., 2018) implemented by
+the multi-pass decoder is proposed to tackle this
+problem. However, how to adapt this approach for
+text-to-text pretraining, which can be further ap-
+plied to the table-to-text task, is another challenge.
+To this end, we propose a table structure under-
+standing and text deliberating approach, namely
+TASD, to solve the table-to-text task with the pre-
+trained language model enhanced by the deliber-
+ation mechanism. Specifically, we first serialize
+1
+arXiv:2301.02071v1  [cs.CL]  5 Jan 2023
+
+Published as a conference paper at EMNLP 2022
+the table input with customized templates which
+do not acquire the target cells to be labeled. Then,
+we employ the multi-head attention in a hierarchi-
+cal way to learn the table representation that is
+aware of table structure and apply it to guide the
+fine-tuning of the text-to-text pretrained model. Af-
+terward, we adopt the multi-pass decoder to realize
+text deliberation. More specifically, we treat the
+above table-structure-aware fine-tuned model as
+the first-pass decoder and adopt another pretrained
+model as the second-pass decoder to further polish
+the descriptive text. In the second-pass decoding
+phase, the table representation can be conveniently
+leveraged as the “original text” in the text deliber-
+ation mechanism. The main contributions of this
+work can be summarized as follows:
+• We propose a novel table-to-text generation
+approach (i.e., TASD) to assimilating the com-
+plete table information with the help of table
+structure distillation, the pretrained language
+model, and the text deliberation.
+• We devise a table-structure-aware text gen-
+eration model (TASATG) via the hierarchi-
+cal multi-head attention network, which can
+realize the content selection automatically.
+And we develop an effective text deliberation
+method dedicated to the table-to-text task.
+• Extensive experiments conducted on two dif-
+ferent datasets demonstrate that TASD out-
+performs comparable baselines in terms of
+various metrics.
+2
+Related Work
+2.1
+Table-to-Text Generation
+Encouraged by the success of seq2seq methods
+in machine translation and text summarization, re-
+searchers proposed to formulate the input table as a
+sequence of records (Lebret et al., 2016; Wiseman
+et al., 2017), and further improve the performance
+of table-to-text methods based on seq2seq by mod-
+eling table representation (Liu et al., 2018; Gong
+et al., 2019a). Introducing auxiliary tasks to enrich
+the table representation (Tian et al., 2019; Li et al.,
+2021a) is another promising paradigm to address
+the table-to-text problem. Moreover, there have
+been studies focusing on how to disaggregate the
+table-to-text pipeline effectively to generate more
+faithful and fluent text, e.g. leveraging content
+selection and planning (Puduppully et al., 2019;
+Trisedya et al., 2020; Bai et al., 2021), combin-
+ing autoregressive and non-autoregressive meth-
+ods (Wang et al., 2021). In addition, recent Trans-
+formers were also applied to solve the table-to-text
+task (Gong et al., 2019b; Wang et al., 2020; Obeid
+and Hoque, 2020). However, current table-to-text
+methods may fail to tackle the overfitting problem
+aroused by the lack of diversity in small datasets.
+Fine-tuning the model pretrained in a large cor-
+pus and adapting to a specific task is an effective
+approach to tackling the generation issues disturbed
+by small data and large parameters (Radford et al.,
+2019). (Kale and Rastogi, 2020) explored the feasi-
+bility of applying the text-to-text pretrained model
+to the table-to-text task, (Gong et al., 2020) applied
+multi-task learning to solve the table-to-text task
+with pretrained language model, and (Suadaa et al.,
+2021) leveraged pretrained language model for fact
+inference in numerical table contents. However,
+these approaches seldom perceived and integrated
+the complete table information into the fine-tuning
+of the pretrained model. A table-to-text pretrained
+model (Xing and Wan, 2021) was proposed though,
+the large and diversified table corpus is often un-
+available. In addition, recent works on fact verifica-
+tion taking tabular as input (Yin et al., 2020; Dong
+and Smith, 2021) have suggested the effectiveness
+of the table-structure-aware pretrained model.
+2.2
+Text Deliberation
+The encoder-decoder framework has been widely
+applied to neural machine translation, while the
+subsequent words are often invisible on the target
+side when decoding a sequence. To alleviate this, re-
+searchers proposed to decode and refine the output
+sequence in multiple passes, like human cognitive
+behavior when polishing an article. Studies have
+been made on text deliberation, such as the solu-
+tion with two separate stages (i.e., generating and
+polishing) (Niehues et al., 2016), combining two
+separate stages as one framework (Xia et al., 2017),
+and deliberating generated text in multiple passes
+adaptively via reinforcement learning (Geng et al.,
+2018) or customized evaluating architecture (Li
+and Yao, 2021). To the best of our knowledge, we
+are the first to apply the deliberation mechanism to
+the table-to-text problem.
+3
+Preliminaries
+3.1
+Problem Formulation
+Our table-to-text problem takes a table as input,
+and we formulate a table as a sequence of records:
+T = {τ1,1, τ1,2, · · · , τi,j, · · · , τm,n}, where m and n
+denote the number of rows and columns of T, re-
+2
+
+Published as a conference paper at EMNLP 2022
+Figure 1: The framework overview of TASD.
+spectively. Then, we aim to generate a document Y
+containing words Y = y1y2 · · · yl that can describe
+the content of T precisely, where l is the document
+length. Formally, given a table T, the table-to-text
+model is excepted to generate a descriptive docu-
+ment Y in an auto-regressive way
+yi = arg max P(yi | T, y1y2 · · · yi−1; Θ), i = 1, · · · , l
+where Θ is the set of model parameters.
+3.2
+Data
+NumericNLG Dataset. The numericNLG dataset
+was released by (Suadaa et al., 2021).
+In this
+dataset, the tables demonstrate experimental re-
+sults in research papers, thus, most of the table
+contents are numerical values. We use this dataset
+to evaluate the accuracy and smoothness of the
+generated descriptions for the table with numerical
+content. In particular, for each table of numer-
+icNLG, <table_id> acts as the pronoun of the
+table, and <caption> is the descriptive text of the
+table. Moreover, for each cell of a table, there are
+<metric>, (row and column) <header>, and
+<value> as different views of a cell.
+Totto Dataset. The Totto dataset (Parikh et al.,
+2020) is an open-domain table-to-text dataset
+collected from Wikipedia. The table contents
+are mainly in text form. The metadata of
+the
+Totto
+dataset
+includes
+<page_title>,
+<section_title> and <section_text>. In
+detail, each cell of a table has corresponding
+<header> and <value>. Unlike numericNLG,
+textual content in our Totto dataset accounts for
+62.4%, which can evaluate the text generation
+effectiveness for the tables with textual records.
+4
+Methodology
+In this section, we introduce the proposed frame-
+work in detail. As shown in Fig. 1, our framework
+mainly consists of three components, i.e., template-
+based table serialization, table-structure-aware
+fine-tuning, and text deliberation. Specifically, we
+first produce a sequence describing the table con-
+tents with customized templates. The templates we
+adopted do not require the target cells to be labeled.
+Then, to generate informative text, we adopt full ta-
+ble representation learning to guide the description
+generation, such that the outcome text is capable
+of emphasizing and delineating the facts in the ta-
+ble from a macroscopic perspective. Finally, we
+employ and adapt the multi-pass decoder to our
+data-to-text problem, which can further fine-tune
+the generated table description. Technical details
+for all three modules will be introduced separately
+in the following subsections.
+4.1
+Template-based Table Serialization
+To well harness the expressive power of the text-to-
+text pretrained model for the input table, it is nec-
+essary to serialize the raw table first. The template-
+based representation offers us a simple yet effective
+linearization approach to generating descriptive
+texts which can reflect the facts in a table without
+yielding an intractable downstream model.
+In particular, the templates we adopted in this
+work are devised to mention all the available facts
+in the table without knowing the emphasized cells
+in advance, which is different from (Suadaa et al.,
+2021). The template for describing facts consists
+of two parts:
+1. The title or descriptive text that comes with
+the table.
+2. A series of expressions, in which each one
+describes the content of a cell.
+More specifically, for the numericNLG dataset,
+we apply the following template:
+<table_id> shows <caption>. <metric1,1
+> of <header1,1> is <value1,1>, · · · , <me
+trici,j> of <headeri,j> is <valuei,j>, · · · .
+For the Totto dataset, we apply another template:
+As <page_title> <section_title>, <se
+ction_text>. <header1,1> is <value1,1>,
+· · · , <headeri,j> is <valuei, j>, · · · .
+The second part of the template enumerates all the
+cells in the table. This preliminary table represen-
+tation, denoted by TS , covers all the available facts
+in a raw table. Note that, the templates we adopt
+may encounter the content selection problem. In
+table-to-text applications, target cells in the input
+table are often not highlighted and the generated
+table description should emphasize certain cells.
+3
+
+table 2 shows the overall mention
+detection results on the test set of
+ontonotes. prec. of our full model
+prec. is 89.6. rec. of our full model
+rec. is 82.2. f1 of our full model f1 is
+table 2 shows the results on the ontonotes
+85..7.....
+test set. f1 score of our full model is 86.2
+our full model outperforms the state-of-
+the-art by 1.8 points in f1 score.
+table 2 shows the results on the ontonotes
+test set. we can see that our full model
+outperforms the state-of-the-art on all
+metrics.Published as a conference paper at EMNLP 2022
+Figure 2: The architecture of table-structure-aware text
+generation model (i.e., TASATG).
+4.2
+Table-Structure-Aware Text Generation
+A text-to-text pretrained model can take the large-
+scale corpus as input to possess vast knowledge
+and generate texts in an unsupervised way so that
+it has been widely applied to text-generation tasks.
+When handling a specific text generation task, it is
+effective to fine-tune the pretrained model on new
+data. However, for the table-to-text task, some hid-
+den information, like table structure, is most likely
+to be overlooked, though the drafted TS mentions
+all the available facts in the table. Thus, we pro-
+pose to exploit table structure information to guide
+fine-tuning of the text-to-text pretrained model.
+As shown in Fig. 2, we first encode the table
+content in a multi-view fashion. To be specific,
+given a cell τi,j in a table T, it can be viewed from
+different perspectives, such as the value of τi,j, the
+row header of τi,j, and the column header of τi,j,
+etc. Then, we treat the k-th view of τi,j as a to-
+ken sequence which is denoted by x(k)
+i,j . Afterward,
+we pad x(k)
+i,j with placeholders (if necessary) and
+concatenate these token sequences as follows:
+xi,j = x(1)
+i,j ⊛ x(2)
+i,j ⊛ · · · ,
+(1)
+where ⊛ denotes the concatenation operator, and
+the multi-viewed representation of a table T is de-
+noted as X = [x1,1, · · · , xi,j, · · · , xm,n]. Each to-
+ken of x(k)
+i,j can be encoded as a d-dimensional em-
+bedding by looking up the text-to-text pretrained
+model and updated accordingly when fine-tuning
+the pretrained model. In this way, we can obtain
+the semantic representation of table T, which is
+denoted by E(0) ∈ Rm×n×s×d, where s is the length
+of concatenated sequence xi,j.
+To realize TASATG for table-to-text, we pro-
+pose to employ multi-head attention (Vaswani et al.,
+2017) to guide fine-tuning of the text-to-text pre-
+trained model. In particular, we adopt three multi-
+head attention (MHA) layers to interactively extract
+the information in the table in a hierarchical way.
+Specifically, the MHA layer is defined as:
+Qi = QWQ
+i , Ki = KWK
+i , Vi = VWV
+i
+head i = Attention (Qi, Ki, Vi) = softmax
+�QiK⊤
+i
+√
+d
+�
+Vi,
+MHA(Q, K, V) = [ head 1, · · · , head h] WO,
+where Q, K, V represent the query, key and value
+in the attention mechanism, respectively.
+As illustrated in Fig. 2, in the first MHA layer,
+we add a cell text position embedding (E(ctpe) ∈
+Rs×d) to each cell of the aforementioned E(0), and
+feed it to the multi-head attention to implement cell
+text self-attention,
+�
+E0 = E(0) ⊕ E(ctpe),
+E(1) = MHA(�
+E(0), �
+E(0), �
+E(0)),
+E(1) = 1
+s
+s
+�
+i=1
+(E(1)[:, :, i, :]) ,
+(2)
+where ⊕ denotes the element-wise addition opera-
+tion. Consequently, E(1) ∈ Rm×n×d can be deemed
+as an initial aggregated table representation. Next,
+in the second MHA layer, we add a table position
+embedding (E(tpe) ∈ Rm×n×d) to E(1) to implement
+table structure self-attention,
+�
+E(1) = E(1) ⊕ E(tpe),
+E(2) = MHA(�
+E(1), �
+E(1), �
+E(1)).
+(3)
+E(2) ∈ Rm×n×d is the table-structure-aware represen-
+tation. Moreover, in the third MHA layer, we ap-
+ply a multi-head cross-attention to take the hidden
+state of the text-to-text pretrained model (denoted
+by H ∈ Rs×d) as the attention query, such that we
+can focus on the important cells of the table,
+�H = MHA(H, E(2), E(2)) ⊕ H.
+(4)
+This new hidden state �H guided by the table repre-
+sentation will replace the original hidden state H
+in the text-to-text pretrained model to generate the
+probability of the next word.
+Note that, the cross attention weights on differ-
+ent table cells based on the previous words can
+realize the content selection automatically. In ad-
+dition, we implement the text-to-text pretrained
+model with GPT2 (Radford et al., 2019), which
+adopts a decoder-only Transformer architecture.
+4
+
+Text-to-Text Pretrained Model
+Multi-Head Self-Attention
+Multi-Head Self-Attention
+Multi-Head Cross-Attention
+Next Word
+Previous
+Probability
+WordsPublished as a conference paper at EMNLP 2022
+(a) Training.
+(b) First and second fine-tuning of TASATG with vali-
+dation data.
+(c) Testing.
+Figure 3: Training, validation and testing procedures of the proposed TASD approach.
+4.3
+Text Deliberation
+The encoder-decoder framework applied in many
+sequence generation tasks often adopts a one-pass
+process while decoding a sequence. Though effi-
+cient, the one-pass decoder cannot perceive future
+context for further text deliberation. Multi-pass de-
+coder extends the capability of generating more
+refined text by exploring global information in the
+sequence (Niehues et al., 2016; Xia et al., 2017).
+For the text-to-text pretrained model, due to the
+huge amount of parameters of the pretrained lan-
+guage model, it is unwise to directly combine the
+models in different passes. A common solution is
+to concatenate the original serialized table content
+and the text generated in the previous pass to fine-
+tune the pretrained model in the next-pass decoding.
+However, in this way, the length of input text prob-
+ably exceeds the limit of the text-to-text pretrained
+model, and the time complexity is too high.
+To effectively implement the fine-tuning of the
+text-to-text pretrained model in multiple passes,
+as shown in Figs. 3a and 3b, we take the table
+representation as the “original text” and feed the
+text generated in the first-pass fine-tuning plus
+the table representation to the second-pass fine-
+tuning. Note that, as shown in Fig. 3a, we sep-
+arately fine-tune the table-to-text generation task
+and the text-to-text deliberation task with two inde-
+pendent TASATG models, and each of them takes
+a text-to-text pretrained model as the backbone.
+5
+Experiments
+5.1
+Experimental Settings
+Data. We conducted experiments on the aforemen-
+tioned datasets, i.e., numericNLG and Totto. The
+statistics of the numericNLG dataset can be found
+in (Suadaa et al., 2021). Besides, the size of the
+original Totto dataset is 120K, which is much larger
+than the numericNLG dataset. To evaluate differ-
+ent methods for table-to-text with comparable data
+size, for the Totto dataset, we filtered out the tables
+with fewer rows and columns, i.e., #rows < 8 and
+#columns < 8, such that the filtered Totto dataset
+contains 1.8K tables. Then, we randomly selected
+1.2K1 tables to generate the new Totto dataset.
+Evaluation Metrics. We calculated BLEU (from
+gram-1 to gram-4) (Papineni et al., 2002), ROUGE-
+L (Lin, 2004) and METEOR (Denkowski and
+Lavie, 2014) to evaluate the quality of the gen-
+erated text. The BLEU-n with a small value of n
+measures the accuracy of the word level, and the
+BLEU-n with a large n can measure the fluency
+of the sentence. The ROUGE-L measures the re-
+call rate based on the longest common sequence
+between source and target texts. The METEOR is
+based on the harmonic mean of unigram precision
+and recall, with recall weighted higher than preci-
+sion. These metrics are widely used to measure the
+accuracy and fluency of the generated sentence.
+Baselines. We compare TASD with the following
+baselines.
+• Template-based Table Serialization.
+We
+use the template designed for table serial-
+ization as a baseline. Note that, the token
+sequence generated by the template-based
+method is denoted as TS .
+• Pointer Generator (See et al., 2017). This
+is a seq2seq model with the attention and
+copy mechanism. We take TS as input for
+the pointer generator model.
+• TRM. We implemented a simplified version
+of the proposed TASD that omits the pos-
+sessed knowledge in the pretrained language
+model and removes text deliberation for focus-
+ing on table representation modeling, namely
+TRM. In particular, TRM adopts the architec-
+ture of GPT2 but initializes the parameters
+randomly and trains 100 epochs at most for
+fine-tuning. Besides, TRM takes TS plus the
+table structure representation as input and is
+fed with TS in the inference phase.
+1The size of numericNLG data is 1.3K.
+5
+
+2nd Fine-tuned TASATG
+TASATG
+TASATG
+1st Fine-tuned TASATG1st Fine-tuned TASATG
+Candidate Fine-tuned TASATG 1
+Candidate Fine-tuned TASATG 1
+Candidate Fine-tuned TASATG 2
+TASATG
+Candidate Fine-tuned TASATG 2
+TASATG
+Candidate Fine-tuned TASATG 3
+Candidate Fine-tuned TASATG 3
+1st Fine-tuned TASATG
+2nd Fine-tuned TASATG2ndFine-tunedTASATG
+1st Fine-tunedTASATGPublished as a conference paper at EMNLP 2022
+Table 1: Performance comparisons of the automatic evaluation on the numericNLG dataset.
+Method
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR ROUGE-L
+Template-based Method
+10.28
+5.52
+2.83
+1.14
+11.31
+11.49
+Pointer Generator
+5.10±0.59
+2.71±0.19
+1.16±0.17
+0.56±0.04
+7.82±0.15
+15.21±0.14
+TRM
+14.16±0.97
+6.05±0.50
+2.11±0.13
+0.80±0.12
+9.72±0.94
+12.72±0.80
+Fine-tuned GPT2
+16.13±0.56
+9.02±0.31
+4.68±0.22
+2.20±0.22
+10.14±0.32
+17.48±0.36
+TableGPT
+18.69±0.39
+8.21±0.24
+3.31±0.19
+1.51±0.14
+11.06±0.18
+16.90±0.27
+TASD w/o TAS
+18.20±2.40
+9.74±1.01
+4.38±0.31
+1.98±0.39
+10.64±0.86
+19.29±1.77
+TASD w/o D
+18.02±0.50
+10.06±0.25
+5.20±0.13
+2.47±0.20
+10.99±0.29
+18.57±0.27
+TASD w/o 1st-TAS
+20.07±1.94
+10.35±0.69
+4.67±0.35
+2.05±0.34
+11.52±0.80
+20.10±0.62
+TASD
+21.81±1.13
+11.03±0.11
+4.92±0.22
+2.15±0.39
+11.87±0.40
+20.40±0.80
+Table 2: Performance comparisons of the automatic evaluation on the Totto dataset.
+Method
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR ROUGE-L
+Template-based Method
+0.84
+0.43
+0.23
+0.09
+4.59
+1.51
+Pointer Generator
+11.34±1.57
+2.05±0.83
+0.45±0.27
+0.35±0.13
+5.38±0.78
+14.46±1.46
+TRM
+10.21±1.79
+3.44±0.88
+1.21±0.48
+0.54±0.25
+9.30±1.16
+11.52±2.03
+Fine-tuned GPT2
+9.53±0.51
+3.65±0.34
+1.18±0.37
+0.40±0.26
+9.89±0.39
+10.69±0.27
+TableGPT
+6.80±0.26
+3.51±0.22
+1.33±0.21
+0.76±0.12
+11.10±0.42
+11.73±0.44
+TASD w/o TAS
+13.70±0.90
+4.44±0.69
+1.28±0.47
+0.65±0.35
+10.79±0.83
+14.47±1.11
+TASD w/o D
+10.03±0.39
+4.42±0.29
+1.64±0.36
+0.71±0.38
+10.29±0.49
+10.67±0.34
+TASD w/o 1st-TAS
+13.90±0.60
+5.07±0.61
+1.68±0.52
+0.79±0.25
+10.98±0.40
+14.88±0.71
+TASD
+14.19±1.08
+5.17±0.38
+1.71±0.32
+0.78±0.21
+11.65±0.71
+14.96±1.10
+• Fine-tuned GPT2 (Radford et al., 2019). We
+take the concatenation of TS and Y as the in-
+put for fine-tuning. In the inference phase,
+we only feed TS to the model to generate Y
+starting after the last token of TS .
+• TableGPT (Gong et al., 2020). TableGPT is
+a state-of-the-art table-to-text method. To im-
+prove the text fidelity and exploit the struc-
+tural information at the same time, TableGPT
+employs a multi-task learning paradigm con-
+sisting of two auxiliary tasks, that is, one task
+reconstructs the table structure from represen-
+tations of GPT2, and the other aligns the tables
+and the information in the generated text.
+Implementation Details. The split settings for
+training, validation and, testing were 1084:136:135
+2 for the numericNLG dataset and 960:120:120
+for the Totto dataset, respectively. Regarding auto-
+matic evaluation, all results of deep models were
+obtained by conducting experiments on a Linux
+machine with Nvidia A100 GPU, and the averaged
+results of 5 runs were reported. Besides, an Adam
+2This setting follows the experiments of (Suadaa et al.,
+2021).
+optimizer was utilized (with an initial learning rate
+of 3e-5) for GPT2 fine-tuning, and the training was
+iterated in 20 epochs at most. A beam search algo-
+rithm was adopted when decoding a sequence and
+the beam width was set to 5 3.
+5.2
+Automatic Evaluation
+The comparisons of automatic evaluation results
+between TASD and other baselines can be found
+in Tables 1 and 2. In general, TASD outperforms
+the baselines for all the metrics on two datasets. In
+particular, compared to the reported best result of
+all the baselines, TASD achieves improvements of
+3.12 for BLEU-1 (18.69 → 21.81), 2.01 for BLEU-
+2 (9.02 → 11.03), 0.24 for BLEU-3 (4.68 → 4.92),
+0.56 for METEOR (11.31 → 11.87), and 2.92 for
+ROUGE-L (17.48 → 20.40) on the numericNLG
+dataset, and 2.85 for BLEU-1 (11.34 → 14.19),
+1.52 for BLEU-2 (3.65 → 5.17), 0.38 for BLEU-3
+(1.33 → 1.71), 0.02 for BLEU-4 (0.76 → 0.78),
+0.55 for METEOR (11.10 → 11.65), and 0.50 for
+ROUGE-L (14.46 → 14.96) on the Totto dataset.
+3Our implementation is available at https://github.
+com/ramber1836/TASD.
+6
+
+Published as a conference paper at EMNLP 2022
+In other words, for different types of source tables,
+TASD generates better descriptive texts w.r.t. accu-
+racy at the word level, recall of the sequence, and
+fluency of sentences.
+Besides, we have the following observations: 1)
+The template-based method performs much bet-
+ter on the numericNLG dataset compared to the
+Totto dataset, since the referenced table descrip-
+tions in numericNLG were collected from scientific
+papers, however, the table summaries in the Totto
+dataset are more diverse. 2) In the Totto dadaset,
+the pointer generator model tends to cover more
+words in descriptive text and generate more fluent
+sentences than the template-based method, as the
+contents in source tables of the Totto dataset are
+mostly linguistic. This can also explain why the
+pointer generator performs worse than the template-
+based method on the numericNLG dataset w.r.t.
+BLEU and METEOR. 3) Fine-tuned GPT2 can
+generate more faithful and fluent text than other
+baselines (refer to Tables 1 and 2) most of the time,
+which validates the effectiveness of the pretrained
+language model. 4) In general, TableGPT performs
+better, and even the best, among all the baselines.
+In the numericNLG dataset, the headers of the
+input tables (a.k.a. the attributes of records for
+TableGPT) are more diverse, which may explain
+why the performance of TableGPT is not promising
+as expected on the numericNLG dataset. 5) TRM
+can generate comparable, or even better descriptive
+text as fined-tuned GPT2, which further suggests
+the effectiveness of table structure understanding.
+5.3
+Ablation Analysis
+Moreover, to verify the effectiveness of different
+modules, we compare TASD with its variants.
+• After generating text with fine-tuned GPT2,
+we fed the generated text concatenated with
+TS to another fine-tuned GPT2 to realize the
+second-pass decoder without table structure
+representation.
+• We implemented TASD without deliberating
+on the outcome text, which means that we
+realized TASATG based on GPT2 in a one-
+pass forward process.
+• TASD w/o 1st-TAS. We removed table struc-
+ture modeling in the first-pass decoding from
+TASD, which was implemented by taking the
+fine-tuned GPT2 as the first-pass decoder and
+the table-structure-aware fine-tuned GPT2 as
+the second-pass decoder.
+As can be seen in Tables 1 and 2, TASD w/o TAS
+performs worse than TASD under all metrics, since
+the table structure modeling can benefit the fine-
+tuning of GPT2. This can also be validated by com-
+paring fine-tuned GPT2 to TASD w/o D. Besides,
+the effectiveness of deliberating text can be proven
+by comparing TASD w/o D to TASD (this can also
+be validated by comparing fine-tuned GPT2 to
+TASD w/o TAS). While text deliberation may harm
+sentence fluency as depicted by the results of these
+methods w.r.t. BLEU-3 & 4 in Table 1. In addition,
+TASD w/o 1st-TAS outperforms TASD w/o TAS under
+all metrics suggesting that taking the table repre-
+sentation as the “original text” in the deliberation
+mechanism is also effective.
+5.4
+Qualitative Analysis
+Figs. 4(a) and (b) show two selected source ta-
+bles and corresponding descriptive texts (i.e., cap-
+tion and section_text) in numericNLG and Totto
+datasets. Fig. 4(c) demonstrates the generated de-
+scriptions by different methods. The text that cor-
+rectly reflects the facts of the source table is in
+green, the erroneous text is in red, and the con-
+fusing text is in blue. We can see that, there are
+many grammatical errors in the text produced by
+the pointer generator. Fine-tuned GPT2 tends to
+repeat phrases and sentences due to the limited
+knowledge about the input table, which can also
+explain why the fine-tuned GPT2 can obtain a false
+high score in BLEU-n as n grows. Thanks to the
+semantic knowledge brought by pretraining, fine-
+tuned GPT2 can generate more natural descriptions,
+in which, however, perplexing factual errors ex-
+ist. Compared to fine-tuned GPT2, the description
+generated by TASD is more relevant to the table
+contents. Since the target cells are not known in
+advance, the generated text may miss the empha-
+sized points described in the reference. The text
+generated by TableGPT is also fluent, though coun-
+terfactual descriptions may exist.
+5.5
+Human Evaluation
+We randomly selected 30 samples from the test set
+in numericNLG and Totto datasets, respectively,
+and invited 10 volunteers to evaluate the quality of
+the outcome text by considering three criteria, i.e.,
+grammar, coherence & concise, and factual per-
+spective (correct and relevant). Each criterion has
+scores of five degrees, ranging from 1 (the worst) to
+5 (the best). The averaged scores were reported in
+Table 3, which show that TASD can generate more
+7
+
+Published as a conference paper at EMNLP 2022
+Figure 4: Two examples of the generated table descriptions.
+Table 3: Result of Human Evaluation
+Dataset
+Method
+Grammar Coherence
+& Concise
+Factual per-
+spective
+numericNLG
+Pointer Genera-
+tor
+3.16±0.99
+2.73±1.20
+1.54±0.69
+Fine-tuned
+GPT2
+3.42±0.56
+3.11±0.58
+2.51±0.45
+TASD w/o D
+3.72±0.61
+3.48±0.55
+2.82±0.45
+TASD
+4.17±0.72
+3.98±0.64
+3.15±0.73
+Totto
+Pointer Genera-
+tor
+2.03±0.71
+1.89±0.82
+1.56±0.55
+Fine-tuned
+GPT2
+2.60±0.55
+2.36±0.64
+1.85±0.46
+TASD w/o D
+2.63±0.52
+2.46±0.60
+1.89±0.46
+TASD
+3.4±0.66
+3.18±0.70
+2.25±0.69
+readable and coherent texts, and describe more
+correct facts. Moreover, the pretrained models con-
+sistently achieve better scores than the pointer gen-
+erator on grammar and coherence because of the
+expressive power learned from the large-scale cor-
+pus. In the Totto dataset, the improvement of the
+table structure modeling is smaller than that of the
+polishing mechanism, which is consistent with the
+automatic evaluation results in Table 2.
+6
+Discussion
+In our work, we devised a two-pass decoder frame-
+work dedicated to the table-to-text task with the
+help of the table-structure-aware text generation
+model (i.e., TASATG). However, the effectiveness
+of the text deliberation for the table-to-text task
+should be further explored and integrated into the
+table-structure-aware modeling in a more harmonic
+Figure 5:
+Table reconstruction for table-structure-
+aware modeling enhancement.
+manner. To discuss the limitation of the text de-
+liberation of TASD, we additionally developed a
+table content reconstruction loss and integrate it
+into TASD in a multi-task learning fashion.
+Specifically, given the table-structure-aware em-
+bedding E(2) generated with Eq. (3), we randomly
+mask certain cells of the input table and yield a
+partially corrupted embedding of the input table,
+denoted by �
+E(2). Then, a two-layer MLP (i.e., multi-
+layer perceptron) is adopted to restore the table-
+structure-aware embedding. Afterward, an MSE
+(i.e., mean square error) loss is adopted to mea-
+sure the effectiveness of table reconstruction and
+further integrated into the TASD framework in the
+multi-task learning paradigm. The process of table
+reconstruction is demonstrated in Fig. 5.
+We carried out a series of experiments to evalu-
+ate the performance of TASD w/ and w/o the help
+of table reconstruction loss (i.e., TRLoss) on nu-
+mericNLG and Totto datasets in terms of BLEU-n
+(1 to 4), METEOR, and ROUGE-L. The results can
+be found in Tables 4 and 5.
+8
+
+our model also outperforms transformer
+our model.
+compares.
+transformer significantly outperforms transformer
+the effectiveness of transformer
+the comparison of our model with transformer on
+caption
+frenchenglish translation task.
+the size of our
+model is slightly larger than the size of the transformer model
+the evaluation metrics of our model and the transformer. our
+model achieves the best results on all the metrics
+suvarnabhumi is the busiest airport in thailand
+international airport international airport
+ international airport international airport...
+2018, airport is airport is airport...
+busiest
+after
+suvarnabhumi airport
+is the third-busiest airport in thailand after
+suvarnabhumi airport
+section_titleTASATG
+GPT2 Fine-Tuning Loss
+Table Reconstruction Loss
+MLPPublished as a conference paper at EMNLP 2022
+Table 4: The performances of TASD w/ and w/o the table reconstruction on the numericNLG dataset.
+Method
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR ROUGE-L
+TASD w/o D
+18.02±0.50
+10.06±0.25
+5.20±0.13
+2.47±0.20
+10.99±0.29
+18.57±0.27
+TASD w/o D w/ TRLoss
+20.56±0.25
+11.57±0.21
+5.90±0.23
+2.98±0.17
+12.00±0.48
+20.50±0.39
+TASD w/ TRLoss
+19.29±0.38
+10.12±0.24
+5.32±0.25
+2.62±0.22
+12.18±0.90
+18.95±0.69
+TASD w/ TRLoss in 1st pass
+18.23±0.68
+9.39±0.52
+4.64±0.26
+2.36±0.24
+11.51±0.78
+18.13±0.45
+TASD w/ TRLoss in 2nd pass
+19.38±2.21
+10.33±1.34
+5.11±0.73
+2.40±0.38
+11.35±0.92
+18.69±1.05
+TASD
+21.81±1.13
+11.03±0.11
+4.92±0.22
+2.15±0.39
+11.87±0.40
+20.40±0.80
+Table 5: The performances of TASD w/ and w/o the table reconstruction on the Totto dataset.
+Method
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+METEOR ROUGE-L
+TASD w/o D
+10.03±0.39
+4.42±0.29
+1.64±0.36
+0.71±0.38
+10.29±0.49
+10.67±0.34
+TASD w/o D w/ TRLoss
+9.94±0.43
+4.35±0.31
+1.63±0.31
+0.75±0.13
+10.37±0.22
+10.62±0.60
+TASD w/ TRLoss
+14.57±0.87
+5.22±0.42
+1.70±0.49
+0.89±0.38
+11.79±0.77
+15.28±0.86
+TASD w/ TRLoss in 1st pass
+14.00±0.82
+5.31±0.27
+1.72±0.25
+0.75±0.13
+11.02±0.77
+14.74±0.51
+TASD w/ TRLoss in 2nd pass
+13.89±0.58
+4.78±0.61
+1.47±0.14
+0.52±0.20
+11.07±0.66
+14.73±0.79
+TASD
+14.19±1.08
+5.17±0.38
+1.71±0.32
+0.78±0.21
+11.65±0.71
+14.96±1.10
+According to the results reported on the nu-
+mericNLG dataset, the TRLoss is helpful in en-
+hancing the capability of table comprehension
+though, the best performance is achieved by TASD
+w/o D w/ TRLoss. It seems that the performance im-
+provement gained by the table comprehension en-
+hancement is sacrificed after the text deliberation
+is adopted. Meanwhile, on the Totto dataset, TASD
+with the table reconstruction (i.e., TASD w/ TRLoss)
+does achieve the best performance in terms of
+BLEU-1, BLEU-2, METEOR, and ROUGE-L,
+though the improvement is not remarkable. The
+contents of the input tables are mainly linguistic
+and the table structures are not too diverse might
+be able to explain the performance improvement
+of TASD w/ TRLoss on the Totto dataset. With the
+above comparisons, we can conclude that, for the
+input tables with diverse structures, the limitation
+of the current text deliberation mechanism cannot
+be neglected if one aims to enhance the capability
+of table comprehension for the table-to-text task.
+Moreover, this also suggests that the generalization
+capability of text deliberation of TASD should be
+improved in the future.
+Limitations. In this work, long tables in the
+Totto dataset are removed since the efficiency and
+performance of TASD on large tables could be low-
+ered. In the future, the capability of handling long
+tables for table-to-text models should be further ex-
+plored. Besides, a large-scale and more exhaustive
+human evaluation is necessary. We plan to recruit
+more volunteers to conduct the human annotation.
+7
+Conclusion
+In this paper, to realize table-to-text with the pre-
+trained language model, we proposed a table struc-
+ture understanding and text deliberating approach,
+namely TASD. The table structure understanding
+was realized by developing a hierarchical multi-
+head attention network, which can benefit the fine-
+tuning of the text-to-text pretrained model. The
+fully represented table information benefits not
+only the pretrained language model but also the
+text deliberation process since the structure infor-
+mation with rich semantics could be fed into the
+second-pass decoding naturally. We carried out ex-
+tensive experiments on two public datasets with
+different table types. Automatic and human-based
+evaluations, as well as qualitative analysis, vali-
+date the effectiveness of our approach to generating
+faithful and fluent table descriptions. In the future,
+we will improve text deliberation by devising a
+unified framework to integrate the multi-pass de-
+coder and refine the descriptive text paying more
+attention to sentence fluency.
+Acknowledgements
+This work is supported in part by Foshan
+HKUST Projects (FSUST21-FYTRI01A, FSUST
+21-FYTRI02A).
+9
+
+Published as a conference paper at EMNLP 2022
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+Peng Wang, Junyang Lin, An Yang, Chang Zhou,
+Yichang Zhang, Jingren Zhou, and Hongxia Yang.
+2021. Sketch and refine: Towards faithful and in-
+formative table-to-text generation. In ACL-IJCNLP,
+pages 4831–4843.
+Zhenyi Wang, Xiaoyang Wang, Bang An, Dong Yu,
+and Changyou Chen. 2020. Towards faithful neural
+table-to-text generation with content-matching con-
+straints. In ACL, pages 1072–1086.
+Sam Wiseman, Stuart M Shieber, and Alexander M
+Rush. 2017.
+Challenges in data-to-document gen-
+eration. In EMNLP, pages 2253–2263.
+Yingce Xia, Fei Tian, Lijun Wu, Jianxin Lin, Tao Qin,
+Nenghai Yu, and Tie-Yan Liu. 2017. Deliberation
+networks: Sequence generation beyond one-pass de-
+coding. NeurIPS, 30:1784–1794.
+Xinyu Xing and Xiaojun Wan. 2021. Structure-aware
+pre-training for table-to-text generation.
+In ACL-
+IJCNLP, pages 2273–2278.
+Pengcheng Yin, Graham Neubig, Wen-tau Yih, and Se-
+bastian Riedel. 2020. Tabert: Pretraining for joint
+understanding of textual and tabular data. In ACL,
+pages 8413–8426.
+Hongzhi Zhang, Yingyao Wang, Sirui Wang, Xuezhi
+Cao, Fuzheng Zhang, and Zhongyuan Wang. 2020.
+Table fact verification with structure-aware trans-
+former. In EMNLP, pages 1624–1629.
+A
+Human Evaluation Settings
+The criteria adopted in our human-based evaluation
+are (1) Grammar (e.g., is this paragraph grammat-
+ical?), (2) Coherence & Concise (e.g., is this para-
+graph coherent and contextually consistent? does
+it repeat redundant information?), and (3) Factual
+perspective (e.g., are the facts that this paragraph
+describes correct? are these facts related to refer-
+ences and tables?). More specifically, we list the
+detailed justifications on how to score the generated
+text in each criterion as follows.
+Grammar
+• 1 It is more like garbled code than a paragraph.
+• 2 There are many obvious grammatical mis-
+takes.
+• 3 There are a few obvious grammatical mistakes.
+• 4 There are few grammatical mistakes.
+• 5 There are no grammatical mistakes.
+Coherence & Concise
+• 1 The logic of text expression is chaotic and
+nonsense.
+• 2 There are a lot of logical inconsistencies or
+redundant information.
+• 3 There are some logical inconsistencies or re-
+dundant information.
+• 4 There are a few logical inconsistencies or
+redundant information, but it does not affect
+browsing.
+• 5 The logic of the text is smooth without redun-
+dant information.
+Factual Perspective
+• 1 The paragraph does not coincide with the ref-
+erence or table, and it is full of information
+inconsistent with the facts.
+• 2 The paragraph describes the facts incorrectly
+and has a low correlation with reference, but is
+related to the information in the table.
+• 3 The paragraph description is incorrect, but it
+is highly coincident with the reference.
+• 4 The paragraph description is basically correct,
+and the coincidence with the reference is low,
+but it also describes the information in the table.
+• 5 The paragraph description is correct and
+highly coincident with the reference.
+B
+Illustrative Examples of Generated
+Descriptions
+We additionally selected another two examples of
+the generated table descriptions from the numeric-
+NLG and Totto datasets, respectively. The results
+are shown in Figs. 6 and 7. From these four ex-
+amples, we can see that TASD can generate more
+accurate and fluent descriptive texts. While incor-
+rect descriptions can be found in the outcome texts
+generated by different models for cases D and F,
+which suggests that generating faithful descriptions
+for open-domain tables is much more challenging
+and requires more powerful and, thus larger, pre-
+trained language models.
+C
+Extra Implementation Details
+The learning rate of GPT2 was searched from {3e−
+4, 3e − 5, 3e − 6}. In the evaluation of discussing
+the limitation of text deliberation (see Section 6),
+a trade-off parameter for balancing the GPT2 fine-
+tuning loss and the TRLoss was adopted, then the
+trade-off parameter was searched from {1e−1, 5e−
+2, 1e − 2, 5e − 3, 1e − 3}, and 1e-2 was selected for
+the reported performance. Besides, the reported
+results in Tables 4 and 5 were averaged in 3 runs.
+11
+
+Published as a conference paper at EMNLP 2022
+Figure 6: Generated table descriptions on cases C and D.
+Figure 7: Generated table descriptions on cases E and F.
+12
+
+robusttc-fsl,
+et al.（85.8 et al., 2018)
+matching and prototypical networks
+relation networks
+the graph networks
+arsc mean acc dataset
+ snorkel network providing the best performance
+our model outperforms the other baselines on mean acc
+it leads to a significant improvement in mean accuracy for the compared models
+on the other hand, the absence of semantic representation (a) leads to a significant
+caption
+improvement in mean accuracy
+our proposed model achieves competitive accuracy, comparable to
+the state-of-the-art models from yu et al. (2018).
+ steve gregory
+ steve gregory
+seven interceptions seven
+13 games
+35 tackles
+sacks
+tour touchdowns
+ steve gregory
+357 tackles seven interceptions three sacks
+forced fumble.0 yards per game
+steve gregory ended his career with 357 tackles, seven
+page_title
+interceptions, three sacks and 1 forced fumble.0 yards per game ... (repeat)
+section title
+steve gregory
+357 tackles seven sacks
+seven interceptionsconsistency score
+the w/o results
+consistency and novelty scores
+w/o adversarial learning results in a full model
+consistency score of 3.84 and a novelty score of 3.24
+baseline
+adversarial adaptation
+results in a much better consistency than the w/o adaptation
+caption
+combined with the w/o Istm, the coherence improves significantly
+consistency, novelty, diversity,
+and coherence
+same consistency
+dynamic strategy leads to a higher novelty
+110 years
+the district index performed last 57 district index ohio 67 district ( death ) until
+district used holder index ( years district)
+alice sanders (12 may 1897 - 7 november 2007)
+survivors
+alice sanders (12 mav 1897 - 7 november 2007)
+alice sargent
+over 40 years
+page_title
+breast
+and breast breast
+section title
\ No newline at end of file
diff --git a/hdA0T4oBgHgl3EQfIP8h/content/tmp_files/load_file.txt b/hdA0T4oBgHgl3EQfIP8h/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1040 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf,len=1039
+page_content='Published as a conference paper at EMNLP 2022  Towards Table-to-Text Generation with Pretrained Language Model: A  Table Structure Understanding and Text Deliberating Approach  Miao Chen§ ∗, Xinjiang Lu¶ �, Tong Xu§, Yanyan Li ¶, Jingbo Zhou¶, Dejing Dou¶, Hui Xiong† ‡  ¶BIL, Baidu Research,  §University of Science and Technology of China  †Hong Kong University of Science and Technology (Guangzhou)  ‡Guangzhou HKUST Fok Ying Tung Research Institute  cmer@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='cn, {luxinjiang,liyanyanliyanyan,zhoujingbo}@baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='com,  tongxu@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='cn, doudejing@baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='com, xionghui@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='hk  Abstract  Although remarkable progress on the neural  table-to-text methods has been made, the gen-  eralization issues hinder the applicability of  these models due to the limited source tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Large-scale pretrained language models sound  like a promising solution to tackle such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  However, how to effectively bridge the gap  between the structured table and the text in-  put by fully leveraging table information to  fuel the pretrained model is still not well ex-  plored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Besides, another challenge of integrat-  ing the deliberation mechanism into the text-  to-text pretrained model for solving the table-  to-text task remains seldom studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In this  paper, to implement the table-to-text genera-  tion with pretrained language model, we pro-  pose a table structure understanding and text  deliberating approach, namely TASD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' To be  specific, we devise a three-layered multi-head  attention network to realize the table-structure-  aware text generation model with the help of  the pretrained language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Furthermore,  a multi-pass decoder framework is adopted to  enhance the capability of polishing generated  text for table descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The empirical stud-  ies, as well as human evaluation, on two public  datasets, validate that our approach can gener-  ate faithful and fluent descriptive texts for dif-  ferent types of tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  1  Introduction  The task of learning to generate natural language  descriptions from non-linguistic input, which is  referred to as data-to-text, is important for many  applications, such as weather forecast genera-  tion (Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2016), sports news writing (Wise-  man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2017), biography writing (Lebret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2016), market comments writing (Murakami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2017) and automatic question-answering (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The input data can be in various forms  ∗ This work was done when the first author was an intern  at Baidu Research under the supervision of the second author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  for data-to-text though, here we focus on the text  generation task that takes the table as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Inspired by neural machine translation models,  previous studies on table-to-text tasks mainly adopt  traditional seq2seq methods to generate table de-  scriptions (Lebret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Wiseman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Despite generating  text with high fluency, lacking numerous source  tables leads to lower generalizability of the table-  to-text model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Recent progress in the pretrained  language model (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2019) shows remarkable performance in solving  natural language processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The model pre-  trained on large-scale data possesses rich knowl-  edge, which inspires us with the potential for solv-  ing generalization issues of the text generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  To exploit the expressive power of the pretrained  model for the table-to-text task, it is necessary to  serialize the input table effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Several works  have put efforts to bridge this gap, such as serial-  izing the table into a token sequence (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Suadaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Xing and Wan, 2021),  or introducing an extra task to control the table  representation (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' However, none  of these leveraged the table structure information  effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Furthermore, the text-to-text pretrained  model decodes and generates a sequence in a one-  pass forward process, which means it cannot per-  ceive the future words in advance on the target side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Recently, the deliberation mechanism (Niehues  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2018) implemented by  the multi-pass decoder is proposed to tackle this  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' However, how to adapt this approach for  text-to-text pretraining, which can be further ap-  plied to the table-to-text task, is another challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  To this end, we propose a table structure under-  standing and text deliberating approach, namely  TASD, to solve the table-to-text task with the pre-  trained language model enhanced by the deliber-  ation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Specifically, we first serialize  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='02071v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='CL]  5 Jan 2023  Published as a conference paper at EMNLP 2022  the table input with customized templates which  do not acquire the target cells to be labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Then,  we employ the multi-head attention in a hierarchi-  cal way to learn the table representation that is  aware of table structure and apply it to guide the  fine-tuning of the text-to-text pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Af-  terward, we adopt the multi-pass decoder to realize  text deliberation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' More specifically, we treat the  above table-structure-aware fine-tuned model as  the first-pass decoder and adopt another pretrained  model as the second-pass decoder to further polish  the descriptive text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In the second-pass decoding  phase, the table representation can be conveniently  leveraged as the “original text” in the text deliber-  ation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The main contributions of this  work can be summarized as follows:  We propose a novel table-to-text generation  approach (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', TASD) to assimilating the com-  plete table information with the help of table  structure distillation, the pretrained language  model, and the text deliberation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  We devise a table-structure-aware text gen-  eration model (TASATG) via the hierarchi-  cal multi-head attention network, which can  realize the content selection automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  And we develop an effective text deliberation  method dedicated to the table-to-text task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Extensive experiments conducted on two dif-  ferent datasets demonstrate that TASD out-  performs comparable baselines in terms of  various metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  2  Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='1  Table-to-Text Generation  Encouraged by the success of seq2seq methods  in machine translation and text summarization, re-  searchers proposed to formulate the input table as a  sequence of records (Lebret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Wiseman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2017), and further improve the performance  of table-to-text methods based on seq2seq by mod-  eling table representation (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Gong  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Introducing auxiliary tasks to enrich  the table representation (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2021a) is another promising paradigm to address  the table-to-text problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Moreover, there have  been studies focusing on how to disaggregate the  table-to-text pipeline effectively to generate more  faithful and fluent text, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' leveraging content  selection and planning (Puduppully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Trisedya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2021), combin-  ing autoregressive and non-autoregressive meth-  ods (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In addition, recent Trans-  formers were also applied to solve the table-to-text  task (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Obeid  and Hoque, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' However, current table-to-text  methods may fail to tackle the overfitting problem  aroused by the lack of diversity in small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Fine-tuning the model pretrained in a large cor-  pus and adapting to a specific task is an effective  approach to tackling the generation issues disturbed  by small data and large parameters (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' (Kale and Rastogi, 2020) explored the feasi-  bility of applying the text-to-text pretrained model  to the table-to-text task, (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2020) applied  multi-task learning to solve the table-to-text task  with pretrained language model, and (Suadaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2021) leveraged pretrained language model for fact  inference in numerical table contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' However,  these approaches seldom perceived and integrated  the complete table information into the fine-tuning  of the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' A table-to-text pretrained  model (Xing and Wan, 2021) was proposed though,  the large and diversified table corpus is often un-  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In addition, recent works on fact verifica-  tion taking tabular as input (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Dong  and Smith, 2021) have suggested the effectiveness  of the table-structure-aware pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='2  Text Deliberation  The encoder-decoder framework has been widely  applied to neural machine translation, while the  subsequent words are often invisible on the target  side when decoding a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' To alleviate this, re-  searchers proposed to decode and refine the output  sequence in multiple passes, like human cognitive  behavior when polishing an article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Studies have  been made on text deliberation, such as the solu-  tion with two separate stages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', generating and  polishing) (Niehues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2016), combining two  separate stages as one framework (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2017),  and deliberating generated text in multiple passes  adaptively via reinforcement learning (Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2018) or customized evaluating architecture (Li  and Yao, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' To the best of our knowledge, we  are the first to apply the deliberation mechanism to  the table-to-text problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  3  Preliminaries  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='1  Problem Formulation  Our table-to-text problem takes a table as input,  and we formulate a table as a sequence of records:  T = {τ1,1, τ1,2, · · · , τi,j, · · · , τm,n}, where m and n  denote the number of rows and columns of T, re-  2  Published as a conference paper at EMNLP 2022  Figure 1: The framework overview of TASD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Then, we aim to generate a document Y  containing words Y = y1y2 · · · yl that can describe  the content of T precisely, where l is the document  length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Formally, given a table T, the table-to-text  model is excepted to generate a descriptive docu-  ment Y in an auto-regressive way  yi = arg max P(yi | T, y1y2 · · · yi−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Θ), i = 1, · · · , l  where Θ is the set of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='2  Data  NumericNLG Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The numericNLG dataset  was released by (Suadaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  In this  dataset, the tables demonstrate experimental re-  sults in research papers, thus, most of the table  contents are numerical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We use this dataset  to evaluate the accuracy and smoothness of the  generated descriptions for the table with numerical  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In particular, for each table of numer-  icNLG, <table_id> acts as the pronoun of the  table, and <caption> is the descriptive text of the  table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Moreover, for each cell of a table, there are  <metric>, (row and column) <header>, and  <value> as different views of a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Totto Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The Totto dataset (Parikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2020) is an open-domain table-to-text dataset  collected from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The table contents  are mainly in text form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The metadata of  the  Totto  dataset  includes  <page_title>,  <section_title> and <section_text>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In  detail, each cell of a table has corresponding  <header> and <value>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Unlike numericNLG,  textual content in our Totto dataset accounts for  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='4%, which can evaluate the text generation  effectiveness for the tables with textual records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  4  Methodology  In this section, we introduce the proposed frame-  work in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 1, our framework  mainly consists of three components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', template-  based table serialization, table-structure-aware  fine-tuning, and text deliberation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Specifically, we  first produce a sequence describing the table con-  tents with customized templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The templates we  adopted do not require the target cells to be labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Then, to generate informative text, we adopt full ta-  ble representation learning to guide the description  generation, such that the outcome text is capable  of emphasizing and delineating the facts in the ta-  ble from a macroscopic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Finally, we  employ and adapt the multi-pass decoder to our  data-to-text problem, which can further fine-tune  the generated table description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Technical details  for all three modules will be introduced separately  in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='1  Template-based Table Serialization  To well harness the expressive power of the text-to-  text pretrained model for the input table, it is nec-  essary to serialize the raw table first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The template-  based representation offers us a simple yet effective  linearization approach to generating descriptive  texts which can reflect the facts in a table without  yielding an intractable downstream model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  In particular, the templates we adopted in this  work are devised to mention all the available facts  in the table without knowing the emphasized cells  in advance, which is different from (Suadaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The template for describing facts consists  of two parts:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The title or descriptive text that comes with  the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' A series of expressions, in which each one  describes the content of a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  More specifically, for the numericNLG dataset,  we apply the following template:  <table_id> shows <caption>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' <metric1,1  > of <header1,1> is <value1,1>, · · · , <me  trici,j> of <headeri,j> is <valuei,j>, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  For the Totto dataset, we apply another template:  As <page_title> <section_title>, <se  ction_text>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' <header1,1> is <value1,1>,  · · , <headeri,j> is <valuei, j>, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  The second part of the template enumerates all the  cells in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' This preliminary table represen-  tation, denoted by TS , covers all the available facts  in a raw table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Note that, the templates we adopt  may encounter the content selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In  table-to-text applications, target cells in the input  table are often not highlighted and the generated  table description should emphasize certain cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  3  table 2 shows the overall mention  detection results on the test set of  ontonotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' of our full model  prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' is 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' of our full model  rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' is 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' f1 of our full model f1 is  table 2 shows the results on the ontonotes  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='.7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' f1 score of our full model is 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='2  our full model outperforms the state-of-  the-art by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='8 points in f1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  table 2 shows the results on the ontonotes  test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' we can see that our full model  outperforms the state-of-the-art on all  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='Published as a conference paper at EMNLP 2022  Figure 2: The architecture of table-structure-aware text  generation model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', TASATG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='2  Table-Structure-Aware Text Generation  A text-to-text pretrained model can take the large-  scale corpus as input to possess vast knowledge  and generate texts in an unsupervised way so that  it has been widely applied to text-generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  When handling a specific text generation task, it is  effective to fine-tune the pretrained model on new  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' However, for the table-to-text task, some hid-  den information, like table structure, is most likely  to be overlooked, though the drafted TS mentions  all the available facts in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Thus, we pro-  pose to exploit table structure information to guide  fine-tuning of the text-to-text pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2, we first encode the table  content in a multi-view fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' To be specific,  given a cell τi,j in a table T, it can be viewed from  different perspectives, such as the value of τi,j, the  row header of τi,j, and the column header of τi,j,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Then, we treat the k-th view of τi,j as a to-  ken sequence which is denoted by x(k)  i,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Afterward,  we pad x(k)  i,j with placeholders (if necessary) and  concatenate these token sequences as follows:  xi,j = x(1)  i,j ⊛ x(2)  i,j ⊛ · · · ,  (1)  where ⊛ denotes the concatenation operator, and  the multi-viewed representation of a table T is de-  noted as X = [x1,1, · · · , xi,j, · · · , xm,n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Each to-  ken of x(k)  i,j can be encoded as a d-dimensional em-  bedding by looking up the text-to-text pretrained  model and updated accordingly when fine-tuning  the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In this way, we can obtain  the semantic representation of table T, which is  denoted by E(0) ∈ Rm×n×s×d, where s is the length  of concatenated sequence xi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  To realize TASATG for table-to-text, we pro-  pose to employ multi-head attention (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2017) to guide fine-tuning of the text-to-text pre-  trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In particular, we adopt three multi-  head attention (MHA) layers to interactively extract  the information in the table in a hierarchical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Specifically, the MHA layer is defined as:  Qi = QWQ  i , Ki = KWK  i , Vi = VWV  i  head i = Attention (Qi, Ki, Vi) = softmax  �QiK⊤  i  √  d  �  Vi,  MHA(Q, K, V) = [ head 1, · · · , head h] WO,  where Q, K, V represent the query, key and value  in the attention mechanism, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2, in the first MHA layer,  we add a cell text position embedding (E(ctpe) ∈  Rs×d) to each cell of the aforementioned E(0), and  feed it to the multi-head attention to implement cell  text self-attention,  �  E0 = E(0) ⊕ E(ctpe),  E(1) = MHA(�  E(0), �  E(0), �  E(0)),  E(1) = 1  s  s  �  i=1  (E(1)[:, :, i, :]) ,  (2)  where ⊕ denotes the element-wise addition opera-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Consequently, E(1) ∈ Rm×n×d can be deemed  as an initial aggregated table representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Next,  in the second MHA layer, we add a table position  embedding (E(tpe) ∈ Rm×n×d) to E(1) to implement  table structure self-attention,  �  E(1) = E(1) ⊕ E(tpe),  E(2) = MHA(�  E(1), �  E(1), �  E(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  (3)  E(2) ∈ Rm×n×d is the table-structure-aware represen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Moreover, in the third MHA layer, we ap-  ply a multi-head cross-attention to take the hidden  state of the text-to-text pretrained model (denoted  by H ∈ Rs×d) as the attention query, such that we  can focus on the important cells of the table,  �H = MHA(H, E(2), E(2)) ⊕ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  (4)  This new hidden state �H guided by the table repre-  sentation will replace the original hidden state H  in the text-to-text pretrained model to generate the  probability of the next word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Note that, the cross attention weights on differ-  ent table cells based on the previous words can  realize the content selection automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In ad-  dition, we implement the text-to-text pretrained  model with GPT2 (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2019), which  adopts a decoder-only Transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  4  Text-to-Text Pretrained Model  Multi-Head Self-Attention  Multi-Head Self-Attention  Multi-Head Cross-Attention  Next Word  Previous  Probability  WordsPublished as a conference paper at EMNLP 2022  (a) Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  (b) First and second fine-tuning of TASATG with vali-  dation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  (c) Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Figure 3: Training, validation and testing procedures of the proposed TASD approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='3  Text Deliberation  The encoder-decoder framework applied in many  sequence generation tasks often adopts a one-pass  process while decoding a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Though effi-  cient, the one-pass decoder cannot perceive future  context for further text deliberation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Multi-pass de-  coder extends the capability of generating more  refined text by exploring global information in the  sequence (Niehues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  For the text-to-text pretrained model, due to the  huge amount of parameters of the pretrained lan-  guage model, it is unwise to directly combine the  models in different passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' A common solution is  to concatenate the original serialized table content  and the text generated in the previous pass to fine-  tune the pretrained model in the next-pass decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  However, in this way, the length of input text prob-  ably exceeds the limit of the text-to-text pretrained  model, and the time complexity is too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  To effectively implement the fine-tuning of the  text-to-text pretrained model in multiple passes,  as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 3a and 3b, we take the table  representation as the “original text” and feed the  text generated in the first-pass fine-tuning plus  the table representation to the second-pass fine-  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Note that, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 3a, we sep-  arately fine-tune the table-to-text generation task  and the text-to-text deliberation task with two inde-  pendent TASATG models, and each of them takes  a text-to-text pretrained model as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  5  Experiments  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='1  Experimental Settings  Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We conducted experiments on the aforemen-  tioned datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', numericNLG and Totto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The  statistics of the numericNLG dataset can be found  in (Suadaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Besides, the size of the  original Totto dataset is 120K, which is much larger  than the numericNLG dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' To evaluate differ-  ent methods for table-to-text with comparable data  size, for the Totto dataset, we filtered out the tables  with fewer rows and columns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', #rows < 8 and  #columns < 8, such that the filtered Totto dataset  contains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='8K tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Then, we randomly selected  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='2K1 tables to generate the new Totto dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We calculated BLEU (from  gram-1 to gram-4) (Papineni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2002), ROUGE-  L (Lin, 2004) and METEOR (Denkowski and  Lavie, 2014) to evaluate the quality of the gen-  erated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The BLEU-n with a small value of n  measures the accuracy of the word level, and the  BLEU-n with a large n can measure the fluency  of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The ROUGE-L measures the re-  call rate based on the longest common sequence  between source and target texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The METEOR is  based on the harmonic mean of unigram precision  and recall, with recall weighted higher than preci-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' These metrics are widely used to measure the  accuracy and fluency of the generated sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We compare TASD with the following  baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Template-based Table Serialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  We  use the template designed for table serial-  ization as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Note that, the token  sequence generated by the template-based  method is denoted as TS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Pointer Generator (See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' This  is a seq2seq model with the attention and  copy mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We take TS as input for  the pointer generator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  TRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We implemented a simplified version  of the proposed TASD that omits the pos-  sessed knowledge in the pretrained language  model and removes text deliberation for focus-  ing on table representation modeling, namely  TRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In particular, TRM adopts the architec-  ture of GPT2 but initializes the parameters  randomly and trains 100 epochs at most for  fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Besides, TRM takes TS plus the  table structure representation as input and is  fed with TS in the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  1The size of numericNLG data is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='Candidate Fine-tuned TASATG 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='Candidate Fine-tuned TASATG 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='Candidate Fine-tuned TASATG 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='Candidate Fine-tuned TASATG 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='10  Fine-tuned GPT2 (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We  take the concatenation of TS and Y as the in-  put for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In the inference phase,  we only feed TS to the model to generate Y  starting after the last token of TS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  TableGPT (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' TableGPT is  a state-of-the-art table-to-text method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' To im-  prove the text fidelity and exploit the struc-  tural information at the same time, TableGPT  employs a multi-task learning paradigm con-  sisting of two auxiliary tasks, that is, one task  reconstructs the table structure from represen-  tations of GPT2, and the other aligns the tables  and the information in the generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The split settings for  training, validation and, testing were 1084:136:135  2 for the numericNLG dataset and 960:120:120  for the Totto dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Regarding auto-  matic evaluation, all results of deep models were  obtained by conducting experiments on a Linux  machine with Nvidia A100 GPU, and the averaged  results of 5 runs were reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Besides, an Adam  2This setting follows the experiments of (Suadaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  optimizer was utilized (with an initial learning rate  of 3e-5) for GPT2 fine-tuning, and the training was  iterated in 20 epochs at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' A beam search algo-  rithm was adopted when decoding a sequence and  the beam width was set to 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='2  Automatic Evaluation  The comparisons of automatic evaluation results  between TASD and other baselines can be found  in Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In general, TASD outperforms  the baselines for all the metrics on two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In  particular, compared to the reported best result of  all the baselines, TASD achieves improvements of  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='12 for BLEU-1 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='69 → 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='81), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='01 for BLEU-  2 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='02 → 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='03), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='24 for BLEU-3 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='68 → 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='92),  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='56 for METEOR (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='31 → 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='87), and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='92 for  ROUGE-L (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='48 → 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='40) on the numericNLG  dataset, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='85 for BLEU-1 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='34 → 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='19),  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='52 for BLEU-2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='65 → 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='17), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='38 for BLEU-3  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='33 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='71), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='02 for BLEU-4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='76 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='78),  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='55 for METEOR (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='10 → 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='65), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='50 for  ROUGE-L (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='46 → 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='96) on the Totto dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  3Our implementation is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  com/ramber1836/TASD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  6  Published as a conference paper at EMNLP 2022  In other words, for different types of source tables,  TASD generates better descriptive texts w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' accu-  racy at the word level, recall of the sequence, and  fluency of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Besides, we have the following observations: 1)  The template-based method performs much bet-  ter on the numericNLG dataset compared to the  Totto dataset, since the referenced table descrip-  tions in numericNLG were collected from scientific  papers, however, the table summaries in the Totto  dataset are more diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2) In the Totto dadaset,  the pointer generator model tends to cover more  words in descriptive text and generate more fluent  sentences than the template-based method, as the  contents in source tables of the Totto dataset are  mostly linguistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' This can also explain why the  pointer generator performs worse than the template-  based method on the numericNLG dataset w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  BLEU and METEOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 3) Fine-tuned GPT2 can  generate more faithful and fluent text than other  baselines (refer to Tables 1 and 2) most of the time,  which validates the effectiveness of the pretrained  language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 4) In general, TableGPT performs  better, and even the best, among all the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  In the numericNLG dataset, the headers of the  input tables (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' the attributes of records for  TableGPT) are more diverse, which may explain  why the performance of TableGPT is not promising  as expected on the numericNLG dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 5) TRM  can generate comparable, or even better descriptive  text as fined-tuned GPT2, which further suggests  the effectiveness of table structure understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='3  Ablation Analysis  Moreover, to verify the effectiveness of different  modules, we compare TASD with its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  After generating text with fine-tuned GPT2,  we fed the generated text concatenated with  TS to another fine-tuned GPT2 to realize the  second-pass decoder without table structure  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  We implemented TASD without deliberating  on the outcome text, which means that we  realized TASATG based on GPT2 in a one-  pass forward process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  TASD w/o 1st-TAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We removed table struc-  ture modeling in the first-pass decoding from  TASD, which was implemented by taking the  fine-tuned GPT2 as the first-pass decoder and  the table-structure-aware fine-tuned GPT2 as  the second-pass decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  As can be seen in Tables 1 and 2, TASD w/o TAS  performs worse than TASD under all metrics, since  the table structure modeling can benefit the fine-  tuning of GPT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' This can also be validated by com-  paring fine-tuned GPT2 to TASD w/o D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Besides,  the effectiveness of deliberating text can be proven  by comparing TASD w/o D to TASD (this can also  be validated by comparing fine-tuned GPT2 to  TASD w/o TAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' While text deliberation may harm  sentence fluency as depicted by the results of these  methods w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' BLEU-3 & 4 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In addition,  TASD w/o 1st-TAS outperforms TASD w/o TAS under  all metrics suggesting that taking the table repre-  sentation as the “original text” in the deliberation  mechanism is also effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='4  Qualitative Analysis  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 4(a) and (b) show two selected source ta-  bles and corresponding descriptive texts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', cap-  tion and section_text) in numericNLG and Totto  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 4(c) demonstrates the generated de-  scriptions by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The text that cor-  rectly reflects the facts of the source table is in  green, the erroneous text is in red, and the con-  fusing text is in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We can see that, there are  many grammatical errors in the text produced by  the pointer generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Fine-tuned GPT2 tends to  repeat phrases and sentences due to the limited  knowledge about the input table, which can also  explain why the fine-tuned GPT2 can obtain a false  high score in BLEU-n as n grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Thanks to the  semantic knowledge brought by pretraining, fine-  tuned GPT2 can generate more natural descriptions,  in which, however, perplexing factual errors ex-  ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Compared to fine-tuned GPT2, the description  generated by TASD is more relevant to the table  contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Since the target cells are not known in  advance, the generated text may miss the empha-  sized points described in the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The text  generated by TableGPT is also fluent, though coun-  terfactual descriptions may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='5  Human Evaluation  We randomly selected 30 samples from the test set  in numericNLG and Totto datasets, respectively,  and invited 10 volunteers to evaluate the quality of  the outcome text by considering three criteria, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',  grammar, coherence & concise, and factual per-  spective (correct and relevant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Each criterion has  scores of five degrees, ranging from 1 (the worst) to  5 (the best).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The averaged scores were reported in  Table 3, which show that TASD can generate more  7  Published as a conference paper at EMNLP 2022  Figure 4: Two examples of the generated table descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Table 3: Result of Human Evaluation  Dataset  Method  Grammar Coherence  & Concise  Factual per-  spective  numericNLG  Pointer Genera-  tor  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='99  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='73±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='69  Fine-tuned  GPT2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='56  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='58  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='45  TASD w/o D  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='61  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='45  TASD  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='72  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='64  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='73  Totto  Pointer Genera-  tor  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='82  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='55  Fine-tuned  GPT2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='60±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='64  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='46  TASD w/o D  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='52  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='46  TASD  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='66  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='69  readable and coherent texts, and describe more  correct facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Moreover, the pretrained models con-  sistently achieve better scores than the pointer gen-  erator on grammar and coherence because of the  expressive power learned from the large-scale cor-  pus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In the Totto dataset, the improvement of the  table structure modeling is smaller than that of the  polishing mechanism, which is consistent with the  automatic evaluation results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  6  Discussion  In our work, we devised a two-pass decoder frame-  work dedicated to the table-to-text task with the  help of the table-structure-aware text generation  model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', TASATG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' However, the effectiveness  of the text deliberation for the table-to-text task  should be further explored and integrated into the  table-structure-aware modeling in a more harmonic  Figure 5:  Table reconstruction for table-structure-  aware modeling enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' To discuss the limitation of the text de-  liberation of TASD, we additionally developed a  table content reconstruction loss and integrate it  into TASD in a multi-task learning fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Specifically, given the table-structure-aware em-  bedding E(2) generated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' (3), we randomly  mask certain cells of the input table and yield a  partially corrupted embedding of the input table,  denoted by �  E(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Then, a two-layer MLP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', multi-  layer perceptron) is adopted to restore the table-  structure-aware embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Afterward, an MSE  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', mean square error) loss is adopted to mea-  sure the effectiveness of table reconstruction and  further integrated into the TASD framework in the  multi-task learning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The process of table  reconstruction is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  We carried out a series of experiments to evalu-  ate the performance of TASD w/ and w/o the help  of table reconstruction loss (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', TRLoss) on nu-  mericNLG and Totto datasets in terms of BLEU-n  (1 to 4), METEOR, and ROUGE-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The results can  be found in Tables 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  8  our model also outperforms transformer  our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  compares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  transformer significantly outperforms transformer  the effectiveness of transformer  the comparison of our model with transformer on  caption  frenchenglish translation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  the size of our  model is slightly larger than the size of the transformer model  the evaluation metrics of our model and the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' our  model achieves the best results on all the metrics  suvarnabhumi is the busiest airport in thailand  international airport international airport  international airport international airport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  2018, airport is airport is airport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  busiest  after  suvarnabhumi airport  is the third-busiest airport in thailand after  suvarnabhumi airport  section_titleTASATG  GPT2 Fine-Tuning Loss  Table Reconstruction Loss  MLPPublished as a conference paper at EMNLP 2022  Table 4: The performances of TASD w/ and w/o the table reconstruction on the numericNLG dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Method  BLEU-1  BLEU-2  BLEU-3  BLEU-4  METEOR ROUGE-L  TASD w/o D  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='20  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='29  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='27  TASD w/o D w/ TRLoss  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='17  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='48  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='39  TASD w/ TRLoss  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='22  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='90  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='69  TASD w/ TRLoss in 1st pass  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='68  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='52  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='26  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='05  TASD  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='80  Table 5: The performances of TASD w/ and w/o the table reconstruction on the Totto dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Method  BLEU-1  BLEU-2  BLEU-3  BLEU-4  METEOR ROUGE-L  TASD w/o D  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='34  TASD w/o D w/ TRLoss  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='60  TASD w/ TRLoss  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='38  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='77  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='86  TASD w/ TRLoss in 1st pass  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='51  TASD w/ TRLoss in 2nd pass  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='58  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='20  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='66  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='79  TASD  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='19±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='08  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='21  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='71  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='96±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='10  According to the results reported on the nu-  mericNLG dataset, the TRLoss is helpful in en-  hancing the capability of table comprehension  though, the best performance is achieved by TASD  w/o D w/ TRLoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' It seems that the performance im-  provement gained by the table comprehension en-  hancement is sacrificed after the text deliberation  is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Meanwhile, on the Totto dataset, TASD  with the table reconstruction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', TASD w/ TRLoss)  does achieve the best performance in terms of  BLEU-1, BLEU-2, METEOR, and ROUGE-L,  though the improvement is not remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The  contents of the input tables are mainly linguistic  and the table structures are not too diverse might  be able to explain the performance improvement  of TASD w/ TRLoss on the Totto dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' With the  above comparisons, we can conclude that, for the  input tables with diverse structures, the limitation  of the current text deliberation mechanism cannot  be neglected if one aims to enhance the capability  of table comprehension for the table-to-text task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Moreover, this also suggests that the generalization  capability of text deliberation of TASD should be  improved in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In this work, long tables in the  Totto dataset are removed since the efficiency and  performance of TASD on large tables could be low-  ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In the future, the capability of handling long  tables for table-to-text models should be further ex-  plored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Besides, a large-scale and more exhaustive  human evaluation is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We plan to recruit  more volunteers to conduct the human annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  7  Conclusion  In this paper, to realize table-to-text with the pre-  trained language model, we proposed a table struc-  ture understanding and text deliberating approach,  namely TASD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The table structure understanding  was realized by developing a hierarchical multi-  head attention network, which can benefit the fine-  tuning of the text-to-text pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The  fully represented table information benefits not  only the pretrained language model but also the  text deliberation process since the structure infor-  mation with rich semantics could be fed into the  second-pass decoding naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' We carried out ex-  tensive experiments on two public datasets with  different table types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Automatic and human-based  evaluations, as well as qualitative analysis, vali-  date the effectiveness of our approach to generating  faithful and fluent table descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In the future,  we will improve text deliberation by devising a  unified framework to integrate the multi-pass de-  coder and refine the descriptive text paying more  attention to sentence fluency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Acknowledgements  This work is supported in part by Foshan  HKUST Projects (FSUST21-FYTRI01A, FSUST  21-FYTRI02A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Sticking to the facts: Con-  fident decoding for faithful data-to-text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Bayu Trisedya, Jianzhong Qi, and Rui Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Sentence generation for entity description with  content-plan attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In AAAI, volume 34, pages  9057–9064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  10  Published as a conference paper at EMNLP 2022  Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob  Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz  Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Attention is all  you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In NeurIPS, pages 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Peng Wang, Junyang Lin, An Yang, Chang Zhou,  Yichang Zhang, Jingren Zhou, and Hongxia Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Sketch and refine: Towards faithful and in-  formative table-to-text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In ACL-IJCNLP,  pages 4831–4843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Zhenyi Wang, Xiaoyang Wang, Bang An, Dong Yu,  and Changyou Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Towards faithful neural  table-to-text generation with content-matching con-  straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In ACL, pages 1072–1086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Sam Wiseman, Stuart M Shieber, and Alexander M  Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Challenges in data-to-document gen-  eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In EMNLP, pages 2253–2263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Yingce Xia, Fei Tian, Lijun Wu, Jianxin Lin, Tao Qin,  Nenghai Yu, and Tie-Yan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Deliberation  networks: Sequence generation beyond one-pass de-  coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' NeurIPS, 30:1784–1794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Xinyu Xing and Xiaojun Wan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Structure-aware  pre-training for table-to-text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  In ACL-  IJCNLP, pages 2273–2278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Pengcheng Yin, Graham Neubig, Wen-tau Yih, and Se-  bastian Riedel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Tabert: Pretraining for joint  understanding of textual and tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In ACL,  pages 8413–8426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Hongzhi Zhang, Yingyao Wang, Sirui Wang, Xuezhi  Cao, Fuzheng Zhang, and Zhongyuan Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Table fact verification with structure-aware trans-  former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In EMNLP, pages 1624–1629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  A  Human Evaluation Settings  The criteria adopted in our human-based evaluation  are (1) Grammar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', is this paragraph grammat-  ical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' ), (2) Coherence & Concise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', is this para-  graph coherent and contextually consistent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' does  it repeat redundant information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' ), and (3) Factual  perspective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=', are the facts that this paragraph  describes correct?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' are these facts related to refer-  ences and tables?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' More specifically, we list the  detailed justifications on how to score the generated  text in each criterion as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Grammar  1 It is more like garbled code than a paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  2 There are many obvious grammatical mis-  takes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  3 There are a few obvious grammatical mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  4 There are few grammatical mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  5 There are no grammatical mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Coherence & Concise  1 The logic of text expression is chaotic and  nonsense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  2 There are a lot of logical inconsistencies or  redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  3 There are some logical inconsistencies or re-  dundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  4 There are a few logical inconsistencies or  redundant information, but it does not affect  browsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  5 The logic of the text is smooth without redun-  dant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Factual Perspective  1 The paragraph does not coincide with the ref-  erence or table, and it is full of information  inconsistent with the facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  2 The paragraph describes the facts incorrectly  and has a low correlation with reference, but is  related to the information in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  3 The paragraph description is incorrect, but it  is highly coincident with the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  4 The paragraph description is basically correct,  and the coincidence with the reference is low,  but it also describes the information in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  5 The paragraph description is correct and  highly coincident with the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  B  Illustrative Examples of Generated  Descriptions  We additionally selected another two examples of  the generated table descriptions from the numeric-  NLG and Totto datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' The results  are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' From these four ex-  amples, we can see that TASD can generate more  accurate and fluent descriptive texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' While incor-  rect descriptions can be found in the outcome texts  generated by different models for cases D and F,  which suggests that generating faithful descriptions  for open-domain tables is much more challenging  and requires more powerful and, thus larger, pre-  trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  C  Extra Implementation Details  The learning rate of GPT2 was searched from {3e−  4, 3e − 5, 3e − 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' In the evaluation of discussing  the limitation of text deliberation (see Section 6),  a trade-off parameter for balancing the GPT2 fine-  tuning loss and the TRLoss was adopted, then the  trade-off parameter was searched from {1e−1, 5e−  2, 1e − 2, 5e − 3, 1e − 3}, and 1e-2 was selected for  the reported performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' Besides, the reported  results in Tables 4 and 5 were averaged in 3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  11  Published as a conference paper at EMNLP 2022  Figure 6: Generated table descriptions on cases C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  Figure 7: Generated table descriptions on cases E and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  12  robusttc-fsl,  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='（85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='8 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' 2018)  matching and prototypical networks  relation networks  the graph networks  arsc mean acc dataset  snorkel network providing the best performance  our model outperforms the other baselines on mean acc  it leads to a significant improvement in mean accuracy for the compared models  on the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' the absence of semantic representation (a) leads to a significant  caption  improvement in mean accuracy  our proposed model achieves competitive accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' comparable to  the state-of-the-art models from yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  steve gregory  steve gregory  seven interceptions seven  13 games  35 tackles  sacks  tour touchdowns  steve gregory  357 tackles seven interceptions three sacks  forced fumble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='0 yards per game  steve gregory ended his career with 357 tackles, seven  page_title  interceptions, three sacks and 1 forced fumble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='0 yards per game .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' (repeat)  section title  steve gregory  357 tackles seven sacks  seven interceptionsconsistency score  the w/o results  consistency and novelty scores  w/o adversarial learning results in a full model  consistency score of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='84 and a novelty score of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='24  baseline  adversarial adaptation  results in a much better consistency than the w/o adaptation  caption  combined with the w/o Istm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' the coherence improves significantly  consistency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' novelty,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content=' diversity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
+page_content='  and coherence  same consistency  dynamic strategy leads to a higher novelty  110 years  the district index performed last 57 district index ohio 67 district ( death ) until  district used holder index ( years district)  alice sanders (12 may 1897 - 7 november 2007)  survivors  alice sanders (12 mav 1897 - 7 november 2007)  alice sargent  over 40 years  page_title  breast  and breast breast  section title' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf'}
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+Task-sequencing Simulator: Integrated Machine Learning to Execution
+Simulation for Robot Manipulation
+Kazuhiro Sasabuchi1, Daichi Saito1, Atsushi Kanehira1, Naoki Wake1, Jun Takamatsu1, Katsushi Ikeuchi1
+Abstract— A task-sequencing simulator in robotics manip-
+ulation to integrate simulation-for-learning and simulation-
+for-execution is introduced. Unlike existing machine-learning
+simulation where a non-decomposed simulation is used to
+simulate a training scenario, the task-sequencing simulator runs
+a composed simulation using building blocks. This way, the
+simulation-for-learning is structured similarly to a multi-step
+simulation-for-execution. To compose both learning and execu-
+tion scenarios, a unified trainable-and-composable description
+of blocks called a concept model is proposed and used. Using the
+simulator design and concept models, a reusable simulator for
+learning different tasks, a common-ground system for learning-
+to-execution, simulation-to-real is achieved and shown.
+I. INTRODUCTION
+Simulators are important in robotics. Compared to the
+real world, simulators can run endlessly, safely, remove
+stochasticity, and provide ground truth data. One direction
+for using simulators in robotics is to help check collisions
+and safe executions before real robot executions. Another
+direction is using simulators as a tool for machine learning.
+Simulators that suit either one of the above directions are
+available today, however, it is desirable to have a simulator
+which fulfills both purposes to better integrate learning and
+execution, especially in the multi-step manipulation domain.
+In multi-step manipulation, a series of tasks which occur
+sequentially must be simulated. An example of a sequential
+series of tasks is bringing an object, which is composed
+of tasks: grasp, pick, bring. An execution simulator must
+be able to trigger these different tasks and connect them
+into a sequenced simulation. When the simulator is used
+for checking robot executions, the simulation is a matter of
+combining programmed or trained building blocks (e.g., run
+“grasp” then “pick” then “bring”).
+In contrast, machine learning simulators often ignore the
+task-sequence composition and are structured to train a
+specific problem or benchmark (e.g., a non-decomposed
+“pick-and-place” simulation)[1][2]. This structural differ-
+ence causes a gap between simulation-for-execution and
+simulation-for-learning. The learned results become specific
+to the trained scenario and contradicts with the simulation-
+for-execution where scenarios are non-fixed.
+Instead, a machine learning simulator could be designed
+similar to an execution simulator. The “pick-and-place”
+scenario can be decomposed into a sequenced simulation
+of “grasp then pick then bring then place then release.”
+The difference compared to the execution simulator is that
+1All
+authors
+are
+with
+Microsoft,
+Redmond,
+WA,
+USA
+Kazuhiro.Sasabuchi@microsoft.com
+Physics
+Kinematics
+Rendering
+Post-process
+Task-sequencing simulator
+Concept Interface
+Environment Engine Pipeline
+CM Grasp
+state
+action
+selected task 
+at step t
+demonstrated 
+execution sequence
+configured 
+training sequence
+CM Open
+CM Release
+training 
+algorithms
+Fig. 1.
+The proposed task-sequencing simulator which enables scenario
+composition for both learning and execution in robotics manipulation.
+some of these blocks are “under-training” and are updated
+as data is collected. Once the update has finished, the trained
+block can be combined and reused for a different scenario,
+thus, the simulation-for-learning can directly transition to the
+simulation-for-execution. In addition, such design enables
+learning new manipulation skills on top of programmed or
+prior-trained building blocks. For example, a “grasp” could
+be trained using the sequence “grasp then pick,” where
+“pick” is a programmed task to provide a supervised signal
+(i.e., teach that the “grasp” was successful if the “pick” was
+successful).
+This article introduces a task-sequencing simulator struc-
+ture which enables integrated learning-to-execution simula-
+tion. At its core, the simulator uses a unified block design
+called the “concept model,” which is proposed within this
+article and defines the necessary descriptions for training
+a task, collecting trained tasks, and running the tasks to
+compose a sequence.
+The rest of the article is outlined as below: Section II
+provides a background on existing robotic simulators. Sec-
+tion III explains the overall simulator structure for achieving
+machine learning to robot execution simulation. Section IV
+explains the concept model core component of the simulator
+and Section V provides some detailed example implementa-
+tions of the model. Section VI shows the capabilities of the
+arXiv:2301.01382v1  [cs.RO]  3 Jan 2023
+
+C16simulator in machine-learning-to-robot-execution followed
+by conclusions in Section VII.
+II. BACKGROUND
+While there are many existing simulators for robotics,
+existing simulators may not achieve the integrated learning-
+to-execution multi-step manipulation purpose for one of
+the following reasons: 1) the simulator targets a different
+domain other than manipulation, 2) the simulator can be
+used for manipulation but misses a capability in simulation-
+for-learning, 3) the simulator can be used for manipulation
+but misses a capability in simulation-for-execution, 4) the
+simulator can be used for manipulation and both learning-
+and-execution purposes but not specifically for learning-to-
+execution purposes.
+Popular
+robotics
+simulators[3][4]
+include
+Gazebo[5],
+MuJoCo[6], CoppeliaSim[7], CARLA[8], AirSim[9], and
+Webots[10]. Gazebo has its advantage in its capability to
+simulate executions using ROS integrated sensors and ac-
+tuators but is not the best choice when it comes to data
+collection and machine learning due to its slow simulation
+performance and inconsistency in physics simulation. Thus,
+Gazebo falls into the second category. Engines like MuJoCo
+on the other hand, are suitable for stable physics simulation
+in machine learning but miss some robotics simulation
+capabilities such as inverse-kinematics and visual feedback
+(realistic rendering). The focus is on physics simulation
+rather than an integrated simulator for robot executions,
+therefore falls into the third category. CARLA and Airsim
+mainly target automobiles such as drones and cars therefore
+miss some important features such as kinematics required for
+manipulation and falls into the first category.
+The CoppeliaSim is an integrated simulator with a kine-
+matics and physics engine, and the PyRep toolkit[11] can
+be used with the simulator for machine learning. WeBots
+is also an integrated simulator and frameworks such as
+Deepbots[12] help the simulator to be used for machine
+learning. The machine learning features of these simulators
+are external features that have been developed within the
+community. While it is possible to use these simulators
+for both execution and learning purposes, they have not
+been designed for integrated learning-to-execution but rather
+using for one-or-the-other purpose. That is, these simulators
+are not designed to connect learning-and-execution, rather,
+learning and execution are separate use cases where one uses
+a community provided wrapper for machine learning, and the
+other uses the integrated features to simulate a robotic system
+execution.
+Compared to the existing simulators, the task-sequencing
+simulator was designed to connect simulation-for-learning
+and simulation-for-execution The simulator uses a con-
+cept model which enables composition of pre-trained, pro-
+grammed, or trained tasks, which is a powerful feature for
+going from machine learning to real robot execution. (e.g.,
+such as plugging-in to machine learning platforms but then
+connecting to execute on ROS). More importantly, tied-
+integration allows features such as training using pre-sequent
+and post-sequent task executions, but also collecting reusable
+execution modules through training.
+III. TASK-SEQUENCING SIMULATOR OVERVIEW
+The task-sequencing simulator has two layers: the Concept
+Interface for “action decision” and the Environment Engine
+Pipeline for “state observing” (Figure 1). However, unlike
+a typical learning simulator, where a specific problem has
+a non-decomposable structure and the action decision is
+a single policy being updated as data is collected for the
+problem, the task-sequencing simulator adds an abstraction
+to this action decision so that the problem is composed of
+a sequence of tasks (i.e., switches between a collection of
+tasks, where each task runs an individual policy), This way,
+a learned task policy can become part of a collection of
+policies for execution once the training has finished. Further
+details of each layer are described below.
+A. concept interface
+At each simulation time-step, a robot decides the next
+action depending on the current state of the world. This
+decision is referred to as a policy. When the relation be-
+tween the state, action, and next state (system dynamics) is
+completely known, this policy can be directly programmed.
+When the system dynamics are unknown, either the learning
+of the policy or system dynamics is required through data
+collection. Data collection is efficient if collected only for
+the unknown dynamics and if known dynamics are directly
+computed. Therefore, it is preferable to break down a robot’s
+execution to a series of tasks, where each task executes its
+own policy optimal for the system dynamics the task is cover-
+ing. In addition, breaking down a robot’s execution increases
+the reusability of each task policy and allows composing
+different execution scenarios from the task building blocks.
+The Concept Interface layer chooses and switches between
+the tasks for a training or execution scenario assuming
+(1) the series of tasks to simulate is known (ways for
+knowing are shown in the experiments), and (2) each task
+policy can indicate when the task has been completed (i.e.,
+has a learned or programmed completion signal). During
+simulation-for-execution, the Concept Interface layer chooses
+the according task in a sequence and switches to the next
+task once the current policy returns a task completion signal.
+The simulation-for-learning can be conducted in a similar
+way, except, the policy’s output of the completion signal is
+evaluated (e.g., by the success of a subsequent task). This
+interchangeable structure enables an integrated learning-to-
+execution simulation. To learn the termination signal under
+an arbitrary training sequence, the tasks share a unified
+design called a “concept model,” which is further explained
+in the later sections.
+B. environment engine pipeline
+At a time-step control scale, a robot behaves under a
+sense-plan-act with a physical embodiment. Thus, simulation
+in robotics requires three important engines: kinematics,
+physics (contact dynamics), and rendering. The role of
+
+the kinematics engine is to do the simulation between the
+robot’s action plan and the movement of the actual body.
+The role of the physics engine is to do the simulation
+between the robot’s body and the environment. The role of
+the rendering engine is to do the simulation between the
+environment and the robot’s sensing. In more technical terms,
+the kinematics engine solves the mapping between cartesian
+space and configuration space, the physics engine solves the
+differential algebraic equation[13] using techniques such as
+velocity-impulse linear complementarity-based time-stepping
+methods[14][15], the rendering engine solves the rendering
+equation[16] about lighting paths for pixel color generation.
+While rendering engines simulate the robot’s sensing by
+generating images, there is a gap between sensing and
+perceiving (extracting meaningful states from the gener-
+ated images). Rather than directly learning the sensing-to-
+planning, sometimes it is more efficient to perceive-before-
+planning and extract visual features. Moreover, in robotics
+it is important to combine both visual and force feedback;
+the visual features help compress the feedback so that the
+vision and force have an aligned state dimension. Thus, the
+proposed simulator adds a fourth “post-process engine” in
+conjunction with the rendering engine.
+These different-role engines are triggered in an ordered
+pipeline to calculate the current state of the simulation world.
+Often, simulators package specific engines to produce a
+single simulation world, but in general, these engines could
+run separately and combine/orchestrate multiple simulation
+worlds to produce better simulation. For example, ROS
+MoveIt could be used for accurate inverse kinematics sim-
+ulation, PyBullet for reproducible physics simulation, and
+the Unreal Engine for photo-realistic ray-traced rendering.
+Combining different engines is possible as long as each
+engine is able to load the same models of the robot/objects
+and is able to share the robot and object states among each
+engine (which can be done using TCP connections etc.).
+In addition, instead of using simulation engines, it is also
+possible to connect “real” engines which replace simulated
+physics with the real robot’s torque sensors and simulated
+rendering with real images from the robot’s camera.
+IV. CONCEPT MODELS
+A single task block operates under some system dynamics
+and achieves a goal state from an initial state. Thus, the de-
+tails of a task can be described using the actors of the system,
+an initial state, a goal (end) state, and the parameters of the
+system dynamics (Figure 2). This kind of task description
+can be referred to as a “task model”[17]. This description
+is enough for executing a task if the system dynamics are
+completely known. The initial state is usually the end state
+of the previous task. However, when the dynamics are not
+fully known, learning is required, and during learning, the
+initial state must be randomized.
+Instead of fully describing the task, a task can be described
+using actor configurations, an initial state, a necessary goal
+state that is described from observable system states, a
+sufficient goal state that is described from non-observable
+Task Model
+initial state
+Action
+Programmed policy
+Actors
+obtained before execution
+Parameters
+completely known and obtained 
+before execution
+defined goal state
+Fig. 2.
+An illustration of a task model.
+Concept Model
+initial state
+Action
+Policy
+Actors
+a set of configurated settings
+Parameters
+partially known and randomized at 
+training, estimated at execution
+necessary goal state
+sufficient goal state
+Fig. 3.
+An illustration of a concept model.
+system states, and the partially known parameters of the
+system dynamics (Figure 3). This kind of task description
+will be referred to as a “concept model” which compared to
+the task model may not be concrete enough for execution, but
+describes the concepts of the task to learn the execution. In
+the special case where the system dynamics are fully known
+and the task is programmable, the descriptions of the concept
+model is identical to the task model.
+By providing a structured description format, the Con-
+cept Interface layer can access these blocks interchangeably
+within a training sequence as the structures are the same and
+only differ in the details.
+A. concept model usage in learning
+The concept model descriptions are used to learn the
+task completion signal as well as the actions to handle the
+unknown system dynamics. The necessary goal state and
+sufficient goal state descriptions are used to evaluate whether
+the task completion signal and actions are appropriate. The
+evaluation is done by minimizing the cost of the current
+states to the goal states. Note that the cost to the necessary
+goal state is evaluated after every action decision, whereas
+the cost to the sufficient goal state is only evaluated once a
+task completion signal is chosen during training.
+The actor configurations describe the possible environ-
+ments (the world including the robot and any manipulating
+target) for training the task. If there are no pre-sequent tasks
+involved for training, then one of the actor configurations is
+used to define the initial state of the tasks. Otherwise, the
+end state of the previous task is the initial state. Unlike the
+initial state, the actor configuration is independent from the
+states of the previous task, thus is configurable and can be
+used for randomizing the states for training.
+
+B. concept model usage in execution
+When the dynamics are fully known, the concept model
+acts the same as the task model. The state of the task changes
+from the initial state using actions from a programmed policy
+based on the system dynamics and actors. The initial state
+of the task during execution is the end state of the previous
+task and the task ends once the goal state is achieved.
+When the dynamics are not fully known, the observable
+system states of the task changes using a learned policy.
+Similar to the programmed case, the initial state is the
+observable states at the end of the previous task. However,
+since part of the goal state is non-observable, the end of the
+task cannot be identified just with the model descriptions.
+Instead, the learned task completion signal from the concept
+model descriptions is used to identify the end of the task.
+V. MODEL IMPLEMENTATIONS
+By following the concept model structure, a task is im-
+plemented in a way that can be trained but then collected
+as a building block for execution. In this article, the screw
+theory[18] based separation of dynamics[17] is used to
+separate a task from another task. That is, once the relation
+between the manipulation target and the robot’s end-effector
+is initialized, a task breaks or maintains a contact state
+between the target and the environment. By classifying the
+inequality equation patterns of contact points, this leads to
+seven pure translation tasks and seven pure rotation tasks.
+Figure 4 shows implementation examples of some of these
+tasks as concept models. Below describes the details of some
+of the examples in the figure. Note that as the task classifica-
+tion only depends on the relation between the end-effector,
+manipulation target, and the environment, the movement of
+the arm (configuration space) can be ignored[19] and the task
+only focuses on the movement of the end-effector (cartesian
+space).
+A. grasping
+The
+grasp
+task
+initiates
+the
+relation
+between
+the
+manipulation-target and the robot’s end-effector. The actors
+are a target object, an environment (e.g., table), and the
+end-effector. The initial state is where the target object is
+attached to the environment but not attached to the end-
+effector (including shape of the finger joints). The goal state
+is where the target object is also attached to the end-effector
+in a way such that enough force is exerted for performing a
+subsequent task. The parameters are the distance between the
+target and end-effector as well as the approaching direction
+of the end-effector.
+When details of the target object are completely known, a
+task model can be defined and programmed from the above
+details. However, in the real world, there is uncertainty in the
+shape of the object and distance to the target, distinguishing
+whether enough force is exerted is intractable due to the
+inaccuracy in the real contact sensors or lack of sensors to
+detect slipping, a grasp-failure due to finger-object collision
+during approach may occur if the policy is not carefully
+designed under the uncertainties of the object properties.
+Grasp Concept Model
+freed OBJ-EEF
+Action
+Policy
+Actors
+end-effector (EEF), environment (ENV), a set 
+of objects (OBJ) with randomized shapes
+Parameters
+estimated approach distance OBJ-EEF,
+approach direction of EEF
+NGS: attached OBJ-EEF
+SGS: next task success
+Open Concept Model
+initial OBJ pose
+Action
+Policy
+Actors
+end-effector (EEF), environment (ENV), a set 
+of objects (OBJ) with randomized parameters
+Parameters
+estimated radius, axis center, axis direction 
+to maintain constraint OBJ-ENV
+NGS: maintain OBJ-ENV
+SGS: desired OBJ pose
+Pick Concept Model
+attached OBJ-ENV
+Action
+Programmed
+Actors
+end-effector (EEF), environment (ENV), target
+object (OBJ)
+Parameters
+detach direction to break constraint OBJ-ENV
+freed OBJ, ENV
+Bring Concept Model
+initial OBJ pose
+Action
+Programmed
+Actors
+end-effector (EEF), target object (OBJ)
+Parameters
+move direction and distance of OBJ
+desired OBJ pose
+Place Concept Model
+freed OBJ, ENV
+Action
+Programmed
+Actors
+end-effector (EEF), environment (ENV), target
+object (OBJ)
+Parameters
+attach direction to create constraint OBJ-ENV
+attached OBJ-ENV
+Release Concept Model
+attached OBJ-EEF
+Action
+Programmed
+Actors
+end-effector (EEF), target object (OBJ)
+Parameters
+release distance OBJ-EEF,
+release direction of EEF
+freed OBJ, EEF
+Pour Concept Model
+initial ENV state
+Action
+Policy
+Actors
+end-effector (EEF), set of environment (ENV) 
+and object (OBJs) with randomized parameters
+Parameters
+estimated ENV (cup) filled state,
+estimated OBJ (pitcher) size,
+estimated axis center, direction of OBJ-ENV
+NGS: maintain OBJ-ENV
+SGS: desired ENV state
+Wipe Concept Model
+initial ENV state
+Action
+Policy
+Actors
+end-effector (EEF), set of environment (ENV) 
+and object (OBJs) with randomized parameters
+Parameters
+estimated ENV (plane) clean state,
+estimated normal axis of OBJ-ENV
+NGS: maintain OBJ-ENV
+SGS: desired ENV state
+Fig. 4. Example concept models of eight different tasks in the screw-theory
+based classification.
+Instead, a set of actor configurations is defined as object
+shapes from a randomized range, a necessary goal state is
+defined as the end-effector to be in contact with the target
+object on an appropriate surface (which can be obtained on
+the real robot with the finger configurations and finger-torque
+sensors with a threshold to determine a binary contacted-or-
+not-contacted state), a sufficient goal state is defined as a
+successful performance of a subsequent task, the estimated
+distance is used for the parameters. The defined goal states
+are used to formulate the reward (cost-to-go) for learning the
+policy.
+Using this concept model, the approaching strategy (ad-
+justed movement around the approaching direction of the
+end-effector) and the enough amount of “closing” of the
+fingers to perform a subsequent task is learned. The learned
+policy chooses the sufficient amount based on the object
+shape which can partially be inferred by the shape of the end-
+effector finger joints once touching the object. The policy
+returns a termination signal once reached the enough amount
+of closing.
+B. door-opening
+The door-opening task is a one degrees-of-freedom pure
+rotation task. The actors are a target object (the door), an
+environment (the hinge), and the end-effector (attached to the
+
+door handle). The initial state is where the end-effector and
+target object are at an attached state. The goal state is where
+the target object has moved to some desired orientation. The
+parameters are the rotation radius, the rotation axis center,
+and the rotation axis direction defined by the target and
+environment.
+When details of the target and environment are completely
+known, a task model can be defined and programmed from
+the above details. However, in the real world, there is
+uncertainty in the environment parameters. Instead, a set of
+actor configurations randomizing the radius, a necessary goal
+state that moves the object along the environment constraint
+at each time-step (which can be obtained on the real robot
+by using a force sensor on the wrist and checking against
+a maximum-stress threshold), a sufficient goal state that
+ensures the door has reached the desired orientation, and
+estimation of the parameters are used to describe the concept
+model.
+By using an end-effector with only force-sensor feedback
+on the wrist, this model enables learning a policy which
+updates parameter estimations at each time-step, and then
+generates a hand motion based on the updated parameters.
+The policy returns a termination signal once inferred that the
+desired orientation has been reached.
+C. bringing
+The bringing task is a six degrees-of-freedom translation
+and rotation task. The actors are the target object and the
+end-effector. The initial state is where the end-effector and
+target object are at an attached state. The goal state is where
+the target object has moved to some relative positioning. The
+parameters are the moving direction and distance.
+Since there are no uncertainties in the target or environ-
+ment, the parameters can be manually specified and the goal
+can be directly specified from the parameters, Thus, the
+concept model is identical to the task model and can be
+programmed.
+VI. EXPERIMENTS
+Experiments were conducted using the concept model
+implementations shown in the previous section, and the
+developed task-sequencing simulator.
+For the learning experiments, the series of tasks to simulate
+were predefined. These experiments were performed to show
+the effectiveness of the simulator and its reusability for
+training different tasks.
+By running the pre-sequent tasks of the task-to-train at
+the start of an episode, and by running the subsequent tasks
+at reward return, the simulator is compatible with common
+reinforcement learning platforms, and utilizes off-the-shelf
+learning algorithms. For the experiments, the simulator was
+connected to the Bonsai platform and used the PPO algo-
+rithm.
+For the execution experiments, the series of tasks were
+obtained from human demonstrations and the actions were
+generated using the learned task policies. These experiments
+were performed to show the effectiveness of the simulator for
+execution simulations (execution of different composed sce-
+narios). Note that the exact same simulator and task blocks
+used for training was used for this experiment, showing
+the simulator’s capability to transition from simulation-for-
+learning to simulation-for-execution.
+For simulation, states were obtained by plugging to the
+environment engine pipeline the PyBullet engine for the
+physics engine role, and the Unreal Engine for rendering. For
+the real robot, states were obtained plugging ROS (connected
+via roslibpy) to the environment engine pipeline. For the arm
+kinematics, the ROS MoveIt package was used.
+A. training
+Figure 5-(A) shows the task-sequencing simulator config-
+urations for running the grasp training. A configured training
+sequence “grasp then pick” is passed to the simulator. The
+“grasp” task is the task-to-train, and a programmed “pick”
+is used as the subsequent task for evaluating the sufficient
+goal state.
+Figure 5-(B) shows the trained grasp results performed
+on a real robot. The concept model was designed with
+the object shape parameters as unknown. Regardless of
+such uncertainty, the learned policy successfully grasps the
+different shaped objects including but not limited to a box,
+a cylindrical cup, an oval rice pack, and a diamond-shaped
+candy box.
+Figure 5-(C) shows a learned grasp for a different robot
+hand, which was trained using the same simulator and
+concept models but with a different actor configuration
+(end-effector) setting. The results show the reusability of
+the simulator for training different robots with different
+mechanics (a hand with multiple fingers and a gripper with
+limited degrees of freedom).
+Figure 5-(D) shows that by changing the configured train-
+ing sequence to “grasp then open,” the door-opening task is
+trained using the same simulator. The “open” task is the task-
+to-train, and the trained “grasp” is reused as the pre-sequent
+task for initiating the relation between the end-effector and
+the target door. Regardless of uncertainty in the rotation
+radius, center, and axis direction, the real robot performed the
+door-opening using the learned policy. Although the policy
+was trained only using simulated data, the policy is directly
+applicable to the real robot as the sufficient goal state does
+not require observability on the real robot and because the
+policy action decisions only rely on the states with very small
+sim-to-real gaps.
+B. execution
+Figure 6-(A) shows the task-sequencing simulator used
+with a demonstrated sequence by a human. Instead of a
+configured sequence as in the previous training experiments,
+the sequence is automatically generated through demonstra-
+tion decomposition using the method described in [20]. The
+same concept models from the training experiments are used
+with the policy-update being disabled (the simulator is not
+connected to any training algorithm and instead uses a fixed
+learned policy without updates).
+
+Task-sequencing simulator
+Concept Interface
+Environment Engine Pipeline
+CM Grasp
+state
+action
+configured 
+training sequence
+CM Pick
+training 
+algorithms
+(A)
+(B)
+(C)
+connect learned policy 
+to the real robot
+train policy on a 
+different robot hand
+CM Grasp
+CM Open
+training sequence 
+for door-opening
+(D)
+configure to a different 
+training scenario
+Fig. 5.
+Results of the task-sequencing simulator when used for learning.
+Task-sequencing simulator
+Concept Interface
+Environment Engine Pipeline
+CM Grasp
+state
+action
+demonstrated execution sequence
+CM Pick
+(A)
+simulation
+CM Bring
+CM Bring
+CM Place
+CM Release
+real
+another 
+scenario
+physics
+rendering
+(B)
+(C)
+(D)
+Fig. 6.
+Results of the task-sequencing simulator when used for execution.
+Figure 6-(B) shows a simulated execution of the demon-
+strated sequence. The first row shows the outputs of the
+physics engine and the second row shows the outputs of
+the rendering engine. As both training and execution run
+on the same system, the learned policy can easily be used
+as a simulation-for-execution. The learned policy is already
+a building block that can be combined with other tasks to
+generate an application such as “pick up a cup from the upper
+shelf and re-place it to the bottom shelf.”
+Figure 6-(C) shows a real robot execution of the demon-
+strated sequence by switching the engines in the environment
+engine pipeline to connect with ROS. This shows how the
+simulator can go from the simulated robot execution to the
+real robot execution by using the same policy connections
+but by changing the engines in which the states are obtained,
+and the actions are performed against. Usually, going from
+simulation to real introduces a sim-to-real gap. However,
+only part of the scenario sequence uses a learned policy and
+due to the careful design of the concept models to divide
+learning observable dynamics (necessary goal states) from
+learning hidden dynamics (sufficient goal states), no such
+gap was encountered.
+Figure 6-(D) shows an execution of a different sequence
+“pick up a cup from the table and throw it in the trash.”
+This scenario uses the same concept models and only differs
+in the demonstrated input, showing how using the simulator
+and concept model descriptions enable reusing the learned
+policies for different execution scenarios. If a policy was
+learned against a full “pick-and-place” scenario, the policy
+would not easily scale to the “pick-and-throw” scenario as
+the problem dynamics are different.
+
+VII. CONCLUSIONS
+This article introduced the task-sequencing simulator
+which bridges simulation-for-learning to simulation-for-
+execution. The simulation scenario for learning is created us-
+ing a sequence of tasks. This way the simulation-for-learning
+has the same structure as the simulation-for-execution. At
+its core, the simulator uses a concept model which enables
+sequencing mixed programmed, trained, and under-training
+building blocks. While the simulator has a large advantage
+in terms of integrated system development, the simulator
+also provides new directions for simulation in execution and
+simulation in learning.
+From an execution perspective, the simulator allows com-
+posing a task-sequence using both programmed and trained
+tasks. Unlike programmed-only sequences, the advantage of
+mixing trained blocks is that, some of the tasks can contain
+uncertainty and the goal state of a task can be described using
+implicit system parameters (the goal state does not have to
+be obtained directly from the real robot). The key is that,
+whether the observed state and selected actions suffice the
+goal state is learned as a termination signal through training.
+From a learning perspective, the simulator and concept
+model design have the following advantages: First, the sim-
+ulation is reusable and easily applicable to slight changes
+in the scenario. A policy for a different end-effector can
+be learned by just changing the actor configurations in the
+concept model. A policy can be optimized for different
+scenarios by just changing the subsequent task in the suffi-
+cient goal state. Second, defining the learning problem using
+the concept model design enables a hierarchical learning-
+structure as well as a structure for reducing sim-to-real gaps.
+Any state parameters that do not have a large gap when
+observed with the real robot is used for defining the necessary
+goal state, whereas any state parameters that have a large gap
+when observed with the real robot is a sufficient goal state
+(implicitly learned in simulation but no need to be observed
+with the real robot). This type of formulation is possible
+as only the parts with uncertainty are being learned instead
+of learning the entire scenario sequence. Following this
+structured formulation has allowed going from simulation
+to real without any extra real-world data collection and
+achieving a reusable policy applicable to different execution
+scenarios.
+ACKNOWLEDGMENT
+The authors thank Brice Chung’s team, Aydan Aksoylar
+and Kartavya Neema for their help in the reward designs and
+training of the concept models used in the experiments.
+REFERENCES
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf,len=349
+page_content='Task-sequencing Simulator: Integrated Machine Learning to Execution  Simulation for Robot Manipulation  Kazuhiro Sasabuchi1, Daichi Saito1, Atsushi Kanehira1, Naoki Wake1, Jun Takamatsu1, Katsushi Ikeuchi1  Abstract— A task-sequencing simulator in robotics manip-  ulation to integrate simulation-for-learning and simulation-  for-execution is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Unlike existing machine-learning  simulation where a non-decomposed simulation is used to  simulate a training scenario, the task-sequencing simulator runs  a composed simulation using building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This way, the  simulation-for-learning is structured similarly to a multi-step  simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' To compose both learning and execu-  tion scenarios, a unified trainable-and-composable description  of blocks called a concept model is proposed and used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Using the  simulator design and concept models, a reusable simulator for  learning different tasks, a common-ground system for learning-  to-execution, simulation-to-real is achieved and shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' INTRODUCTION  Simulators are important in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Compared to the  real world, simulators can run endlessly, safely, remove  stochasticity, and provide ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' One direction  for using simulators in robotics is to help check collisions  and safe executions before real robot executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Another  direction is using simulators as a tool for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Simulators that suit either one of the above directions are  available today, however, it is desirable to have a simulator  which fulfills both purposes to better integrate learning and  execution, especially in the multi-step manipulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  In multi-step manipulation, a series of tasks which occur  sequentially must be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' An example of a sequential  series of tasks is bringing an object, which is composed  of tasks: grasp, pick, bring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' An execution simulator must  be able to trigger these different tasks and connect them  into a sequenced simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' When the simulator is used  for checking robot executions, the simulation is a matter of  combining programmed or trained building blocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', run  “grasp” then “pick” then “bring”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  In contrast, machine learning simulators often ignore the  task-sequence composition and are structured to train a  specific problem or benchmark (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', a non-decomposed  “pick-and-place” simulation)[1][2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This structural differ-  ence causes a gap between simulation-for-execution and  simulation-for-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The learned results become specific  to the trained scenario and contradicts with the simulation-  for-execution where scenarios are non-fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Instead, a machine learning simulator could be designed  similar to an execution simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The “pick-and-place”  scenario can be decomposed into a sequenced simulation  of “grasp then pick then bring then place then release.”  The difference compared to the execution simulator is that  1All  authors  are  with  Microsoft,  Redmond,  WA,  USA  Kazuhiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='Sasabuchi@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='com  Physics  Kinematics  Rendering  Post-process  Task-sequencing simulator  Concept Interface  Environment Engine Pipeline  CM Grasp  state  action  selected task  at step t  demonstrated  execution sequence  configured  training sequence  CM Open  CM Release  training  algorithms  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The proposed task-sequencing simulator which enables scenario  composition for both learning and execution in robotics manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  some of these blocks are “under-training” and are updated  as data is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Once the update has finished, the trained  block can be combined and reused for a different scenario,  thus, the simulation-for-learning can directly transition to the  simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In addition, such design enables  learning new manipulation skills on top of programmed or  prior-trained building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For example, a “grasp” could  be trained using the sequence “grasp then pick,” where  “pick” is a programmed task to provide a supervised signal  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', teach that the “grasp” was successful if the “pick” was  successful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  This article introduces a task-sequencing simulator struc-  ture which enables integrated learning-to-execution simula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' At its core, the simulator uses a unified block design  called the “concept model,” which is proposed within this  article and defines the necessary descriptions for training  a task, collecting trained tasks, and running the tasks to  compose a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The rest of the article is outlined as below: Section II  provides a background on existing robotic simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Sec-  tion III explains the overall simulator structure for achieving  machine learning to robot execution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Section IV  explains the concept model core component of the simulator  and Section V provides some detailed example implementa-  tions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Section VI shows the capabilities of the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='01382v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='RO]  3 Jan 2023  C16simulator in machine-learning-to-robot-execution followed  by conclusions in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' BACKGROUND  While there are many existing simulators for robotics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  existing simulators may not achieve the integrated learning-  to-execution multi-step manipulation purpose for one of  the following reasons: 1) the simulator targets a different  domain other than manipulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 2) the simulator can be  used for manipulation but misses a capability in simulation-  for-learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 3) the simulator can be used for manipulation  but misses a capability in simulation-for-execution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 4) the  simulator can be used for manipulation and both learning-  and-execution purposes but not specifically for learning-to-  execution purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Popular  robotics  simulators[3][4]  include  Gazebo[5],  MuJoCo[6], CoppeliaSim[7], CARLA[8], AirSim[9], and  Webots[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Gazebo has its advantage in its capability to  simulate executions using ROS integrated sensors and ac-  tuators but is not the best choice when it comes to data  collection and machine learning due to its slow simulation  performance and inconsistency in physics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Thus,  Gazebo falls into the second category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Engines like MuJoCo  on the other hand, are suitable for stable physics simulation  in machine learning but miss some robotics simulation  capabilities such as inverse-kinematics and visual feedback  (realistic rendering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The focus is on physics simulation  rather than an integrated simulator for robot executions,  therefore falls into the third category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' CARLA and Airsim  mainly target automobiles such as drones and cars therefore  miss some important features such as kinematics required for  manipulation and falls into the first category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The CoppeliaSim is an integrated simulator with a kine-  matics and physics engine, and the PyRep toolkit[11] can  be used with the simulator for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' WeBots  is also an integrated simulator and frameworks such as  Deepbots[12] help the simulator to be used for machine  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The machine learning features of these simulators  are external features that have been developed within the  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' While it is possible to use these simulators  for both execution and learning purposes, they have not  been designed for integrated learning-to-execution but rather  using for one-or-the-other purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' That is, these simulators  are not designed to connect learning-and-execution, rather,  learning and execution are separate use cases where one uses  a community provided wrapper for machine learning, and the  other uses the integrated features to simulate a robotic system  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Compared to the existing simulators, the task-sequencing  simulator was designed to connect simulation-for-learning  and simulation-for-execution The simulator uses a con-  cept model which enables composition of pre-trained, pro-  grammed, or trained tasks, which is a powerful feature for  going from machine learning to real robot execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=',  such as plugging-in to machine learning platforms but then  connecting to execute on ROS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' More importantly, tied-  integration allows features such as training using pre-sequent  and post-sequent task executions, but also collecting reusable  execution modules through training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' TASK-SEQUENCING SIMULATOR OVERVIEW  The task-sequencing simulator has two layers: the Concept  Interface for “action decision” and the Environment Engine  Pipeline for “state observing” (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However, unlike  a typical learning simulator, where a specific problem has  a non-decomposable structure and the action decision is  a single policy being updated as data is collected for the  problem, the task-sequencing simulator adds an abstraction  to this action decision so that the problem is composed of  a sequence of tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', switches between a collection of  tasks, where each task runs an individual policy), This way,  a learned task policy can become part of a collection of  policies for execution once the training has finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Further  details of each layer are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' concept interface  At each simulation time-step, a robot decides the next  action depending on the current state of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This  decision is referred to as a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' When the relation be-  tween the state, action, and next state (system dynamics) is  completely known, this policy can be directly programmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  When the system dynamics are unknown, either the learning  of the policy or system dynamics is required through data  collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Data collection is efficient if collected only for  the unknown dynamics and if known dynamics are directly  computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Therefore, it is preferable to break down a robot’s  execution to a series of tasks, where each task executes its  own policy optimal for the system dynamics the task is cover-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In addition, breaking down a robot’s execution increases  the reusability of each task policy and allows composing  different execution scenarios from the task building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The Concept Interface layer chooses and switches between  the tasks for a training or execution scenario assuming  (1) the series of tasks to simulate is known (ways for  knowing are shown in the experiments), and (2) each task  policy can indicate when the task has been completed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=',  has a learned or programmed completion signal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' During  simulation-for-execution, the Concept Interface layer chooses  the according task in a sequence and switches to the next  task once the current policy returns a task completion signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The simulation-for-learning can be conducted in a similar  way, except, the policy’s output of the completion signal is  evaluated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', by the success of a subsequent task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This  interchangeable structure enables an integrated learning-to-  execution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' To learn the termination signal under  an arbitrary training sequence, the tasks share a unified  design called a “concept model,” which is further explained  in the later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment engine pipeline  At a time-step control scale, a robot behaves under a  sense-plan-act with a physical embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Thus, simulation  in robotics requires three important engines: kinematics,  physics (contact dynamics), and rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The role of  the kinematics engine is to do the simulation between the  robot’s action plan and the movement of the actual body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The role of the physics engine is to do the simulation  between the robot’s body and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The role of  the rendering engine is to do the simulation between the  environment and the robot’s sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In more technical terms,  the kinematics engine solves the mapping between cartesian  space and configuration space, the physics engine solves the  differential algebraic equation[13] using techniques such as  velocity-impulse linear complementarity-based time-stepping  methods[14][15], the rendering engine solves the rendering  equation[16] about lighting paths for pixel color generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  While rendering engines simulate the robot’s sensing by  generating images, there is a gap between sensing and  perceiving (extracting meaningful states from the gener-  ated images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Rather than directly learning the sensing-to-  planning, sometimes it is more efficient to perceive-before-  planning and extract visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Moreover, in robotics  it is important to combine both visual and force feedback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  the visual features help compress the feedback so that the  vision and force have an aligned state dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Thus, the  proposed simulator adds a fourth “post-process engine” in  conjunction with the rendering engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  These different-role engines are triggered in an ordered  pipeline to calculate the current state of the simulation world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Often, simulators package specific engines to produce a  single simulation world, but in general, these engines could  run separately and combine/orchestrate multiple simulation  worlds to produce better simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For example, ROS  MoveIt could be used for accurate inverse kinematics sim-  ulation, PyBullet for reproducible physics simulation, and  the Unreal Engine for photo-realistic ray-traced rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Combining different engines is possible as long as each  engine is able to load the same models of the robot/objects  and is able to share the robot and object states among each  engine (which can be done using TCP connections etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  In addition, instead of using simulation engines, it is also  possible to connect “real” engines which replace simulated  physics with the real robot’s torque sensors and simulated  rendering with real images from the robot’s camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' CONCEPT MODELS  A single task block operates under some system dynamics  and achieves a goal state from an initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Thus, the de-  tails of a task can be described using the actors of the system,  an initial state, a goal (end) state, and the parameters of the  system dynamics (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This kind of task description  can be referred to as a “task model”[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This description  is enough for executing a task if the system dynamics are  completely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state is usually the end state  of the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However, when the dynamics are not  fully known, learning is required, and during learning, the  initial state must be randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Instead of fully describing the task, a task can be described  using actor configurations, an initial state, a necessary goal  state that is described from observable system states, a  sufficient goal state that is described from non-observable  Task Model  initial state  Action  Programmed policy  Actors  obtained before execution  Parameters  completely known and obtained  before execution  defined goal state  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  An illustration of a task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Concept Model  initial state  Action  Policy  Actors  a set of configurated settings  Parameters  partially known and randomized at  training, estimated at execution  necessary goal state  sufficient goal state  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  An illustration of a concept model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  system states, and the partially known parameters of the  system dynamics (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This kind of task description  will be referred to as a “concept model” which compared to  the task model may not be concrete enough for execution, but  describes the concepts of the task to learn the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In  the special case where the system dynamics are fully known  and the task is programmable, the descriptions of the concept  model is identical to the task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  By providing a structured description format, the Con-  cept Interface layer can access these blocks interchangeably  within a training sequence as the structures are the same and  only differ in the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' concept model usage in learning  The concept model descriptions are used to learn the  task completion signal as well as the actions to handle the  unknown system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The necessary goal state and  sufficient goal state descriptions are used to evaluate whether  the task completion signal and actions are appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  evaluation is done by minimizing the cost of the current  states to the goal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Note that the cost to the necessary  goal state is evaluated after every action decision, whereas  the cost to the sufficient goal state is only evaluated once a  task completion signal is chosen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The actor configurations describe the possible environ-  ments (the world including the robot and any manipulating  target) for training the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' If there are no pre-sequent tasks  involved for training, then one of the actor configurations is  used to define the initial state of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Otherwise, the  end state of the previous task is the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Unlike the  initial state, the actor configuration is independent from the  states of the previous task, thus is configurable and can be  used for randomizing the states for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' concept model usage in execution  When the dynamics are fully known, the concept model  acts the same as the task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The state of the task changes  from the initial state using actions from a programmed policy  based on the system dynamics and actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state  of the task during execution is the end state of the previous  task and the task ends once the goal state is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  When the dynamics are not fully known, the observable  system states of the task changes using a learned policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Similar to the programmed case, the initial state is the  observable states at the end of the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However,  since part of the goal state is non-observable, the end of the  task cannot be identified just with the model descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Instead, the learned task completion signal from the concept  model descriptions is used to identify the end of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' MODEL IMPLEMENTATIONS  By following the concept model structure, a task is im-  plemented in a way that can be trained but then collected  as a building block for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' In this article, the screw  theory[18] based separation of dynamics[17] is used to  separate a task from another task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' That is, once the relation  between the manipulation target and the robot’s end-effector  is initialized, a task breaks or maintains a contact state  between the target and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' By classifying the  inequality equation patterns of contact points, this leads to  seven pure translation tasks and seven pure rotation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 4 shows implementation examples of some of these  tasks as concept models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Below describes the details of some  of the examples in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Note that as the task classifica-  tion only depends on the relation between the end-effector,  manipulation target, and the environment, the movement of  the arm (configuration space) can be ignored[19] and the task  only focuses on the movement of the end-effector (cartesian  space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' grasping  The  grasp  task  initiates  the  relation  between  the  manipulation-target and the robot’s end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The actors  are a target object, an environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=', table), and the  end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state is where the target object is  attached to the environment but not attached to the end-  effector (including shape of the finger joints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The goal state  is where the target object is also attached to the end-effector  in a way such that enough force is exerted for performing a  subsequent task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The parameters are the distance between the  target and end-effector as well as the approaching direction  of the end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  When details of the target object are completely known, a  task model can be defined and programmed from the above  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However, in the real world, there is uncertainty in the  shape of the object and distance to the target, distinguishing  whether enough force is exerted is intractable due to the  inaccuracy in the real contact sensors or lack of sensors to  detect slipping, a grasp-failure due to finger-object collision  during approach may occur if the policy is not carefully  designed under the uncertainties of the object properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Grasp Concept Model  freed OBJ-EEF  Action  Policy  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment (ENV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a set  of objects (OBJ) with randomized shapes  Parameters  estimated approach distance OBJ-EEF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  approach direction of EEF  NGS: attached OBJ-EEF  SGS: next task success  Open Concept Model  initial OBJ pose  Action  Policy  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment (ENV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a set  of objects (OBJ) with randomized parameters  Parameters  estimated radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' axis center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' axis direction  to maintain constraint OBJ-ENV  NGS: maintain OBJ-ENV  SGS: desired OBJ pose  Pick Concept Model  attached OBJ-ENV  Action  Programmed  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment (ENV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' target  object (OBJ)  Parameters  detach direction to break constraint OBJ-ENV  freed OBJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' ENV  Bring Concept Model  initial OBJ pose  Action  Programmed  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' target object (OBJ)  Parameters  move direction and distance of OBJ  desired OBJ pose  Place Concept Model  freed OBJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' ENV  Action  Programmed  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' environment (ENV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' target  object (OBJ)  Parameters  attach direction to create constraint OBJ-ENV  attached OBJ-ENV  Release Concept Model  attached OBJ-EEF  Action  Programmed  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' target object (OBJ)  Parameters  release distance OBJ-EEF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  release direction of EEF  freed OBJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' EEF  Pour Concept Model  initial ENV state  Action  Policy  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' set of environment (ENV)  and object (OBJs) with randomized parameters  Parameters  estimated ENV (cup) filled state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  estimated OBJ (pitcher) size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  estimated axis center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' direction of OBJ-ENV  NGS: maintain OBJ-ENV  SGS: desired ENV state  Wipe Concept Model  initial ENV state  Action  Policy  Actors  end-effector (EEF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' set of environment (ENV)  and object (OBJs) with randomized parameters  Parameters  estimated ENV (plane) clean state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  estimated normal axis of OBJ-ENV  NGS: maintain OBJ-ENV  SGS: desired ENV state  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Example concept models of eight different tasks in the screw-theory  based classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Instead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a set of actor configurations is defined as object  shapes from a randomized range,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a necessary goal state is  defined as the end-effector to be in contact with the target  object on an appropriate surface (which can be obtained on  the real robot with the finger configurations and finger-torque  sensors with a threshold to determine a binary contacted-or-  not-contacted state),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' a sufficient goal state is defined as a  successful performance of a subsequent task,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' the estimated  distance is used for the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The defined goal states  are used to formulate the reward (cost-to-go) for learning the  policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Using this concept model, the approaching strategy (ad-  justed movement around the approaching direction of the  end-effector) and the enough amount of “closing” of the  fingers to perform a subsequent task is learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The learned  policy chooses the sufficient amount based on the object  shape which can partially be inferred by the shape of the end-  effector finger joints once touching the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The policy  returns a termination signal once reached the enough amount  of closing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' door-opening  The door-opening task is a one degrees-of-freedom pure  rotation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The actors are a target object (the door), an  environment (the hinge), and the end-effector (attached to the  door handle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state is where the end-effector and  target object are at an attached state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The goal state is where  the target object has moved to some desired orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  parameters are the rotation radius, the rotation axis center,  and the rotation axis direction defined by the target and  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  When details of the target and environment are completely  known, a task model can be defined and programmed from  the above details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However, in the real world, there is  uncertainty in the environment parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Instead, a set of  actor configurations randomizing the radius, a necessary goal  state that moves the object along the environment constraint  at each time-step (which can be obtained on the real robot  by using a force sensor on the wrist and checking against  a maximum-stress threshold), a sufficient goal state that  ensures the door has reached the desired orientation, and  estimation of the parameters are used to describe the concept  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  By using an end-effector with only force-sensor feedback  on the wrist, this model enables learning a policy which  updates parameter estimations at each time-step, and then  generates a hand motion based on the updated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  The policy returns a termination signal once inferred that the  desired orientation has been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' bringing  The bringing task is a six degrees-of-freedom translation  and rotation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The actors are the target object and the  end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The initial state is where the end-effector and  target object are at an attached state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The goal state is where  the target object has moved to some relative positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  parameters are the moving direction and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Since there are no uncertainties in the target or environ-  ment, the parameters can be manually specified and the goal  can be directly specified from the parameters, Thus, the  concept model is identical to the task model and can be  programmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' EXPERIMENTS  Experiments were conducted using the concept model  implementations shown in the previous section, and the  developed task-sequencing simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  For the learning experiments, the series of tasks to simulate  were predefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' These experiments were performed to show  the effectiveness of the simulator and its reusability for  training different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  By running the pre-sequent tasks of the task-to-train at  the start of an episode, and by running the subsequent tasks  at reward return, the simulator is compatible with common  reinforcement learning platforms, and utilizes off-the-shelf  learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For the experiments, the simulator was  connected to the Bonsai platform and used the PPO algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  For the execution experiments, the series of tasks were  obtained from human demonstrations and the actions were  generated using the learned task policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' These experiments  were performed to show the effectiveness of the simulator for  execution simulations (execution of different composed sce-  narios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Note that the exact same simulator and task blocks  used for training was used for this experiment, showing  the simulator’s capability to transition from simulation-for-  learning to simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  For simulation, states were obtained by plugging to the  environment engine pipeline the PyBullet engine for the  physics engine role, and the Unreal Engine for rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For  the real robot, states were obtained plugging ROS (connected  via roslibpy) to the environment engine pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' For the arm  kinematics, the ROS MoveIt package was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' training  Figure 5-(A) shows the task-sequencing simulator config-  urations for running the grasp training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' A configured training  sequence “grasp then pick” is passed to the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  “grasp” task is the task-to-train, and a programmed “pick”  is used as the subsequent task for evaluating the sufficient  goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 5-(B) shows the trained grasp results performed  on a real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The concept model was designed with  the object shape parameters as unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Regardless of  such uncertainty, the learned policy successfully grasps the  different shaped objects including but not limited to a box,  a cylindrical cup, an oval rice pack, and a diamond-shaped  candy box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 5-(C) shows a learned grasp for a different robot  hand, which was trained using the same simulator and  concept models but with a different actor configuration  (end-effector) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The results show the reusability of  the simulator for training different robots with different  mechanics (a hand with multiple fingers and a gripper with  limited degrees of freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 5-(D) shows that by changing the configured train-  ing sequence to “grasp then open,” the door-opening task is  trained using the same simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The “open” task is the task-  to-train, and the trained “grasp” is reused as the pre-sequent  task for initiating the relation between the end-effector and  the target door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Regardless of uncertainty in the rotation  radius, center, and axis direction, the real robot performed the  door-opening using the learned policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Although the policy  was trained only using simulated data, the policy is directly  applicable to the real robot as the sufficient goal state does  not require observability on the real robot and because the  policy action decisions only rely on the states with very small  sim-to-real gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' execution  Figure 6-(A) shows the task-sequencing simulator used  with a demonstrated sequence by a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Instead of a  configured sequence as in the previous training experiments,  the sequence is automatically generated through demonstra-  tion decomposition using the method described in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The  same concept models from the training experiments are used  with the policy-update being disabled (the simulator is not  connected to any training algorithm and instead uses a fixed  learned policy without updates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Task-sequencing simulator  Concept Interface  Environment Engine Pipeline  CM Grasp  state  action  configured  training sequence  CM Pick  training  algorithms  (A)  (B)  (C)  connect learned policy  to the real robot  train policy on a  different robot hand  CM Grasp  CM Open  training sequence  for door-opening  (D)  configure to a different  training scenario  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Results of the task-sequencing simulator when used for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Task-sequencing simulator  Concept Interface  Environment Engine Pipeline  CM Grasp  state  action  demonstrated execution sequence  CM Pick  (A)  simulation  CM Bring  CM Bring  CM Place  CM Release  real  another  scenario  physics  rendering  (B)  (C)  (D)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Results of the task-sequencing simulator when used for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 6-(B) shows a simulated execution of the demon-  strated sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The first row shows the outputs of the  physics engine and the second row shows the outputs of  the rendering engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' As both training and execution run  on the same system, the learned policy can easily be used  as a simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The learned policy is already  a building block that can be combined with other tasks to  generate an application such as “pick up a cup from the upper  shelf and re-place it to the bottom shelf.”  Figure 6-(C) shows a real robot execution of the demon-  strated sequence by switching the engines in the environment  engine pipeline to connect with ROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This shows how the  simulator can go from the simulated robot execution to the  real robot execution by using the same policy connections  but by changing the engines in which the states are obtained,  and the actions are performed against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Usually, going from  simulation to real introduces a sim-to-real gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' However,  only part of the scenario sequence uses a learned policy and  due to the careful design of the concept models to divide  learning observable dynamics (necessary goal states) from  learning hidden dynamics (sufficient goal states), no such  gap was encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Figure 6-(D) shows an execution of a different sequence  “pick up a cup from the table and throw it in the trash.”  This scenario uses the same concept models and only differs  in the demonstrated input, showing how using the simulator  and concept model descriptions enable reusing the learned  policies for different execution scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' If a policy was  learned against a full “pick-and-place” scenario, the policy  would not easily scale to the “pick-and-throw” scenario as  the problem dynamics are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' CONCLUSIONS  This article introduced the task-sequencing simulator  which bridges simulation-for-learning to simulation-for-  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The simulation scenario for learning is created us-  ing a sequence of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This way the simulation-for-learning  has the same structure as the simulation-for-execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' At  its core, the simulator uses a concept model which enables  sequencing mixed programmed, trained, and under-training  building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' While the simulator has a large advantage  in terms of integrated system development, the simulator  also provides new directions for simulation in execution and  simulation in learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  From an execution perspective, the simulator allows com-  posing a task-sequence using both programmed and trained  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Unlike programmed-only sequences, the advantage of  mixing trained blocks is that, some of the tasks can contain  uncertainty and the goal state of a task can be described using  implicit system parameters (the goal state does not have to  be obtained directly from the real robot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' The key is that,  whether the observed state and selected actions suffice the  goal state is learned as a termination signal through training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  From a learning perspective, the simulator and concept  model design have the following advantages: First, the sim-  ulation is reusable and easily applicable to slight changes  in the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' A policy for a different end-effector can  be learned by just changing the actor configurations in the  concept model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' A policy can be optimized for different  scenarios by just changing the subsequent task in the suffi-  cient goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Second, defining the learning problem using  the concept model design enables a hierarchical learning-  structure as well as a structure for reducing sim-to-real gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  Any state parameters that do not have a large gap when  observed with the real robot is used for defining the necessary  goal state, whereas any state parameters that have a large gap  when observed with the real robot is a sufficient goal state  (implicitly learned in simulation but no need to be observed  with the real robot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' This type of formulation is possible  as only the parts with uncertainty are being learned instead  of learning the entire scenario sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content=' Following this  structured formulation has allowed going from simulation  to real without any extra real-world data collection and  achieving a reusable policy applicable to different execution  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
+page_content='  ACKNOWLEDGMENT  The authors thank Brice Chung’s team, Aydan Aksoylar  and Kartavya Neema for their help in the reward designs and  training of the concept models used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idAzT4oBgHgl3EQfbPxy/content/2301.01382v1.pdf'}
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diff --git a/itAzT4oBgHgl3EQfbPyr/content/tmp_files/2301.01383v1.pdf.txt b/itAzT4oBgHgl3EQfbPyr/content/tmp_files/2301.01383v1.pdf.txt
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+How to get the most out of Twinned Regression
+Methods
+Sebastian J. Wetzel
+University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
+Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
+Homes Plus Magazine Inc., Waterloo, Ontario N2V 2B1, Canada
+Abstract.
+Twinned regression methods are designed to solve the dual problem to
+the original regression problem, predicting differences between regression targets rather
+then the targets themselves. A solution to the original regression problem can be
+obtained by ensembling predicted differences between the targets of an unknown data
+point and multiple known anchor data points. We explore different aspects of twinned
+regression methods: (1) We decompose different steps in twinned regression algorithms
+and examine their contributions to the final performance, (2) We examine the intrinsic
+ensemble quality, (3) We combine twin neural network regression with k-nearest neighbor
+regression to design a more accurate and efficient regression method, and (4) we develop
+a simplified semi-supervised regression scheme.
+Keywords: Artificial Neural Networks, k-Nearest Neighbors, Random Forests, Regression,
+Semi-Supervised Learning
+arXiv:2301.01383v1  [cs.LG]  3 Jan 2023
+
+How to get the most out of Twinned Regression Methods
+2
+1. Introduction
+Regression is one of the most general and common machine-learning tasks, practitioners
+in many different fields of science and industry rely on methods that help them to make
+the most accurate and reliable predictions on new data points inferred from a limited
+amount of training data. Conventional regression methods aim to infer the mapping
+of input features to one or multiple target variables. Twinned regression methods aim
+to solve the dual problem of predicting the difference between target values, a solution
+to the original regression problem can then be obtained by evaluating the predictions
+between a new unknown data point and multiple anchor data points. This process creates
+an ensemble of predictions from a single trained model, which tends to be more accurate
+than solving the original regression problem directly and opens up semi-supervised
+learning and uncertainty estimates for a very low cost [1, 2, 3]. However, the trade-off
+is the poor scaling with large data sets, since the effective training data set size scales
+quadratically with the size of the original training data set.
+Thus, these methods are beneficial in domains where data is either scarce or costly
+to obtain. This is the case in for example real estate where markets differ from city to city
+and data becomes outdated quickly [4, 5, 6]. Another example stems from calculations
+in chemistry where simulations of chemical systems based on quantum mechanics require
+an enormous amount of computational resources [7, 8].
+This article is meant to be a practitioner’s guide to using twinned regression methods
+that guides the reader through advantages and trade-offs and attempts to answer most
+questions that were left open in the recent years.
+The main question on our mind is why can such a simple trick of solving the dual
+problem yield a more accurate prediction than solving the regression problem directly.
+While we do not fully answer this question, we decomposed twin neural network regression
+(TNNR) into different steps each with the potential to enhance the performance over
+traditional algorithms. These include increased effective data set size obtained though
+pairing training data points, or different ensembles of TNNR predictions. Further, by
+mapping extreme cases of TNNR to k-nearest neighbor (k-NN) regression and normal
+artificial neural networks (ANN) we can observe a distinct performance behaviour of
+twinned regression methods different from traditional regression.
+Further, we are eager to improve the accuracy of twinned regression methods. For
+this purpose, we devise an improvement to TNNR based on the idea of weighting the
+predictions from different anchors. This leads us to combine k-nearest neighbors (kNN)
+with TNNR to an even more accurate regression scheme.
+The semi-supervised regression framework for TNNR invented in [2] is specifically
+tailored to neural network-based regression. It is based on enforcing consistency conditions
+on unknown data points through a modified loss function. At the end of this manuscript
+we examine a way to translate a simplified version of this semi-supervised learning scheme
+to a twinned version of random forests (RF) proposed in [3].
+In some projects it is important to apply neural networks with strong memory
+
+How to get the most out of Twinned Regression Methods
+3
+constraints, this might be the case in small chips or autonomous systems [9, 10]. In
+these cases it would normally be very inefficient to store ensembles of machine learning
+models due to the increased number of parameters. In contrast to that, with twinned
+regression methods, one only needs to store additional anchor data points.
+2. Prior Work
+The pairwise comparison inherent to twinned regression methods is inspired by Siamese
+neural networks which were devised to solve the similarity classification problem as
+it occurs in fingerprint recognition or signature verification [11, 12]. Siamese neural
+networks contain two identical neural networks with shared weights which project a
+pair of inputs into a latent space on which the pairwise similarity is determined by the
+distance. Twinned regression methods also take a pair of inputs to predict the difference
+between the labels [1].
+Twin neural network regression [1] was invented as a regression method that solves
+the dual problem of predicting pairwise differences between the target values of pairs of
+input data points. Independently, the same idea has been developed for random forests
+[3]. This kind of regression framework has been shown to have several advantages: (1) it
+allows for a very efficient generation of ensemble predictions [1, 3]. Typically in methods
+that generate ensembles from training a single machine learning model, the predictions
+are strongly correlated [13, 14] since they can be deformed into each other through small
+perturbations. In twinned regression methods however, ensemble members are separated
+by the distance of the input data points themselves. (2) Twinned regression methods
+tend to be more accurate than the underlying base algorithm on many data sets [1, 3],
+(3) consistency conditions allow for the formulation of uncertainty estimators in addition
+to the ensemble variance [1, 3] and (4) loops containing unlabelled data points can be
+supplied while training, hence turning the method into a semi-supervised regression
+algorithm [2]. Further, (5) the intrinsic uncertainty estimation lends itself for active
+learning [3].
+A central contribution of this article is the combination of twin neural network
+regression and k-NN regression to increase the accuracy of over standard twin neural
+network regression. Similarly, artificial neural networks have been employed in tandem
+with k-NN regression in different contexts before [15, 16, 17].
+3. Reformulation of the Regression Problem
+A regression problem can be formulated as follows: Given a labelled training data
+set of n data points Xtrain = (xtrain
+1
+,...,xtrain
+n
+) with their corresponding target values
+Y train = (ytrain
+1
+,...,ytrain
+m
+), we are tasked to find a function f such that the deviation
+between f(xi) and yi is minimized with respect to a predefined objective function for all
+data points xi on the data manifold. In this work, this function is the root mean square
+error LRMSE =
+√
+∑n
+i=1(f(xi) − yi)2. Unless stated otherwise, all performance measures
+
+How to get the most out of Twinned Regression Methods
+4
+x
+y=f(x)
+x1
+x2
+y2-y1=F(x2,x1)
+F(x1,x2)
+f(x2)
+F(x2,x3)
+F(x3,x1)
+f(x3)
+f(x1)
+Traditional Regression  
+Twinned Regression  
+Figure 1: Dual formulation of a regression problem: A traditional solution to a regression problem
+consists of finding an approximation to the function that maps a data point x to its target value f(x) = y.
+Twinned regression methods solve the dual problem of mapping a pair of inputs x1 and x2 to the
+difference between the target values F(x2,x1) = y2 − y1. The resulting function can then be employed
+as an estimator for the original regression problem y2 = F(x2,x1) + y1 given a labelled anchor point
+(x1,y1). Twinned regression methods must satisfy loop consistency: predictions along each loop sum to
+zero: F(x1,x2) + F(x2,x3) + F(x3,x1) = 0.
+are evaluated on unknown test data (Xtest,Y test).
+Twinned regression methods aim to solve a reformulation of the original regression
+problem which is visualized in Fig. 1. For each pair of data points (xtrain
+i
+,xtrain
+j
+) we
+train a regression model to find a function F to predict the difference
+F(xi,xj) = yi − yj
+.
+(1)
+This function F can be used to construct a solution to the original regression problem
+via ypred
+i
+= F(xi,xj) + yj, where (xj,yj) is an anchor whose target value is known. Every
+training data point xtrain
+j
+∈ Xtrain can be used as such an anchor. A more accurate
+estimate for the solution of the original regression problem is obtained by averaging over
+many differences between a fixed unknown data point and different anchor data points
+ypred
+i
+= 1
+n
+n
+∑
+j=1
+(F(xi,xtrain
+j
+) + ytrain
+j
+)
+= 1
+n
+n
+∑
+j=1
+(1
+2F(xi,xtrain
+j
+) − 1
+2F(xtrain
+j
+,xi) + ytrain
+j
+)
+.
+(2)
+The increase in accuracy is based on averaging out the noise from different anchors and
+the reduction of the variance error via an ensemble of predictions. Previous works [1, 3]
+recommended using the whole training data set as anchors, hence creating an ensemble
+of difference predictions yi − yj which is twice as large as the training set for every single
+prediction of yi.
+A major advantage of the dual formulation is the description via loops containing
+multiple data points as can be seen in Fig. 1. In contrast to traditional regression, the
+
+How to get the most out of Twinned Regression Methods
+5
+results of twinned regression methods need to satisfy consistency conditions, for example
+for each three data points x1,x2,x3, summing up the predictions along a closed loop
+should yield zero: F(x1,x2) + F(x2,x3) + F(x3,x1) = 0. During inference, violations of
+these consistency conditions give rise to uncertainty estimates [1, 3]. Enforcing loop
+consistency on predictions involving unlabelled data points in the training phase is what
+makes twinned regression methods into semi-supervised regression algorithms.
+While neural networks are naturally good learners of linear functions this is not the
+case for other algorithms like random forests. For this reason, [3] proposed to augment
+the input features by their difference (xi,xj) → (xi,xj,xi − xj). One might argue that
+this improvement is similar to common data augmentation, however, this is a feature
+that traditional machine learning algorithms don’t have access to, because it requires
+two different data points.
+4. Notes About Experiments
+All experiments in this article are performed on the data sets outlined in Appendix A.
+Since only neural network based methods scale favorably with the data set size, they use
+the full data sets of which 70% are used for training, 10% as validation set and 20% as
+test set. The details of the neural network architectures can be found in the appendix
+Appendix C. Random forests and especially twinned random forests scale poorly with
+the data set size thus only 100 training data points are chosen from the data sets and
+100 data points comprise the test sets. Random forests do not need validation sets, since
+the hyper=parameters are optimized via 5-fold cross-validation. In section Appendix D
+one can find the details about our random forest implementations. All experiments are
+repeated for 25 random but fixed splits of training, test, and if applicable, validation
+data.
+5. Ensemble Performance
+Twinned regression methods have been shown to produce accurate solutions to regression
+problems [1, 3], comparable to or better than other current state-of-the-art algorithms at
+the cost of scaling poorly towards larger data sets. This naturally leads to the question of
+where the increased performance stems from. The reformulation of a regression problem
+into its dual problem of predicting differences between target values opens up several
+potential reasons for improved accuracy. These include increased effective training set
+size, internal ensembling of predictions (see explanation in Appendix B), or the nature of
+solving a different problem. In the following we examine these reasons at the example of
+TNNR, however, we assume the answers will also be valid for other baseline algorithms.
+Let us start with discussing different kinds of ensembles and their effect on accuracy.
+Fig. 2 contains the results of several experiments examining the performance of different
+ensemble types of ANN regression and TNNR. For each data set, the baseline results are
+the solid blue horizontal line, which represents the test RMSE after applying standard
+
+How to get the most out of Twinned Regression Methods
+6
+Figure 2: Comparison of different ensembles of ANNs and TNNR measured by RMSE vs number of
+ensemble members for the dashed lines. Ensembles can be created either by training several models or
+evaluating TNNR for different anchors. The blue solid line corresponds to training and evaluating one
+TNNR model on all possible training pairs and predicting the results with all possible anchors. The
+dashed blue line varies the number of anchors during the inference phase and converges to the solid
+blue line in the limit of increasing the inference anchors to the full training set. The dashed orange line
+indicates traditional ANN ensembles where multiple ANNs are trained independently. The dashed green
+line corresponds to ensembles of independently trained TNNR models, if the ensemble size is 1, this
+is equivalent to the solid blue line. The dashed red line corresponds to independently trained TNNR
+models each having only one inference anchor per prediction.
+full anchor TNNR and the leftmost point of the orange line which represents the results
+of applying a single ANN, confirming that TNNR almost always yields a lower RMSE
+than ANN regression.
+In order to compare traditional ANNs with TNNR, we observe that after training we
+can map TNNR to an ANN for each anchor, since both ANN regression and TNNR use
+the same internal architecture. For each fixed xj ∈ Xtrain an ANN ˜f is defined through
+˜fj(xi) ∶= F(xi,xj) + yj
+(3)
+The features of xj modify the weights of ˜fj while yj is absorbed by the output neuron
+inside its bias. At that point, the only difference between each ˜fj and an equivalent
+ANN is the procedure with which the weights were optimized.
+
+ANNEnsemblevsTNNanchorensemble
+Bio Conservation
+Boston Housing
+Concrete Strength
+3.7
+5.50
+06'0
+3.6 
+5.25
+0.85
+3.5
+RMSE
+5.00
+3.4
+4.75
+08'0
+EE
+3.2
+4.50
+0.75
+3.1
+4.25
+16
+32
+Energy Efficiency
+RCL Circuit
+Test Function
+0.012
+1.0
+0.020
+0.010
+0.9 
+0.018
+0.008
+0.8
+0.016
+0.006
+0.7
+0.014
+0.004
+16
+32
+16
+32
+16
+32
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+0.045
+0.70
+all anchors inference
+0.775
++TNN inference anchors
+0.040
+0.750
+ANN ensemble
+0.65
+0.725
+Ensemble of full anchor TNN
+Ensemble of one anchor TNN
+0.030
+0.700
+0.60
+0.675
+0.025
+0.650
+0.55
+2
+16
+32
+2
+4
+16
+32
+2
+4
+16
+32
+Ensemble Members
+Ensemble Members
+Ensemble MembersHow to get the most out of Twinned Regression Methods
+7
+Table 1: Best estimates for test RMSEs obtained by artificial neural network(ANN) regression compared
+to ensembles of ANN regression. Our confidence on the RMSEs is determined by their standard error.
+We train on 70% of the available data, 10% validation data, 20% test data.
+Single ANN
+32 ANN ensemble
+Gain
+Bio Conservation(BC)
+0.7874±0.012
+0.7438±0.0118
+5.53%
+Boston Housing(BH)
+3.5695±0.1051
+3.3291±0.1137
+6.73%
+Concrete Strength(CS)
+5.3829±0.0964
+4.9842±0.0938
+7.41%
+Energy Efficiency(EE)
+1.0310±0.0313
+0.9649±0.0224
+6.42%
+RCL Circuit(RCL)
+0.0193±0.0002
+0.0157±0.0002
+18.41%
+Test Function(TF)
+0.0076±0.0005
+0.0071±0.0005
+6.35%
+Wheatstone Bridge(WSB)
+0.0443±0.0016
+0.0359±0.0014
+19.1%
+Red Wine Quality(WN)
+0.7681±0.024
+0.6487±0.0068
+15.54%
+Yacht Hydrodynamics(YH)
+0.6723±0.0406
+0.6143±0.0357
+8.63%
+Table 2: Best estimates for test RMSEs obtained by Twin Neural Network Regression (TNNR). We
+measure the improvement between using a single anchor during inference phase and using all anchors.
+Further, the latter is compared to an ensemble of full anchor TNNR.
+1 Anchor TNNR
+All Anchor TNNR
+Gain
+Ensemble of 32 TNNR
+Gain
+BC
+0.9021±0.0131
+0.8140±0.0149
+9.77%
+0.7947±0.0134
+2.37%
+BH
+3.4793±0.1262
+3.2897±0.1293
+5.45%
+3.2232±0.1256
+2.02%
+CS
+4.9602±0.1073
+4.5385±0.1124
+8.5%
+4.2791±0.0995
+5.71%
+EE
+0.7445±0.0224
+0.7071±0.0229
+5.02%
+0.6468±0.0196
+8.53%
+RCL
+0.0212±0.0003
+0.0155±0.0002
+26.6%
+0.0130±0.0001
+16.23%
+TF
+0.0106±0.0004
+0.0053±0.0004
+50.24%
+0.0028±0.0001
+46.41%
+WSB
+0.0309±0.0009
+0.0239±0.0009
+22.81%
+0.0227±0.0008
+5.15%
+WN
+0.7654±0.0059
+0.6985±0.0064
+8.75%
+0.6713±0.0055
+3.89%
+YH
+0.6344±0.0363
+0.5798±0.0362
+8.62%
+0.5723±0.0336
+1.29%
+This gives us access to a framework to directly compare ensembles of ANNs and
+the implicit ensembles generated by TNNR using multiple anchors during inference. We
+examine the results of these both models through the orange and blue dashed lines in
+Fig. 2. While both curves reduce the RMSE as we increase the ensemble size, we come
+to the sobering conclusion that TNNR ensembles and ANN ensembles are not equivalent,
+they neither have a uniform slope nor do they converge to similar RMSEs.
+We have just used a single trained TNNR model for all ensemble members while
+training each ANN model from scratch. What happens if we retrain TNNR for each
+single anchor? The results of these experiments are visualized in the red dashed line.
+Since retraining increases the ensemble diversity the red line is consistently below the blue
+
+How to get the most out of Twinned Regression Methods
+8
+Figure 3: How many different pairings are used for training effects the performance of TNNR measured
+by RMSE. The blue solid line corresponds to training TNNR on all possible training pairs. It is
+compared to training TNNR on a randomly chosen but fixed data set of pairs while still employing all
+training data points as inference anchors. A training set multiplier indicates on how many pairs are
+taken into account compared to the original unpaired training set.
+line. Further, we can see that the performance of these independently trained TNNRs
+increases faster with the number of anchors. If the resources are available one can further
+create an ensemble of different TNNR models each having access to all anchors depicted
+in the green line. On seven out of nine data sets this yields clearly the best performance.
+We note that one-anchor TNNR with retraining (red line) converges towards the green for
+more than 32 ensemble members. This tells us that the full ensemble diversity through
+multiple anchors and multiple models can be captured independently.
+A more quantitative version of the out-performance of TNNR ensembles can be
+seen in Table 1 and Table 2. The magnitude of the % improvement of the combined
+anchor+direct ensembling containing multiple TNNs causes a much larger improvement
+than the ensembling of traditional ANNs.
+6. Effective Training Set Size
+Another improvement over a traditional regression analysis is the increased training set
+size that comes from preparing training sets by pairing each training data point with
+
+TrainingSetMultiplier
+Bio Conservation
+Boston Housing
+Concrete Strength
+0.84
+3.5 
+5.0
+0.82
+RMSE
+3.4
+4.8
+0.80
+EE
+4.6
+0.78
+3.2
+4.4
+4
+8
+16
+2
+4
+8
+16
+32
+1
+2
+8
+16
+32
+Energy Efficiency
+RCL Circuit
+Test Function
+11
+0.0170
+0.0065
+1.0
+0.0165
+0.0060
+0.0160
+0.8
+0.0055
+0.0155
+0.7
+0.0050
+1
+a
+16
+32
+1
+7
+4
+8
+16
+32
+1
+2
+4
+8
+16
+32
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+0E0'0
+0.75
+all anchors training
+0.70
+anchor pairs per
+0.70
+0.028
+training data
+MSE
+0.69
+0.65
+0.026
+0.68
+0.60
+0.024
+0.67
+0.55
+2
+4
+8
+16
+32
+2
+4
+8
+16
+32
+1
+2
+4
+8
+16
+32
+IrainingPairsperTrainingData
+Training Pairs per Training Data
+TrainingPairsperTrainingDataHow to get the most out of Twinned Regression Methods
+9
+Table 3: Best estimates for test RMSEs obtained by Nearest Neighbor Twin Neural Network Regression
+(NNTNNR).
+TNNR
+NN inference
+Gain
+NN train+inference
+Gain
+BC
+0.8234±0.0144
+0.8162±0.0155
+0.87%
+0.8133±0.0152
+1.23%
+BH
+3.3104±0.1202
+3.2898±0.1202
+0.62%
+3.3563±0.1161
+-1.39%
+CS
+4.4731±0.1242
+4.4458±0.1091
+0.61%
+4.4290±0.1174
+0.99%
+EE
+0.7156±0.0204
+0.7056±0.0193
+1.4%
+0.6825±0.0216
+4.63%
+RCL
+0.0158±0.0002
+0.0151±0.0003
+3.98%
+0.0140±0.0002
+11.33%
+TF
+0.0050±0.0001
+0.0028±0.0002
+43.65%
+0.0021±0.0003
+57.26%
+WSB
+0.0233±0.0006
+0.0224±0.0009
+4.06%
+0.0236±0.0009
+-1.01%
+WN
+0.6998±0.006
+0.6951±0.0062
+0.68%
+0.6944±0.006
+0.77%
+YH
+0.5977±0.0344
+0.5184±0.0331
+13.26%
+0.5009±0.0333
+16.19%
+every other training data point. This transforms a training set of size n into a pairwise
+training set of size n2. In this section, we measure if the increase in the number of
+pairings leads to an increase in accuracy.
+To address this question we look at several curves in Fig. 3. In this figure, we
+compare the effect of increasing the effective pairwise training data set on the accuracy.
+For this purpose, we define the training set multiplier as the number of pairs that are
+created from the original training set to produce the paired training set. A training
+set multiplier of one means that each training data point is paired with only one other
+randomly chosen (without replacement) but fixed data point (on average this means each
+training data point is used twice). Increasing the training set multiplier to the size of the
+training set converges to the standard formulation of twinned regression methods. We
+can see that in all data sets, except two, increasing the training set multiplier increases
+the performance of TNNR. More precisely, a training set multiplier of ≈ 8 − 16 seems
+already to be enough to reach the accuracy of standard TNNR. It is important to
+note, that on two data sets, namely Bio Conservation(BC) and Red Wine Quality(WN),
+increasing the training set multiplier has the effect of reducing the performance. This
+coincides with other algorithms beating TNNR (Fig. 2,Fig. 5) and is a sign that TNNR
+might not be suitable for such regression tasks. A data scientist using TNNR might do
+a training set multiplier check, if he finds a decreasing accuracy while increasing the
+multiplier, he can reject TNNR as optimal regression algorithm.
+7. Nearest Neighbor TNNR
+In this manuscript we propose a new regression algorithm based on a combination
+of k-nearest neighbor regression and TNNR, which of course could be implemented
+for various baseline twinned regression algorithms. In standard twinned regression
+methods the model learns to predict differences between the targets of two arbitrary
+
+How to get the most out of Twinned Regression Methods
+10
+Figure 4: Effect of nearest neighbor pairing on TNNR measured in terms of RMSE vs the number
+of nearest neighbors used if applicable. The solid blue line marks the performance of the original
+TNNR. The dashed blue line displays the results of TNNR trained on all possible pairs while performing
+inference only using nearest neighbor anchors. The dashed orange line is produced by restricting pairs
+to nearest neighbors for training and inference.
+data points. This model is then employed to create an ensemble prediction via averaging
+the approximations of the differences between the target value of a new data point and
+all anchor data points, see (2). However, not all of these anchor data points might be of
+equal importance for the prediction. That is why in this section we restrict the anchor
+points to the nearest neighbors. For this purpose, we define the notation NN(i,m) as
+the set of m nearest neighbors of a data point xi ∈ X within the training set xi ∈ Xtrain
+to reformulate the prediction:
+ypred
+i
+= 1
+m
+∑
+j∈NN(i,m)
+(F(xi,xtrain
+j
+) + ytrain
+j
+)
+(4)
+While we have defined the prediction using nearest neighbors during the inference
+phase, it is an open question whether it is better to train the model to predict
+differences between target values of generic data points or between neighboring data
+points corresponding to the same number of nearest neighbors in the inference phase. The
+different training versions are compared in Fig. 4 where the baseline is set by standard
+
+Nearest Neighbor Training
+Bio Conservation
+Boston Housing
+Concrete Strength
+4.75
+5.4
+1.05
+4.50
+5.2
+1.00
+4.25
+5.0
+4.00
+4.8
+06'0
+3.75
+0.85
+3.50
+4.6
+3.25
+4.4
+0.80
+2
+4
+8
+16
+37
+64
+1
+4
+8
+16
+64
+8
+16
+32
+64
+Energy Efficiency
+RCL Circuit
+Test Function
+1.6 
+0.008
+0.030
+1.4
+0.007
+0.025
+0.006
+0.005
+1.0
+0.020
+0.004
+0.8
+E00'0
+0.015
+0.002
+8
+16
+32
+64
+2
+4
+8
+16
+32
+64 
+16
+32
+64
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+0.06
+all anchor training
+0.85
+1.0
+all training pairs
+0.05
+0.80
+nearest neighbor
+0.8
+training pairs
+0.75
+0.03
+0.6
+0.70
+0.02
+2
+A
+8
+16
+32
+64
+2
+4
+8
+16
+32
+64 
+2
+4
+8
+16
+32
+64
+# Nearest Neighbors
+# Nearest Neighbors
+# Nearest NeighborsHow to get the most out of Twinned Regression Methods
+11
+Figure 5: Comparison of k-NN regression and TNNR with different numbers of nearest neighbor
+training pairs measured by RMSE vs the number of neighbors for the dashed lines. The solid blue
+corresponds to normal TNNR with access to all possible pairs during training and inference. The blue
+dashed line restricts the pairs to be nearest neighbors during inference. The orange dashed line restricts
+the pairs to be nearest neighbors during training and inference. The green dashed line describes k-NN.
+TNNR. We emphasize that both versions of training obey the same principle for selecting
+nearest neighbors during inference. When only using the very nearest neighbor as an
+anchor for inference, we can see that for 7 out of 9 data sets both training versions
+underperform traditional TNNR, while training on all possible pairs performs better than
+just training on neighboring training data points. This picture changes as we increase
+the number of nearest neighbors. On all data sets both versions of nearest neighbor
+TNNR converge to standard TNNR in the limit of increasing the number of neighbors to
+the training set size. In 7 out of 9 data sets there is a sweet spot where nearest neighbor
+TNNR with nearest neighbor training outperforms at around 16 to 64 neighbors, in 3
+out of those data sets nearest neighbor training outperforms by a very large margin
+culminating in reducing the RMSE by ≈ 60% on the TF data set, see Table 3. We note
+that this is the data set with zero noise.
+
+NearestNeighborTrainingvsk-NN
+Bio Conservation
+Boston Housing
+Concrete Strength
+12
+1.0
+10
+RMSE
+0.9
+8
+0.8
+i
+16
+32
+64
+4
+64
+2
+8
+16
+32
+64
+Energy Efficiency
+RCL Circuit
+Test Function
+5
+0.25
+0.10
+4
+0.20
+0.08
+E
+0.15
+0.06
+RM
+0.10 
+0.04
+2
+0.05
+0.02
+0.00
+1
+4
+8
+16
+32
+64
+1
+2
+4
+8
+16
+32
+64
+2
+4
+8
+16
+32
+64
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+10
+0.20
+0.85
+8
+all anchor training
+0.15
+0.80
+all training pairs
+MSE
+nearest neighbor
+0.10
+0.75
+training pairs
+k-NN regression
+0.05
+0.70
+2
+4
+8
+16
+32
+64
+4
+8
+16
+32
+64
+2
+4
+8
+16
+32
+64
+# Nearest Neighbors
+# Nearest Neighbors
+# Nearest NeighborsHow to get the most out of Twinned Regression Methods
+12
+8. TNNR vs k-NN
+A natural question is how nearest-neighbor TNNR(NNTNNR) related to k-NN regression.
+Nearest neighbor TNNR can be related to k-NN regression through setting F(xi,xj) ≡ 0,
+then
+ypred
+i
+= 1
+m
+∑
+j∈NN(i,m)
+⎛
+⎜⎜
+⎝
+F(xi,xtrain
+j
+)
+�����������������������������������������������������������
+0
++ytrain
+j
+⎞
+⎟⎟
+⎠
+= 1
+m
+m
+∑
+j∈NN(i)
+ytrain
+j
+(5)
+Assuming F(xi,xj) would just be a minor contribution to k-NN regression we would see
+a qualitatively similar performance of NNTNNR. In order to test this statement, we
+visualize the behavior of k-NN and NNTNNR in Fig. 5. In this figure we can clearly see,
+that NNTNNR beats k-NN regression by an enormous margin on 7 out of 9 data sets.
+However, there are two data sets, namely BC, WN where k-NN is the winner. Again, we
+note that these data sets are exactly where the expected TNNR mechanism fails Fig. 3
+and it coincides with ANN ensembles outperforming TNNR Fig. 2. Further, the number
+of optimal TNNR anchors is much larger than the optimal number of neighbors in k-NN.
+9. Reformulation Benefits
+After having discussed the impacts of ensembling and the increased effective training
+set size, we have now finally the tools to partially answer the question of whether the
+reformulation to the dual problem itself contributes to the increased accuracy of twinned
+regression methods. We have related TNNR to normal ANNs in (3) and connected
+NNTNNR to k-NN regression in (5). If TNNR would be a glorified form of ANN or k-NN
+regression, the performance of TNNR could be related qualitatively to neural networks
+or k-NN. However, as we can see by comparing with ANNs in Fig. 2 or k-NN Fig. 5, it
+is clear that TNNR has a distinct performance profile that beats ANNs and k-NN on
+the same 7 of 9 data sets and underperforms both on the remaining two data sets. As
+we know from testing the impact of the increased training set size Fig. 3, these are the
+data sets where increasing the data sets has an adverse effect on accuracy, which signals
+that the TNNR mechanism fails while ANN and k-NN continue to perform normally.
+All these facts support the conclusion that the reformulation to a dual problem itself
+tends to have a positive effect on accuracy on most data sets.
+10. Miniaturizing accurate networks
+Often it is required to store fully trained neural networks on hardware that has strict
+memory limitations. This might be on chips that allow for autotuning of quantum
+dots [9] or in self driving vehicles [10]. In these cases it is required to consider the
+trade-off between accuracy and memory requirements. Ensembles of neural networks
+
+How to get the most out of Twinned Regression Methods
+13
+Figure 6: Memory requirements for storing real ensembles in terms of number of parameters stored vs
+the ensemble size. The architecture used corresponds to the architecture that was used to produce the
+results in this manuscript. Different architectures cause quantitatively different plots, but qualitatively
+the behave similarly. The solid lines indicate the memory requirements for storing an ensemble of
+independently trained ANN models for different numbers of features f describing each data instance.
+The dashed lines correspond to ensembles generated by storing one TNNR model and a representative
+number of anchors which can be combined to produce ensembles.
+tend to be more accurate than single neural networks, however, storing them requires
+linearly more memory capacity per each ensemble member. TNNR provides an elegant
+solution to this problem, because in order to store an ensemble of predictions it only
+requires the storage of a single set of trained weights and biases together with one
+anchor data point per ensemble member. Our neural network architectures are chosen
+such that they produce accurate results on all nine considered data sets. Thus, we
+have chosen an architecture with two hidden layers, each with 128 neurons. In Fig. 6
+we visualize the number of parameters that are required to be stored in the case of
+traditional ANN and TNNR ensembles for different feature sizes of the input data. It can
+be seen that in all cases TNNR parameters plus anchors need far less storage capacity
+to achieve a similar ensemble size as ANNs. Of course in practice the optimal neural
+network architecture and its number of parameters varies between problems, meaning
+our quantitative analysis of memory requirements might not generalize to other problems.
+However, the qualitative trend remains the same as long as the feature size is smaller
+than the number of parameters of the model.
+11. Making other models semi-supervised
+In this section we explore a simple framework to train any twinned regression method in
+a semi-supervised manner. The idea is based on the semi-supervised regression method
+devised in [2] for TNNR. As we can see in Fig. 1 the dual formulation requires machine
+learning models to predict differences F(xi,xj) = yi − yh between target values instead of
+the targets f(x) = y, themselves. One advantage of this formulation is that a correct
+
+Number of Parameters to store ensemble
+106
+ANN ensemble f=4
+ANN ensemble f=13
+ parameters
+ANN ensemble f=100
+TNN anchors f=4
+TNN anchors f=13
+103
+TNN anchors f=100
+#
+0
+5
+10
+15
+20
+25 
+30
+Ensemble sizeHow to get the most out of Twinned Regression Methods
+14
+Figure 7: Transductive semi-supervised learning with random forests. Random forests have been
+supplied with loops containing unlabelled data points. The magnitude of the influence the loops have
+on the training process is measured by Λ. Tuning Λ ≈ 1 reduces the RMSE on almost all data sets
+compared to purely supervised learning (solid blue line).
+Table 4: Best estimates for test RMSEs obtained by Semi-supervised random forests.
+Supervised RF
+Semi-Supervised RF
+Improvement
+Bio Conservation
+0.7367±0.0081
+0.7357±0.0078
+0.14%
+Boston Housing
+3.8408±0.1322
+3.8281±0.1356
+0.33%
+Concrete Strength
+5.8813±0.2242
+5.8440±0.2276
+0.63%
+Energy Efficiency
+1.9112±0.0417
+1.8919±0.0411
+1.01%
+RCL Circuit
+0.2934±0.0068
+0.2932±0.0068
+0.05%
+Test Function
+0.0857±0.0022
+0.0855±0.0021
+0.19%
+Wheatstone Bridge
+0.0958±0.0025
+0.0958±0.0022
+0.02%
+Red Wine Quality
+0.6066±0.0083
+0.6045±0.0084
+0.35%
+Yacht Hydrodynamics
+1.1138±0.0383
+1.1091±0.0384
+0.42%
+
+RandomForestSemi-SupervisedLearning
+Bio Conservation
+Boston Housing
+Concrete Strength
+0.742
+3.86
+5.88
+0.740
+3.85
+5.87
+RMSE
+0.738
+5.86
+3.84
+5.85
+0.736
+E8E
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+Energy Efficiency
+RCL Circuit
+Test Function
+1910
+0.302
+0.092
+1.905
+0.300
+MSE
+0.090
+0.298
+0.296
+0.088
+1.895
+0.294
+0.086
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+0.09725
+0.608
+115
+TNN
+0.09700
+-+ all training pairs
+0.09675
+0.607
+1.14
+ISE
+0.09650
+0.606
+1.13
+0.09625
+112
+0.09600
+0.605
+0.09575
+1.11
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+0.001
+0.010
+0.100
+1.000
+10.000
+Loop Consistency Weight A
+Loop Consistency Weight A
+Loop Consistency Weight ^How to get the most out of Twinned Regression Methods
+15
+solution would satisfy loop consistency F(x1,x2) + F(x2,x3) + F(x3,x1) = 0.
+Hence, we propose the following algorithm that is applicable to all twinned regression
+methods: At first we train the regression model on the labelled training data. This model
+is then used to predict the differences between targets along loops randomly sampled
+from an unlabelled data set. For each loop an adjustment a defined by
+a = F(x1,x2) + F(x2,x3) + F(x3,x1)
+(6)
+is then used to propose a label yij = F(xi,xj)−Λ×a for each combination of xi,xj within
+the loops. Here, Λ is the loop weight hyper-parameter. The unlabelled data set together
+with the proposed labels is then added to the labelled training set, on which the model
+is retrained. The algorithm is further depicted in Fig. 1.
+We apply this idea to the pairwise/twinned random forest regression proposed in [3],
+which was originally aimed at solving regression problems on small data sets in chemistry.
+Since random forests don’t scale as well with large data sets, we restrict our data sets to
+100 training and 100 test data points. The details of the training process are outlined in
+section Appendix D.
+Before applying the semi-supervised learning strategy, we convince ourselves that
+twinned random forest regression is suitable for the test bed consisting of the nine data
+sets (Appendix A) used in this paper. The corresponding results can be seen in Table D1.
+Twinned random forests perform equally well, or slightly worse, compared to traditional
+random forests on three data sets (BC,EE,WN). It moderately outperforms on four data
+sets (BH,CS,RCL,YH) and it massively outperforms by cutting the RMSE by more than
+35% on two data sets (TF,WSB).
+After having convinced ourselves of the superior performance of twinned random
+forests, we apply our semi-supervised learning framework in a transductive manner.
+Transductive means that the test data is used as unlabelled training data. This is in
+contrast to inductive semi-supervised learning where the unlabelled training data would
+be kept separate from the final test data. The final results are depicted in Fig. 7 for
+various choices of the loop weight Λ. We can clearly see, that the optimal choice of
+Λ ≈ 1 leads to a reduction of RMSE on six out of nine data sets. However, the relative
+improvement from semi-supervised learning is very small, as shown in Table 4 and most of
+the time less than 1%. If we compare these results with other semi-supervised regression
+algorithms on the same data sets from [2], one can observe that this improvement is
+significantly less than semi-supervised TNNR and slightly less than co-training with
+neural networks.
+12. Negative Results
+Let us briefly discuss in this section different ideas that we tried during our experiments
+but did not lead to a consistent improvement of twinned regression methods.
+While exploring the ideal weighting of anchor during the inference phase of twinned
+regression methods, a straightforward idea was to try incorporating the intrinsic
+
+How to get the most out of Twinned Regression Methods
+16
+Figure 8: Training and inference time comparison in seconds between different versions of TNNR and
+ANN regression. The solid orange line indicates the time for training and inference in the case of ANNs.
+The solid blue line is the same for original TNNR. The blue dotted line indicates restricting the possible
+training/inference pairs to nearest neighbors, the x-axis corresponds to the number of neighbors.
+uncertainty metrics [1]. These include the ensemble standard deviation and the violation
+of loop consistency. Anchors with a lower uncertainty metric should be weighted higher
+than anchors with high uncertainty metrics. While we observed some benefit, we could
+not consistently show that this process improved the accuracy in a statistically significant
+manner. We believe that other uncertainty metrics that unrelated to the intrinsic
+consistency metrics might be better suited as for example in Gaussian processes.
+Our initial plan when devising a strategy to adopt the semi-supervised learning
+framework from [2] to other algorithms was based on an iterative algorithm. After
+training the underlying twinned regression algorithm, the model would predict labels
+on unknown data points. Randomly sampling loops containing these data points would
+allow us to check for loop consistency. The unknown data points would then be added
+to the training data set with a label that corresponds to the original prediction slightly
+modified in the direction which satisfies the loop condition. The idea was to iterative
+refine the labels by repeating this process. However, it turned out that many times this
+process would either not converge for Λ > 1/3, or eventually converge to sub-optimal
+solutions, worse than the initial supervised version.
+Combining k-NN regression with TNNR was also aimed at reducing the
+
+Training+InferenceTimeComparison
+Bio Conservation
+Boston Housing
+Concrete Strength
+1750
+seconds
+2000
+1500
+6000
+1500
+1250
+1000
+4000
+Time
+1000
+750
+500
+2000
+500
+2
+4
+8
+16
+32
+64
+i
+2
+8
+16
+32
+64
+i
+4
+8
+16
+32
+64
+Energy Efficiency
+RCL Circuit
+Test Function
+6000
+25000
+3500
+seconds
+5000
+20000
+3000
+4000
+2500
+15000
+3000
+2000
+Time
+10000
+2000
+1500
+5000
+1000
+1000
+16
+32
+64
+2
+4
+8
+1
+:
+4
+8
+16
+32
+64
+1
+2
+4
+8
+16
+32
+64
+Wheatstone Bridge
+Red Wine Quality
+Yacht Hydrodynamics
+800
+5000
+seconds
+1200
+600
+4000
+1000
+ANN
+3000
+800
+TNN
+400
+nearest neighbor
+2000
+600
+training data
+200
+1000
+400
+200
+2
+4
+8
+16
+32
+t9
+1
+2
+4
+8
+16
+32
+64
+1
+2
+4
+8
+16
+32
+64
+# Nearest Neighbors
+# Nearest Neighbors
+# Nearest NeighborsHow to get the most out of Twinned Regression Methods
+17
+computational time. As explored in [1], the training time of twinned regression methods
+scales poorly towards larger data sets, mostly caused by the increase in the effective
+data set size through pairing of data points. While for many baseline algorithms a
+clear relationship between data set size and training time, for neural networks it is less
+known. As neural networks training time scales very favorably with training set size
+we focused on TNNR to test the training time improvement from only using nearest
+neighbor paring during training phase. In Fig. 8 we can see that there is a tendency for a
+reduced computational cost on most data sets. However, the training time scaling is too
+minor and inconsistent to use it as a sole justification to use NNTNNR over traditional
+TNNR.
+13. Conclusion
+Twinned regression is a simple and versatile framework to improve performance through
+intrinsic ensembling and semi-supervised learning on small to medium-sized data sets.
+In this article, we have answered several questions about the nature of the twinned
+regression framework. Further, we devised several improvements to further improve the
+already state-of-the-art performance of twinned regression methods.
+We compared the ensemble behaviour of traditional ANNs and TNNR. For this
+purpose we mapped single anchor TNNR to an equivalent ANN model during inference
+phase. By visualizing the results of these examinations in Fig. 2 we can see that the
+performance an ensemble of single anchor TNNs converges towards an ensemble of full
+anchor TNNs at around 32 ensemble members. Increasing the number of anchors would
+not yield any additional gain. This suggests that the intrinsic ensemble diversity of
+TNNR is a subset of the diversity that can be achieved by retraining the network for
+each anchor. The combined anchor+direct ensemling containing multiple TNNs causes a
+much larger improvement than the ensembling of traditional ANNs as can be seen in
+Table 1 and Table 2. This explains one element of the outperformance of TNNR over
+ANN regression.
+Further, we examined what effect the increased training set size through pairing
+data points has on the TNN accuracy, Fig. 3. Generally, more pairings per training data
+point reduced the RMSE. However, to our surprise not all data sets benefited from this
+pairing, on two data sets (BC,WN) TNNR had the lowest RMSE if only one pairing
+per training data point was allowed. By comparing this malfunction to results in Fig. 2
+and Fig. 5, we can see that it occurs in exactly the data sets where k-NN regression
+and ANN ensembles outperform TNNR, signaling a breakdown of the performance
+increasing factors of TNNR. By looking at the properties of the data sets it seems like
+twinned regression methods perform best on continuous data sets where the label can be
+approximated through a deterministic function.
+We also pointed at another advantage of TNNR in the case where it is impossible to
+store a large number of parameters, but one wants to retain the advantages of ensembles.
+TNNR provides the possibility to generate an ensemble of predictions just by storing
+
+How to get the most out of Twinned Regression Methods
+18
+one model and some anchors, which is usually significantly smaller than storing multiple
+ANN models, see Fig. 6.
+While exploring the ideal weighting of anchors during the inference phase of TNNR,
+we found that nearest-neighbor predictions tend to yield the most accurate results.
+This lead us to develop nearest-neighbor TNNR (NNTNNR) which is a combination
+of the k-nearest neighbor algorithm and TNNR. There are two versions of NNTNNR,
+one which respects nearest neighbors during both training and inference and another
+version that only restricts nearest neighbors during the inference phase. Restricting
+to nearest-neighbor training tends to yield slightly better results Fig. 4. Both versions
+outperform standard TNNR especially on low noise data sets, see Table 3. It is important
+to note that NNTNNR is not just a minor improvement to k-NN regression since it has
+a very different performance profile when it comes to varying the number of anchors, or
+nearest neighbors, respectively, as can be seen in Fig. 5.
+We devised a semi-supervised regression framework based on enforcing loop
+consistency that can be applied to any twinned regression algorithm, but we tested it for
+random forests. This method yielded a clearly visible improvement over their supervised
+counterparts as can be seen in Fig. 7. However, the magnitude of the reduction of the
+RMSE is relatively small and almost always less than 1%, see Table 4. Comparing these
+results with other semi-supervised regression algorithms on the same data sets from [2],
+we can see that this improvement is significantly less than semi-supervised TNNR and
+slightly less than co-training with neural networks.
+The code supporting this publication is available at [18].
+14. Acknowledgements
+Let us thank Zurab Jashi for his help with the random forest code. This work was
+supported by Mitacs and Homes Plus Magazine Inc. through the Mitacs Accelerate
+program. We also acknowledge Compute Canada for computational resources. We thank
+the National Research Council of Canada for their partnership with Perimeter on the
+PIQuIL. Research at Perimeter Institute is supported in part by the Government of
+Canada through the Department of Innovation, Science and Economic Development
+Canada and by the Province of Ontario through the Ministry of Economic Development,
+Job Creation and Trade.
+
+How to get the most out of Twinned Regression Methods
+19
+Appendix A. Data sets
+Table A1: Data sets
+Name
+Key
+Size
+Features
+Type
+Bio Concentration
+BC
+779
+14
+Discrete, Continuous
+Boston Housing
+BH
+506
+13
+Discrete, Continuous
+Concrete Strength
+CS
+1030
+8
+Continuous
+Energy Efficiency
+EF
+768
+8
+Discrete, Continuous
+RCL Circuit Current
+RCL
+4000
+6
+Continuous
+Test Function
+TF
+1000
+2
+Continuous
+Red Wine Quality
+WN
+1599
+11
+Discrete, Continuous
+Wheatstone Bridge Voltage
+WSB
+200
+4
+Continuous
+Yacht Hydrodynamics
+YH
+308
+6
+Discrete
+The test function (TF) data set created from the equation
+F(x1,x2) = x3
+1 + x2
+1 − x1 − 1 + x1x2 + sin(x2)
+(A.1)
+and zero noise.
+The output in the RCL circuit current data set (RCL) is the current through an
+RCL circuit, modeled by the equation
+I0 = V0 cos(ωt)/
+√
+R2 + (ωL − 1/(ωC))2
+(A.2)
+with added Gaussian noise of mean 0 and standard deviation 0.1.
+The output of the Wheatstone Bridge voltage (WSB) is the measured voltage given
+by the equation
+V = U(R2/(R1 + R2) − R3/(R2 + R3))
+(A.3)
+with added Gaussian noise of mean 0 and standard deviation 0.1.
+Appendix B. Bias-Variance Tradeoff and Ensembles
+In a regression problem, one is tasked with finding the true labels on yet unlabelled
+data points through the estimation of a function f(x) = y. Given a finite training data
+set D we denote this approximation ˆf(x;D). The expected mean squared error can
+be decomposed by three sources of error, bias error BiasD[ ˆf(x;D)] , variance error
+VarD [ ˆf(x;D)] and intrinsic error of the data set σ.
+MSE = Ex {BiasD[ ˆf(x;D)]2 + VarD [ ˆf(x;D)]} + σ2.
+(B.1)
+
+How to get the most out of Twinned Regression Methods
+20
+If we replace the estimator by an ensemble of two functions ˆf(x;D) = 1/2 ˆfA(x;D) +
+1/2 ˆfB(x;D), each exhibiting the same bias and variance as the original estimator, then
+we can decompose the MSE
+MSE = Ex {BiasD[1/2 ˆfA(x;D) + 1/2 ˆfB(x;D)]2 + VarD [1/2 ˆfA(x;D) + 1/2 ˆfB(x;D)]} + σ2
+(B.2)
+= Ex {BiasD[ ˆf(x;D)]2 + VarD [1/2 ˆfA(x;D)] + VarD [1/2 ˆfB(x;D)]
+(B.3)
++ 2CovD [1/2 ˆfA(x;D),1/2 ˆfB(x;D)]} + σ2
+(B.4)
+= Ex {BiasD[ ˆf(x;D)]2 + 1/2VarD [ ˆfA(x;D)] + 1/2CovD [ ˆfA(x;D), ˆfB(x;D)]} + σ2
+(B.5)
+The more uncorrelated ˆfA(x;D) and ˆfB(x;D) are, the smaller is the ratio between
+variance and covariance. Thus an ensemble consisting of weakly correlated ensemble
+members reduce the MSE by circumventing the bias-variance tradeoff. By induction this
+argument extends to larger ensemble sizes.
+Appendix C. Neural Network Architectures
+Both our traditional neural network regression and twin neural network regression
+methods are build using the same architecture build using the tensorflow library [19].
+They consist of two hidden layers with 128 neurons each and relu activation functions.
+The final layer contains one single neuron without an activation function. We train
+our neural networks using the adadelta optimizer, and use learning rate and early stop
+callbacks that reduce the learning rate by 50% or stop training if the loss stops decreasing.
+For this reason it is enough to set the number of epochs large enough such that the early
+stopping is always triggered, in our cases this is 2000 for ANNs and 10000 for TNNR.
+The batchsizes are in both cases 16.
+Appendix D. Random Forests
+The random forests in this article use the scikit-learn library [20]. They are trained
+on a subset of all data sets: from each data set, we randomly sample 100 training
+data points and 100 test data points. We use five-fold cross-validation to optimize
+the following hyper-parameters of our random forests: ’max depth’ ∈ [4,8,16,32,64],
+’max features’ ∈ [0.33,0.667,1.0], ’min samples leaf’ ∈ [1,2,5], ’min samples split’
+∈ [2,4,8], ’n estimators’ ∈ [100,300,600]. Both, the traditional and the twinned random
+forests choose their optimal hyper-parameters from the same pool. It is important to
+note that for semi-supervised learning the hyper-parameters are only optimized during
+
+How to get the most out of Twinned Regression Methods
+21
+Table D1: Best estimates for test RMSEs obtained by Random Forest compared to Twinned Random
+Forests
+Random Forest
+Twinned Random Forest
+Improvement
+Bio Conservation
+0.7407± 0.0087
+0.7427± 0.0081
+-0.27%
+Boston Housing
+4.0019± 0.1394
+3.8301± 0.1322
+4.29%
+Concrete Strength
+6.3763± 0.2136
+5.8519± 0.2242
+8.22%
+Energy Efficiency
+1.8773± 0.0422
+1.8906± 0.0417
+-0.71%
+RCL Circuit
+0.3168± 0.0071
+0.2958± 0.0068
+6.63%
+Test Function
+0.1402± 0.0044
+0.0874± 0.0023
+37.66%
+Wheatstone Bridge
+0.1461± 0.0039
+0.0942± 0.0025
+35.52%
+Red Wine Quality
+0.5989± 0.0085
+0.6068± 0.0083
+-1.32%
+Yacht Hydrodynamics
+1.1917± 0.029
+1.1117± 0.0383
+6.71%
+the initial supervised learning step, the optimal parameters are then carried forward to
+be used during semi-supervised learning.
+Algorithm 1: Semi-Supervised Learning through Loop Consistency
+Data: Labelled data set DL = (Xtrain,L,Ytrain,L)
+Unlabelled data set DU = (Xtrain,U)
+Input: Loop weight Λ
+Loop number nl =length(DL)/3
+1 create ˜DL = [((xi,xj,xi − xj),yij = yi − yj) for xi ∈ Xtrain,L for xj ∈ Xtrain,L]
+2 initialize machine learning model M
+3 train M on ˜DL
+4 sample nl loops L=[(xi,xj,xk) where xi ∈ Xtrain,L, xj,xk ∈ Xtrain,U]
+5 for (xi,xj,xk) ∈ L do
+6
+predict M(xi,xj),M(xj,xk),M(xk,xi)
+7
+a = M(xi,xj) + M(xj,xk) + M(xk,xi)
+8
+(yij,yjk,yki) = (M(xi,xj) − Λa,M(xj,xk) − Λa,M(xk,xi) − Λa)
+9
+add ((xi,xj,xi − xj),yij),((xj,xk,xj − xk),yjk),((xk,xi,xk − xi),yki) to ˜DL
+10 train M on ˜DL
+Output: Trained Model M
+
+How to get the most out of Twinned Regression Methods
+22
+[1] Wetzel S J, Ryczko K, Melko R G and Tamblyn I 2022 Applied AI Letters e78
+[2] Wetzel S J, Melko R G and Tamblyn I 2022 Machine Learning: Science and Technology 3 045007
+[3] Tynes M, Gao W, Burrill D J, Batista E R, Perez D, Yang P and Lubbers N 2021 Journal of
+Chemical Information and Modeling 61 3846–3857
+[4] Baldominos A, Blanco I, Moreno A J, Iturrarte R, Bern´ardez ´O and Afonso C 2018 Applied sciences
+8 2321
+[5] Rafiei M H and Adeli H 2016 Journal of Construction Engineering and Management 142 04015066
+[6] Yu Y, Lu J, Shen D and Chen B 2021 Neural Computing and Applications 33 3925–3937
+[7] Ryczko K, Wetzel S J, Melko R G and Tamblyn I 2022 Journal of Chemical Theory and Computation
+18 1122–1128
+[8] Avula N V, Veesam S K, Behera S and Balasubramanian S 2022 Machine Learning: Science and
+Technology
+[9] Czischek S, Yon V, Genest M A, Roux M A, Rochette S, Lemyre J C, Moras M, Pioro-Ladri`ere M,
+Drouin D, Beilliard Y et al. 2021 Machine Learning: Science and Technology 3 015001
+[10] Lechner M, Hasani R M and Grosu R 2018 arXiv preprint arXiv:1803.08554
+[11] Bromley J, Guyon I, LeCun Y, S¨ackinger E and Shah R 1993 Advances in neural information
+processing systems 6 737–744
+[12] Baldi P and Chauvin Y 1993 neural computation 5 402–418
+[13] Srivastava N, Hinton G, Krizhevsky A, Sutskever I and Salakhutdinov R 2014 The journal of
+machine learning research 15 1929–1958
+[14] Wan L, Zeiler M, Zhang S, Le Cun Y and Fergus R 2013 Regularization of neural networks using
+dropconnect International conference on machine learning (PMLR) pp 1058–1066
+[15] Wu J 2009 A novel artificial neural network ensemble model based on k–nearest neighbor
+nonparametric estimation of regression function and its application for rainfall forecasting
+2009 international joint conference on computational sciences and optimization vol 2 (IEEE) pp
+44–48
+[16] Bensaci R, Khaldi B, Aiadi O and Benchabana A 2021 Applied Sciences 11 10176
+[17] Liu Z, Guo J, Cao J, Wei Y and Huang W 2018 Promet-Traffic&Transportation 30 445–456
+[18] Wetzel
+S
+2023
+Public
+github
+repository
+URL
+https://github.com/sjwetzel/
+PublicGetMostOutOfTNNR
+[19] Abadi M, Agarwal A, Barham P, Brevdo E, Chen Z, Citro C, Corrado G S, Davis A, Dean J,
+Devin M, Ghemawat S, Goodfellow I, Harp A, Irving G, Isard M, Jia Y, Jozefowicz R, Kaiser
+L, Kudlur M, Levenberg J, Man´e D, Monga R, Moore S, Murray D, Olah C, Schuster M,
+Shlens J, Steiner B, Sutskever I, Talwar K, Tucker P, Vanhoucke V, Vasudevan V, Vi´egas F,
+Vinyals O, Warden P, Wattenberg M, Wicke M, Yu Y and Zheng X 2015 TensorFlow: Large-
+scale machine learning on heterogeneous systems software available from tensorflow.org URL
+https://www.tensorflow.org/
+[20] Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer
+P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M and
+Duchesnay E 2011 Journal of Machine Learning Research 12 2825–2830
+
diff --git a/itAzT4oBgHgl3EQfbPyr/content/tmp_files/load_file.txt b/itAzT4oBgHgl3EQfbPyr/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ea6bd3d4e0495a0ea123cf55f865b61e9c505c20
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+++ b/itAzT4oBgHgl3EQfbPyr/content/tmp_files/load_file.txt
@@ -0,0 +1,1135 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf,len=1134
+page_content='How to get the most out of Twinned Regression  Methods  Sebastian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Wetzel  University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada  Homes Plus Magazine Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=', Waterloo, Ontario N2V 2B1, Canada  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Twinned regression methods are designed to solve the dual problem to  the original regression problem, predicting differences between regression targets rather  then the targets themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' A solution to the original regression problem can be  obtained by ensembling predicted differences between the targets of an unknown data  point and multiple known anchor data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We explore different aspects of twinned  regression methods: (1) We decompose different steps in twinned regression algorithms  and examine their contributions to the final performance, (2) We examine the intrinsic  ensemble quality, (3) We combine twin neural network regression with k-nearest neighbor  regression to design a more accurate and efficient regression method, and (4) we develop  a simplified semi-supervised regression scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Keywords: Artificial Neural Networks, k-Nearest Neighbors, Random Forests, Regression,  Semi-Supervised Learning  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='01383v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='LG]  3 Jan 2023  How to get the most out of Twinned Regression Methods  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Introduction  Regression is one of the most general and common machine-learning tasks, practitioners  in many different fields of science and industry rely on methods that help them to make  the most accurate and reliable predictions on new data points inferred from a limited  amount of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Conventional regression methods aim to infer the mapping  of input features to one or multiple target variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Twinned regression methods aim  to solve the dual problem of predicting the difference between target values, a solution  to the original regression problem can then be obtained by evaluating the predictions  between a new unknown data point and multiple anchor data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This process creates  an ensemble of predictions from a single trained model, which tends to be more accurate  than solving the original regression problem directly and opens up semi-supervised  learning and uncertainty estimates for a very low cost [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' However, the trade-off  is the poor scaling with large data sets, since the effective training data set size scales  quadratically with the size of the original training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Thus, these methods are beneficial in domains where data is either scarce or costly  to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This is the case in for example real estate where markets differ from city to city  and data becomes outdated quickly [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Another example stems from calculations  in chemistry where simulations of chemical systems based on quantum mechanics require  an enormous amount of computational resources [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  This article is meant to be a practitioner’s guide to using twinned regression methods  that guides the reader through advantages and trade-offs and attempts to answer most  questions that were left open in the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The main question on our mind is why can such a simple trick of solving the dual  problem yield a more accurate prediction than solving the regression problem directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  While we do not fully answer this question, we decomposed twin neural network regression  (TNNR) into different steps each with the potential to enhance the performance over  traditional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' These include increased effective data set size obtained though  pairing training data points, or different ensembles of TNNR predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Further, by  mapping extreme cases of TNNR to k-nearest neighbor (k-NN) regression and normal  artificial neural networks (ANN) we can observe a distinct performance behaviour of  twinned regression methods different from traditional regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Further, we are eager to improve the accuracy of twinned regression methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' For  this purpose, we devise an improvement to TNNR based on the idea of weighting the  predictions from different anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This leads us to combine k-nearest neighbors (kNN)  with TNNR to an even more accurate regression scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The semi-supervised regression framework for TNNR invented in [2] is specifically  tailored to neural network-based regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' It is based on enforcing consistency conditions  on unknown data points through a modified loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' At the end of this manuscript  we examine a way to translate a simplified version of this semi-supervised learning scheme  to a twinned version of random forests (RF) proposed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  In some projects it is important to apply neural networks with strong memory  How to get the most out of Twinned Regression Methods  3  constraints, this might be the case in small chips or autonomous systems [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In  these cases it would normally be very inefficient to store ensembles of machine learning  models due to the increased number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In contrast to that, with twinned  regression methods, one only needs to store additional anchor data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Prior Work  The pairwise comparison inherent to twinned regression methods is inspired by Siamese  neural networks which were devised to solve the similarity classification problem as  it occurs in fingerprint recognition or signature verification [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Siamese neural  networks contain two identical neural networks with shared weights which project a  pair of inputs into a latent space on which the pairwise similarity is determined by the  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Twinned regression methods also take a pair of inputs to predict the difference  between the labels [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Twin neural network regression [1] was invented as a regression method that solves  the dual problem of predicting pairwise differences between the target values of pairs of  input data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Independently, the same idea has been developed for random forests  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This kind of regression framework has been shown to have several advantages: (1) it  allows for a very efficient generation of ensemble predictions [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Typically in methods  that generate ensembles from training a single machine learning model, the predictions  are strongly correlated [13, 14] since they can be deformed into each other through small  perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In twinned regression methods however, ensemble members are separated  by the distance of the input data points themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' (2) Twinned regression methods  tend to be more accurate than the underlying base algorithm on many data sets [1, 3],  (3) consistency conditions allow for the formulation of uncertainty estimators in addition  to the ensemble variance [1, 3] and (4) loops containing unlabelled data points can be  supplied while training, hence turning the method into a semi-supervised regression  algorithm [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Further, (5) the intrinsic uncertainty estimation lends itself for active  learning [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  A central contribution of this article is the combination of twin neural network  regression and k-NN regression to increase the accuracy of over standard twin neural  network regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Similarly, artificial neural networks have been employed in tandem  with k-NN regression in different contexts before [15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Reformulation of the Regression Problem  A regression problem can be formulated as follows: Given a labelled training data  set of n data points Xtrain = (xtrain  1  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=',xtrain  n  ) with their corresponding target values  Y train = (ytrain  1  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=',ytrain  m  ), we are tasked to find a function f such that the deviation  between f(xi) and yi is minimized with respect to a predefined objective function for all  data points xi on the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In this work, this function is the root mean square  error LRMSE =  √  ∑n  i=1(f(xi) − yi)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Unless stated otherwise, all performance measures  How to get the most out of Twinned Regression Methods  4  x  y=f(x)  x1  x2  y2-y1=F(x2,x1)  F(x1,x2)  f(x2)  F(x2,x3)  F(x3,x1)  f(x3)  f(x1)  Traditional Regression  Twinned Regression  Figure 1: Dual formulation of a regression problem: A traditional solution to a regression problem  consists of finding an approximation to the function that maps a data point x to its target value f(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Twinned regression methods solve the dual problem of mapping a pair of inputs x1 and x2 to the  difference between the target values F(x2,x1) = y2 − y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The resulting function can then be employed  as an estimator for the original regression problem y2 = F(x2,x1) + y1 given a labelled anchor point  (x1,y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Twinned regression methods must satisfy loop consistency: predictions along each loop sum to  zero: F(x1,x2) + F(x2,x3) + F(x3,x1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  are evaluated on unknown test data (Xtest,Y test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Twinned regression methods aim to solve a reformulation of the original regression  problem which is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' For each pair of data points (xtrain  i  ,xtrain  j  ) we  train a regression model to find a function F to predict the difference  F(xi,xj) = yi − yj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  (1)  This function F can be used to construct a solution to the original regression problem  via ypred  i  = F(xi,xj) + yj, where (xj,yj) is an anchor whose target value is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Every  training data point xtrain  j  ∈ Xtrain can be used as such an anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' A more accurate  estimate for the solution of the original regression problem is obtained by averaging over  many differences between a fixed unknown data point and different anchor data points  ypred  i  = 1  n  n  ∑  j=1  (F(xi,xtrain  j  ) + ytrain  j  )  = 1  n  n  ∑  j=1  (1  2F(xi,xtrain  j  ) − 1  2F(xtrain  j  ,xi) + ytrain  j  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  (2)  The increase in accuracy is based on averaging out the noise from different anchors and  the reduction of the variance error via an ensemble of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Previous works [1, 3]  recommended using the whole training data set as anchors, hence creating an ensemble  of difference predictions yi − yj which is twice as large as the training set for every single  prediction of yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  A major advantage of the dual formulation is the description via loops containing  multiple data points as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In contrast to traditional regression, the  How to get the most out of Twinned Regression Methods  5  results of twinned regression methods need to satisfy consistency conditions, for example  for each three data points x1,x2,x3, summing up the predictions along a closed loop  should yield zero: F(x1,x2) + F(x2,x3) + F(x3,x1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' During inference, violations of  these consistency conditions give rise to uncertainty estimates [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Enforcing loop  consistency on predictions involving unlabelled data points in the training phase is what  makes twinned regression methods into semi-supervised regression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  While neural networks are naturally good learners of linear functions this is not the  case for other algorithms like random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' For this reason, [3] proposed to augment  the input features by their difference (xi,xj) → (xi,xj,xi − xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' One might argue that  this improvement is similar to common data augmentation, however, this is a feature  that traditional machine learning algorithms don’t have access to, because it requires  two different data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Notes About Experiments  All experiments in this article are performed on the data sets outlined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Since only neural network based methods scale favorably with the data set size, they use  the full data sets of which 70% are used for training, 10% as validation set and 20% as  test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The details of the neural network architectures can be found in the appendix  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Random forests and especially twinned random forests scale poorly with  the data set size thus only 100 training data points are chosen from the data sets and  100 data points comprise the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Random forests do not need validation sets, since  the hyper=parameters are optimized via 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In section Appendix D  one can find the details about our random forest implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' All experiments are  repeated for 25 random but fixed splits of training, test, and if applicable, validation  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Ensemble Performance  Twinned regression methods have been shown to produce accurate solutions to regression  problems [1, 3], comparable to or better than other current state-of-the-art algorithms at  the cost of scaling poorly towards larger data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This naturally leads to the question of  where the increased performance stems from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The reformulation of a regression problem  into its dual problem of predicting differences between target values opens up several  potential reasons for improved accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' These include increased effective training set  size, internal ensembling of predictions (see explanation in Appendix B), or the nature of  solving a different problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In the following we examine these reasons at the example of  TNNR, however, we assume the answers will also be valid for other baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Let us start with discussing different kinds of ensembles and their effect on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 2 contains the results of several experiments examining the performance of different  ensemble types of ANN regression and TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' For each data set, the baseline results are  the solid blue horizontal line, which represents the test RMSE after applying standard  How to get the most out of Twinned Regression Methods  6  Figure 2: Comparison of different ensembles of ANNs and TNNR measured by RMSE vs number of  ensemble members for the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Ensembles can be created either by training several models or  evaluating TNNR for different anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The blue solid line corresponds to training and evaluating one  TNNR model on all possible training pairs and predicting the results with all possible anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The  dashed blue line varies the number of anchors during the inference phase and converges to the solid  blue line in the limit of increasing the inference anchors to the full training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The dashed orange line  indicates traditional ANN ensembles where multiple ANNs are trained independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The dashed green  line corresponds to ensembles of independently trained TNNR models, if the ensemble size is 1, this  is equivalent to the solid blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The dashed red line corresponds to independently trained TNNR  models each having only one inference anchor per prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  full anchor TNNR and the leftmost point of the orange line which represents the results  of applying a single ANN, confirming that TNNR almost always yields a lower RMSE  than ANN regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  In order to compare traditional ANNs with TNNR, we observe that after training we  can map TNNR to an ANN for each anchor, since both ANN regression and TNNR use  the same internal architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' For each fixed xj ∈ Xtrain an ANN ˜f is defined through  ˜fj(xi) ∶= F(xi,xj) + yj  (3)  The features of xj modify the weights of ˜fj while yj is absorbed by the output neuron  inside its bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' At that point, the only difference between each ˜fj and an equivalent  ANN is the procedure with which the weights were optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='29%  This gives us access to a framework to directly compare ensembles of ANNs and  the implicit ensembles generated by TNNR using multiple anchors during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We  examine the results of these both models through the orange and blue dashed lines in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' While both curves reduce the RMSE as we increase the ensemble size, we come  to the sobering conclusion that TNNR ensembles and ANN ensembles are not equivalent,  they neither have a uniform slope nor do they converge to similar RMSEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  We have just used a single trained TNNR model for all ensemble members while  training each ANN model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' What happens if we retrain TNNR for each  single anchor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The results of these experiments are visualized in the red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Since retraining increases the ensemble diversity the red line is consistently below the blue  How to get the most out of Twinned Regression Methods  8  Figure 3: How many different pairings are used for training effects the performance of TNNR measured  by RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The blue solid line corresponds to training TNNR on all possible training pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' It is  compared to training TNNR on a randomly chosen but fixed data set of pairs while still employing all  training data points as inference anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' A training set multiplier indicates on how many pairs are  taken into account compared to the original unpaired training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Further, we can see that the performance of these independently trained TNNRs  increases faster with the number of anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' If the resources are available one can further  create an ensemble of different TNNR models each having access to all anchors depicted  in the green line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' On seven out of nine data sets this yields clearly the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  We note that one-anchor TNNR with retraining (red line) converges towards the green for  more than 32 ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This tells us that the full ensemble diversity through  multiple anchors and multiple models can be captured independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  A more quantitative version of the out-performance of TNNR ensembles can be  seen in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The magnitude of the % improvement of the combined  anchor+direct ensembling containing multiple TNNs causes a much larger improvement  than the ensembling of traditional ANNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Effective Training Set Size  Another improvement over a traditional regression analysis is the increased training set  size that comes from preparing training sets by pairing each training data point with  TrainingSetMultiplier  Bio Conservation  Boston Housing  Concrete Strength  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content=' This transforms a training set of size n into a pairwise  training set of size n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In this section, we measure if the increase in the number of  pairings leads to an increase in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  To address this question we look at several curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In this figure, we  compare the effect of increasing the effective pairwise training data set on the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  For this purpose, we define the training set multiplier as the number of pairs that are  created from the original training set to produce the paired training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' A training  set multiplier of one means that each training data point is paired with only one other  randomly chosen (without replacement) but fixed data point (on average this means each  training data point is used twice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Increasing the training set multiplier to the size of the  training set converges to the standard formulation of twinned regression methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We  can see that in all data sets, except two, increasing the training set multiplier increases  the performance of TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' More precisely, a training set multiplier of ≈ 8 − 16 seems  already to be enough to reach the accuracy of standard TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' It is important to  note, that on two data sets, namely Bio Conservation(BC) and Red Wine Quality(WN),  increasing the training set multiplier has the effect of reducing the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This  coincides with other algorithms beating TNNR (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 2,Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 5) and is a sign that TNNR  might not be suitable for such regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' A data scientist using TNNR might do  a training set multiplier check, if he finds a decreasing accuracy while increasing the  multiplier, he can reject TNNR as optimal regression algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Nearest Neighbor TNNR  In this manuscript we propose a new regression algorithm based on a combination  of k-nearest neighbor regression and TNNR, which of course could be implemented  for various baseline twinned regression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In standard twinned regression  methods the model learns to predict differences between the targets of two arbitrary  How to get the most out of Twinned Regression Methods  10  Figure 4: Effect of nearest neighbor pairing on TNNR measured in terms of RMSE vs the number  of nearest neighbors used if applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The solid blue line marks the performance of the original  TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The dashed blue line displays the results of TNNR trained on all possible pairs while performing  inference only using nearest neighbor anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The dashed orange line is produced by restricting pairs  to nearest neighbors for training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This model is then employed to create an ensemble prediction via averaging  the approximations of the differences between the target value of a new data point and  all anchor data points, see (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' However, not all of these anchor data points might be of  equal importance for the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' That is why in this section we restrict the anchor  points to the nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' For this purpose,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' we define the notation NN(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='m) as  the set of m nearest neighbors of a data point xi ∈ X within the training set xi ∈ Xtrain  to reformulate the prediction:  ypred  i  = 1  m  ∑  j∈NN(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='m)  (F(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xtrain  j  ) + ytrain  j  )  (4)  While we have defined the prediction using nearest neighbors during the inference  phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' it is an open question whether it is better to train the model to predict  differences between target values of generic data points or between neighboring data  points corresponding to the same number of nearest neighbors in the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The  different training versions are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 4 where the baseline is set by standard  Nearest Neighbor Training  Bio Conservation  Boston Housing  Concrete Strength  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='06  all anchor training  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='0  all training pairs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='80  nearest neighbor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='8  training pairs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='02  2  A  8  16  32  64  2  4  8  16  32  64  2  4  8  16  32  64  # Nearest Neighbors  # Nearest Neighbors  # Nearest NeighborsHow to get the most out of Twinned Regression Methods  11  Figure 5: Comparison of k-NN regression and TNNR with different numbers of nearest neighbor  training pairs measured by RMSE vs the number of neighbors for the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The solid blue  corresponds to normal TNNR with access to all possible pairs during training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The blue  dashed line restricts the pairs to be nearest neighbors during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The orange dashed line restricts  the pairs to be nearest neighbors during training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The green dashed line describes k-NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We emphasize that both versions of training obey the same principle for selecting  nearest neighbors during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' When only using the very nearest neighbor as an  anchor for inference, we can see that for 7 out of 9 data sets both training versions  underperform traditional TNNR, while training on all possible pairs performs better than  just training on neighboring training data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This picture changes as we increase  the number of nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' On all data sets both versions of nearest neighbor  TNNR converge to standard TNNR in the limit of increasing the number of neighbors to  the training set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In 7 out of 9 data sets there is a sweet spot where nearest neighbor  TNNR with nearest neighbor training outperforms at around 16 to 64 neighbors, in 3  out of those data sets nearest neighbor training outperforms by a very large margin  culminating in reducing the RMSE by ≈ 60% on the TF data set, see Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We note  that this is the data set with zero noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  NearestNeighborTrainingvsk-NN  Bio Conservation  Boston Housing  Concrete Strength  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='0  10  RMSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='9  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='8  i  16  32  64  4  64  2  8  16  32  64  Energy Efficiency  RCL Circuit  Test Function  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='10  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='08  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='06  RM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='04  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='00  1  4  8  16  32  64  1  2  4  8  16  32  64  2  4  8  16  32  64  Wheatstone Bridge  Red Wine Quality  Yacht Hydrodynamics  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='85  8  all anchor training  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='80  all training pairs  MSE  nearest neighbor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='75  training pairs  k-NN regression  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='70  2  4  8  16  32  64  4  8  16  32  64  2  4  8  16  32  64  # Nearest Neighbors  # Nearest Neighbors  # Nearest NeighborsHow to get the most out of Twinned Regression Methods  12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' TNNR vs k-NN  A natural question is how nearest-neighbor TNNR(NNTNNR) related to k-NN regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Nearest neighbor TNNR can be related to k-NN regression through setting F(xi,xj) ≡ 0,  then  ypred  i  = 1  m  ∑  j∈NN(i,m)  ⎛  ⎜⎜  ⎝  F(xi,xtrain  j  )  �����������������������������������������������������������  0  +ytrain  j  ⎞  ⎟⎟  ⎠  = 1  m  m  ∑  j∈NN(i)  ytrain  j  (5)  Assuming F(xi,xj) would just be a minor contribution to k-NN regression we would see  a qualitatively similar performance of NNTNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In order to test this statement, we  visualize the behavior of k-NN and NNTNNR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In this figure we can clearly see,  that NNTNNR beats k-NN regression by an enormous margin on 7 out of 9 data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  However, there are two data sets, namely BC, WN where k-NN is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Again, we  note that these data sets are exactly where the expected TNNR mechanism fails Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 3  and it coincides with ANN ensembles outperforming TNNR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Further, the number  of optimal TNNR anchors is much larger than the optimal number of neighbors in k-NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Reformulation Benefits  After having discussed the impacts of ensembling and the increased effective training  set size, we have now finally the tools to partially answer the question of whether the  reformulation to the dual problem itself contributes to the increased accuracy of twinned  regression methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We have related TNNR to normal ANNs in (3) and connected  NNTNNR to k-NN regression in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' If TNNR would be a glorified form of ANN or k-NN  regression, the performance of TNNR could be related qualitatively to neural networks  or k-NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' However, as we can see by comparing with ANNs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 2 or k-NN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 5, it  is clear that TNNR has a distinct performance profile that beats ANNs and k-NN on  the same 7 of 9 data sets and underperforms both on the remaining two data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' As  we know from testing the impact of the increased training set size Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 3, these are the  data sets where increasing the data sets has an adverse effect on accuracy, which signals  that the TNNR mechanism fails while ANN and k-NN continue to perform normally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  All these facts support the conclusion that the reformulation to a dual problem itself  tends to have a positive effect on accuracy on most data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Miniaturizing accurate networks  Often it is required to store fully trained neural networks on hardware that has strict  memory limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This might be on chips that allow for autotuning of quantum  dots [9] or in self driving vehicles [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In these cases it is required to consider the  trade-off between accuracy and memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Ensembles of neural networks  How to get the most out of Twinned Regression Methods  13  Figure 6: Memory requirements for storing real ensembles in terms of number of parameters stored vs  the ensemble size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The architecture used corresponds to the architecture that was used to produce the  results in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Different architectures cause quantitatively different plots, but qualitatively  the behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The solid lines indicate the memory requirements for storing an ensemble of  independently trained ANN models for different numbers of features f describing each data instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The dashed lines correspond to ensembles generated by storing one TNNR model and a representative  number of anchors which can be combined to produce ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  tend to be more accurate than single neural networks, however, storing them requires  linearly more memory capacity per each ensemble member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' TNNR provides an elegant  solution to this problem, because in order to store an ensemble of predictions it only  requires the storage of a single set of trained weights and biases together with one  anchor data point per ensemble member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Our neural network architectures are chosen  such that they produce accurate results on all nine considered data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Thus, we  have chosen an architecture with two hidden layers, each with 128 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 6  we visualize the number of parameters that are required to be stored in the case of  traditional ANN and TNNR ensembles for different feature sizes of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' It can  be seen that in all cases TNNR parameters plus anchors need far less storage capacity  to achieve a similar ensemble size as ANNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Of course in practice the optimal neural  network architecture and its number of parameters varies between problems, meaning  our quantitative analysis of memory requirements might not generalize to other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  However, the qualitative trend remains the same as long as the feature size is smaller  than the number of parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Making other models semi-supervised  In this section we explore a simple framework to train any twinned regression method in  a semi-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The idea is based on the semi-supervised regression method  devised in [2] for TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' As we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 1 the dual formulation requires machine  learning models to predict differences F(xi,xj) = yi − yh between target values instead of  the targets f(x) = y, themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' One advantage of this formulation is that a correct  Number of Parameters to store ensemble  106  ANN ensemble f=4  ANN ensemble f=13  parameters  ANN ensemble f=100  TNN anchors f=4  TNN anchors f=13  103  TNN anchors f=100  #  0  5  10  15  20  25  30  Ensemble sizeHow to get the most out of Twinned Regression Methods  14  Figure 7: Transductive semi-supervised learning with random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Random forests have been  supplied with loops containing unlabelled data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The magnitude of the influence the loops have  on the training process is measured by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Tuning Λ ≈ 1 reduces the RMSE on almost all data sets  compared to purely supervised learning (solid blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Table 4: Best estimates for test RMSEs obtained by Semi-supervised random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='000  Loop Consistency Weight A  Loop Consistency Weight A  Loop Consistency Weight ^How to get the most out of Twinned Regression Methods  15  solution would satisfy loop consistency F(x1,x2) + F(x2,x3) + F(x3,x1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Hence, we propose the following algorithm that is applicable to all twinned regression  methods: At first we train the regression model on the labelled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This model  is then used to predict the differences between targets along loops randomly sampled  from an unlabelled data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' For each loop an adjustment a defined by  a = F(x1,x2) + F(x2,x3) + F(x3,x1)  (6)  is then used to propose a label yij = F(xi,xj)−Λ×a for each combination of xi,xj within  the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Here, Λ is the loop weight hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The unlabelled data set together  with the proposed labels is then added to the labelled training set, on which the model  is retrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The algorithm is further depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  We apply this idea to the pairwise/twinned random forest regression proposed in [3],  which was originally aimed at solving regression problems on small data sets in chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Since random forests don’t scale as well with large data sets, we restrict our data sets to  100 training and 100 test data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The details of the training process are outlined in  section Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Before applying the semi-supervised learning strategy, we convince ourselves that  twinned random forest regression is suitable for the test bed consisting of the nine data  sets (Appendix A) used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The corresponding results can be seen in Table D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Twinned random forests perform equally well, or slightly worse, compared to traditional  random forests on three data sets (BC,EE,WN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' It moderately outperforms on four data  sets (BH,CS,RCL,YH) and it massively outperforms by cutting the RMSE by more than  35% on two data sets (TF,WSB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  After having convinced ourselves of the superior performance of twinned random  forests, we apply our semi-supervised learning framework in a transductive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Transductive means that the test data is used as unlabelled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This is in  contrast to inductive semi-supervised learning where the unlabelled training data would  be kept separate from the final test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The final results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 7 for  various choices of the loop weight Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We can clearly see, that the optimal choice of  Λ ≈ 1 leads to a reduction of RMSE on six out of nine data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' However, the relative  improvement from semi-supervised learning is very small, as shown in Table 4 and most of  the time less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' If we compare these results with other semi-supervised regression  algorithms on the same data sets from [2], one can observe that this improvement is  significantly less than semi-supervised TNNR and slightly less than co-training with  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Negative Results  Let us briefly discuss in this section different ideas that we tried during our experiments  but did not lead to a consistent improvement of twinned regression methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  While exploring the ideal weighting of anchor during the inference phase of twinned  regression methods, a straightforward idea was to try incorporating the intrinsic  How to get the most out of Twinned Regression Methods  16  Figure 8: Training and inference time comparison in seconds between different versions of TNNR and  ANN regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The solid orange line indicates the time for training and inference in the case of ANNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The solid blue line is the same for original TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The blue dotted line indicates restricting the possible  training/inference pairs to nearest neighbors, the x-axis corresponds to the number of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  uncertainty metrics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' These include the ensemble standard deviation and the violation  of loop consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Anchors with a lower uncertainty metric should be weighted higher  than anchors with high uncertainty metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' While we observed some benefit, we could  not consistently show that this process improved the accuracy in a statistically significant  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We believe that other uncertainty metrics that unrelated to the intrinsic  consistency metrics might be better suited as for example in Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Our initial plan when devising a strategy to adopt the semi-supervised learning  framework from [2] to other algorithms was based on an iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' After  training the underlying twinned regression algorithm, the model would predict labels  on unknown data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Randomly sampling loops containing these data points would  allow us to check for loop consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The unknown data points would then be added  to the training data set with a label that corresponds to the original prediction slightly  modified in the direction which satisfies the loop condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The idea was to iterative  refine the labels by repeating this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' However, it turned out that many times this  process would either not converge for Λ > 1/3, or eventually converge to sub-optimal  solutions, worse than the initial supervised version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='t9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='# Nearest Neighbors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='# Nearest Neighbors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='# Nearest NeighborsHow to get the most out of Twinned Regression Methods  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' As explored in [1], the training time of twinned regression methods  scales poorly towards larger data sets, mostly caused by the increase in the effective  data set size through pairing of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' While for many baseline algorithms a  clear relationship between data set size and training time, for neural networks it is less  known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' As neural networks training time scales very favorably with training set size  we focused on TNNR to test the training time improvement from only using nearest  neighbor paring during training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 8 we can see that there is a tendency for a  reduced computational cost on most data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' However, the training time scaling is too  minor and inconsistent to use it as a sole justification to use NNTNNR over traditional  TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Conclusion  Twinned regression is a simple and versatile framework to improve performance through  intrinsic ensembling and semi-supervised learning on small to medium-sized data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  In this article, we have answered several questions about the nature of the twinned  regression framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Further, we devised several improvements to further improve the  already state-of-the-art performance of twinned regression methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  We compared the ensemble behaviour of traditional ANNs and TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' For this  purpose we mapped single anchor TNNR to an equivalent ANN model during inference  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' By visualizing the results of these examinations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 2 we can see that the  performance an ensemble of single anchor TNNs converges towards an ensemble of full  anchor TNNs at around 32 ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Increasing the number of anchors would  not yield any additional gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This suggests that the intrinsic ensemble diversity of  TNNR is a subset of the diversity that can be achieved by retraining the network for  each anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The combined anchor+direct ensemling containing multiple TNNs causes a  much larger improvement than the ensembling of traditional ANNs as can be seen in  Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This explains one element of the outperformance of TNNR over  ANN regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Further, we examined what effect the increased training set size through pairing  data points has on the TNN accuracy, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Generally, more pairings per training data  point reduced the RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' However, to our surprise not all data sets benefited from this  pairing, on two data sets (BC,WN) TNNR had the lowest RMSE if only one pairing  per training data point was allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' By comparing this malfunction to results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 2  and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 5, we can see that it occurs in exactly the data sets where k-NN regression  and ANN ensembles outperform TNNR, signaling a breakdown of the performance  increasing factors of TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' By looking at the properties of the data sets it seems like  twinned regression methods perform best on continuous data sets where the label can be  approximated through a deterministic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  We also pointed at another advantage of TNNR in the case where it is impossible to  store a large number of parameters, but one wants to retain the advantages of ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  TNNR provides the possibility to generate an ensemble of predictions just by storing  How to get the most out of Twinned Regression Methods  18  one model and some anchors, which is usually significantly smaller than storing multiple  ANN models, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  While exploring the ideal weighting of anchors during the inference phase of TNNR,  we found that nearest-neighbor predictions tend to yield the most accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  This lead us to develop nearest-neighbor TNNR (NNTNNR) which is a combination  of the k-nearest neighbor algorithm and TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' There are two versions of NNTNNR,  one which respects nearest neighbors during both training and inference and another  version that only restricts nearest neighbors during the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Restricting  to nearest-neighbor training tends to yield slightly better results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Both versions  outperform standard TNNR especially on low noise data sets, see Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' It is important  to note that NNTNNR is not just a minor improvement to k-NN regression since it has  a very different performance profile when it comes to varying the number of anchors, or  nearest neighbors, respectively, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  We devised a semi-supervised regression framework based on enforcing loop  consistency that can be applied to any twinned regression algorithm, but we tested it for  random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This method yielded a clearly visible improvement over their supervised  counterparts as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' However, the magnitude of the reduction of the  RMSE is relatively small and almost always less than 1%, see Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Comparing these  results with other semi-supervised regression algorithms on the same data sets from [2],  we can see that this improvement is significantly less than semi-supervised TNNR and  slightly less than co-training with neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The code supporting this publication is available at [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Acknowledgements  Let us thank Zurab Jashi for his help with the random forest code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' This work was  supported by Mitacs and Homes Plus Magazine Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' through the Mitacs Accelerate  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We also acknowledge Compute Canada for computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We thank  the National Research Council of Canada for their partnership with Perimeter on the  PIQuIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Research at Perimeter Institute is supported in part by the Government of  Canada through the Department of Innovation, Science and Economic Development  Canada and by the Province of Ontario through the Ministry of Economic Development,  Job Creation and Trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  How to get the most out of Twinned Regression Methods  19  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Data sets  Table A1: Data sets  Name  Key  Size  Features  Type  Bio Concentration  BC  779  14  Discrete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Continuous  Boston Housing  BH  506  13  Discrete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Continuous  Concrete Strength  CS  1030  8  Continuous  Energy Efficiency  EF  768  8  Discrete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Continuous  RCL Circuit Current  RCL  4000  6  Continuous  Test Function  TF  1000  2  Continuous  Red Wine Quality  WN  1599  11  Discrete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Continuous  Wheatstone Bridge Voltage  WSB  200  4  Continuous  Yacht Hydrodynamics  YH  308  6  Discrete  The test function (TF) data set created from the equation  F(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='x2) = x3  1 + x2  1 − x1 − 1 + x1x2 + sin(x2)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='1)  and zero noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The output in the RCL circuit current data set (RCL) is the current through an  RCL circuit, modeled by the equation  I0 = V0 cos(ωt)/  √  R2 + (ωL − 1/(ωC))2  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='2)  with added Gaussian noise of mean 0 and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The output of the Wheatstone Bridge voltage (WSB) is the measured voltage given  by the equation  V = U(R2/(R1 + R2) − R3/(R2 + R3))  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='3)  with added Gaussian noise of mean 0 and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Bias-Variance Tradeoff and Ensembles  In a regression problem, one is tasked with finding the true labels on yet unlabelled  data points through the estimation of a function f(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Given a finite training data  set D we denote this approximation ˆf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' The expected mean squared error can  be decomposed by three sources of error, bias error BiasD[ ˆf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)] , variance error  VarD [ ˆf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)] and intrinsic error of the data set σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  MSE = Ex {BiasD[ ˆf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]2 + VarD [ ˆf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]} + σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='1)  How to get the most out of Twinned Regression Methods  20  If we replace the estimator by an ensemble of two functions ˆf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D) = 1/2 ˆfA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D) +  1/2 ˆfB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D), each exhibiting the same bias and variance as the original estimator, then  we can decompose the MSE  MSE = Ex {BiasD[1/2 ˆfA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D) + 1/2 ˆfB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]2 + VarD [1/2 ˆfA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D) + 1/2 ˆfB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]} + σ2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='2)  = Ex {BiasD[ ˆf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]2 + VarD [1/2 ˆfA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)] + VarD [1/2 ˆfB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='3)  + 2CovD [1/2 ˆfA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D),1/2 ˆfB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]} + σ2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='4)  = Ex {BiasD[ ˆf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]2 + 1/2VarD [ ˆfA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)] + 1/2CovD [ ˆfA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D), ˆfB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D)]} + σ2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='5)  The more uncorrelated ˆfA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D) and ˆfB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='D) are, the smaller is the ratio between  variance and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Thus an ensemble consisting of weakly correlated ensemble  members reduce the MSE by circumventing the bias-variance tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' By induction this  argument extends to larger ensemble sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Neural Network Architectures  Both our traditional neural network regression and twin neural network regression  methods are build using the same architecture build using the tensorflow library [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  They consist of two hidden layers with 128 neurons each and relu activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The final layer contains one single neuron without an activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We train  our neural networks using the adadelta optimizer, and use learning rate and early stop  callbacks that reduce the learning rate by 50% or stop training if the loss stops decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  For this reason it is enough to set the number of epochs large enough such that the early  stopping is always triggered, in our cases this is 2000 for ANNs and 10000 for TNNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  The batchsizes are in both cases 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Random Forests  The random forests in this article use the scikit-learn library [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' They are trained  on a subset of all data sets: from each data set, we randomly sample 100 training  data points and 100 test data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' We use five-fold cross-validation to optimize  the following hyper-parameters of our random forests: ’max depth’ ∈ [4,8,16,32,64],  ’max features’ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='33,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='667,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='0], ’min samples leaf’ ∈ [1,2,5], ’min samples split’  ∈ [2,4,8], ’n estimators’ ∈ [100,300,600].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Both, the traditional and the twinned random  forests choose their optimal hyper-parameters from the same pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' It is important to  note that for semi-supervised learning the hyper-parameters are only optimized during  How to get the most out of Twinned Regression Methods  21  Table D1: Best estimates for test RMSEs obtained by Random Forest compared to Twinned Random  Forests  Random Forest  Twinned Random Forest  Improvement  Bio Conservation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='7407± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='0087  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='7427± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='0081  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='27%  Boston Housing  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='0019± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='1394  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='1322  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='71%  the initial supervised learning step, the optimal parameters are then carried forward to  be used during semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Algorithm 1: Semi-Supervised Learning through Loop Consistency  Data: Labelled data set DL = (Xtrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='Ytrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='L)  Unlabelled data set DU = (Xtrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='U)  Input: Loop weight Λ  Loop number nl =length(DL)/3  1 create ˜DL = [((xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xi − xj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='yij = yi − yj) for xi ∈ Xtrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='L for xj ∈ Xtrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='L]  2 initialize machine learning model M  3 train M on ˜DL  4 sample nl loops L=[(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xk) where xi ∈ Xtrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xk ∈ Xtrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='U]  5 for (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xk) ∈ L do  6  predict M(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='M(xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='M(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xi)  7  a = M(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xj) + M(xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xk) + M(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xi)  8  (yij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='yjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='yki) = (M(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xj) − Λa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='M(xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xk) − Λa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='M(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xi) − Λa)  9  add ((xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='yij),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='((xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='xj − xk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content='xk − xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='yki) to ˜DL  10 train M on ˜DL  Output: Trained Model M  How to get the most out of Twinned Regression Methods  22  [1] Wetzel S J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Ryczko K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Melko R G and Tamblyn I 2022 Applied AI Letters e78  [2] Wetzel S J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Melko R G and Tamblyn I 2022 Machine Learning: Science and Technology 3 045007  [3] Tynes M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Gao W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Burrill D J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Batista E R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Perez D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Yang P and Lubbers N 2021 Journal of  Chemical Information and Modeling 61 3846–3857  [4] Baldominos A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content=' Iturrarte R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Bern´ardez ´O and Afonso C 2018 Applied sciences  8 2321  [5] Rafiei M H and Adeli H 2016 Journal of Construction Engineering and Management 142 04015066  [6] Yu Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Lu J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Shen D and Chen B 2021 Neural Computing and Applications 33 3925–3937  [7] Ryczko K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Wetzel S J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Melko R G and Tamblyn I 2022 Journal of Chemical Theory and Computation  18 1122–1128  [8] Avula N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content=' Behera S and Balasubramanian S 2022 Machine Learning: Science and  Technology  [9] Czischek S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Yon V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Genest M A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Roux M A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Rochette S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Lemyre J C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Moras M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Pioro-Ladri`ere M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Drouin D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Beilliard Y et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' 2021 Machine Learning: Science and Technology 3 015001  [10] Lechner M, Hasani R M and Grosu R 2018 arXiv preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='08554  [11] Bromley J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Guyon I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' LeCun Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' S¨ackinger E and Shah R 1993 Advances in neural information  processing systems 6 737–744  [12] Baldi P and Chauvin Y 1993 neural computation 5 402–418  [13] Srivastava N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Hinton G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Krizhevsky A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Sutskever I and Salakhutdinov R 2014 The journal of  machine learning research 15 1929–1958  [14] Wan L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Zeiler M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Zhang S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Le Cun Y and Fergus R 2013 Regularization of neural networks using  dropconnect International conference on machine learning (PMLR) pp 1058–1066  [15] Wu J 2009 A novel artificial neural network ensemble model based on k–nearest neighbor  nonparametric estimation of regression function and its application for rainfall forecasting  2009 international joint conference on computational sciences and optimization vol 2 (IEEE) pp  44–48  [16] Bensaci R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Khaldi B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Aiadi O and Benchabana A 2021 Applied Sciences 11 10176  [17] Liu Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Guo J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Cao J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Wei Y and Huang W 2018 Promet-Traffic&Transportation 30 445–456  [18] Wetzel  S  2023  Public  github  repository  URL  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='com/sjwetzel/  PublicGetMostOutOfTNNR  [19] Abadi M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Agarwal A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Barham P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Brevdo E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Chen Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Citro C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Corrado G S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Davis A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Dean J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Devin M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Ghemawat S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Goodfellow I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Harp A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Irving G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Isard M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Jia Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Jozefowicz R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Kaiser  L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Kudlur M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Levenberg J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Man´e D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Monga R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Moore S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
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+page_content=' Olah C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Schuster M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Shlens J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Steiner B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Sutskever I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Talwar K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Tucker P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Vanhoucke V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Vasudevan V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Vi´egas F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='  Vinyals O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Warden P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Wattenberg M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Wicke M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content=' Yu Y and Zheng X 2015 TensorFlow: Large-  scale machine learning on heterogeneous systems software available from tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='org URL  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
+page_content='org/  [20] Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer  P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M and  Duchesnay E 2011 Journal of Machine Learning Research 12 2825–2830' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfbPyr/content/2301.01383v1.pdf'}
diff --git a/j9E1T4oBgHgl3EQf0QVw/content/tmp_files/2301.03453v1.pdf.txt b/j9E1T4oBgHgl3EQf0QVw/content/tmp_files/2301.03453v1.pdf.txt
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@@ -0,0 +1,900 @@
+1 
+ 
+  
+How do Quantum Effects Influence the Capacitance and Carrier 
+Density of Monolayer MoS2 Transistors? 
+ 
+Robert K. A. Bennett1 and Eric Pop1,* 
+1Department of Electrical Engineering, Stanford University, Stanford, California 94305, U.S.A. 
+*Contact: epop@stanford.edu 
+ 
+ABSTRACT: When transistor gate insulators have nanometer-scale equivalent oxide thickness (EOT), the 
+gate capacitance (CG) becomes smaller than the oxide capacitance (Cox) due to the quantum capacitance and 
+charge centroid capacitance of the channel. Here, we study the capacitance of monolayer MoS2 as a proto-
+typical two-dimensional (2D) channel while considering spatial variations in the potential, charge density, 
+and density of states. At 0.5 nm EOT, the monolayer MoS2 capacitance is smaller than its quantum capaci-
+tance, limiting the single-gated CG of an n-type channel to between 63% and 78% of Cox for gate overdrive 
+voltages between 0.5 and 1 V. Despite these limitations, for dual-gated devices, the on-state CG of monolayer 
+MoS2 is 50% greater than that of silicon at 0.5 nm EOT and more than three times that of InGaAs at 1 nm 
+EOT, indicating that 2D semiconductors are promising for nanoscale devices at future technology nodes. 
+ 
+KEYWORDS: Semiconductor capacitance, quantum capacitance, centroid capacitance, electrostatics, field-
+effect transistor, 2D semiconductor 
+ 
+Two-dimensional (2D) semiconductors have emerged over the last decade as promising candidates for 
+channel materials in sub-10-nm metal-oxide-semiconductor field-effect transistors (MOSFETs).1,2 Using 
+2D monolayer semiconductors in such transistors is appealing from an electrostatic perspective because 
+their ultrathin channel (< 1 nm) reduces the impact of lateral fringing fields while increasing the 2D 
+semiconductor’s out-of-plane capacitance Csc (sometimes called the inversion layer capacitance in bulk 
+semiconductor transistors in the on-state). Although conventional bulk semiconductors (like silicon) suf-
+fer from mobility degradation as their channel thickness is reduced to a few nanometers, 2D semicon-
+ductors maintain good carrier mobilities even at their monolayer limit, allowing them to simultaneously 
+offer excellent electrostatic control and good on-state conductance.3,4 
+In a field-effect transistor, the total gate capacitance CG = q∂nch/∂VG [where q is the elementary charge, 
+nch is the number of charge carriers (electrons or holes) per unit area, and VG is the gate voltage] is given 
+by the series capacitance of Csc with the gate insulator capacitance, denoted here as Cox (acknowledging 
+that gate insulators may have nitrides or other components),5,6 as shown in Figure 1a: 
+ 
+1
+𝐶G =
+1
+𝐶sc +
+1
+𝐶ox. 
+(1) 
+
+2 
+ 
+  
+The semiconductor channel’s contribution to CG is negligible when Csc ≫ Cox, at which point CG ≈ Cox 
+= ϵox/tox, where ϵox and tox are the insulator’s permittivity and thickness, respectively. If Csc is comparable 
+to Cox, however, then Csc can limit CG, thereby limiting the maximum carrier densities achievable in the 
+FET on-state.5 For example, we demonstrate in this work that for monolayer MoS2, Csc is negligible 
+when the gate insulator’s equivalent oxide thickness (EOT) is ≥ 2.5 nm, although the precise EOT at 
+which Csc becomes negligible varies between semiconductors.5,7-9 
+The Csc has two main components: the centroid capacitance (due to the penetration of the charge centroid 
+into the semiconductor channel5,10) and the quantum capacitance Cq (due to Fermi level movement with 
+respect to the energy bands in a semiconductor channel with finite density of states9,11-13). For 2D semi-
+conductors, the centroid capacitance has often been taken as infinite (implicitly assuming Csc ≈ Cq), and 
+Cq is evaluated as12 
+  
+𝐶q = 2𝜋
+𝑞2𝑔𝑠𝑔𝑣𝑚∗
+ℎ2
+(1 +
+exp⁡[𝐸G (2𝑘B𝑇)
+⁄
+]
+2cosh[𝑞𝜓𝑐h (𝑘B𝑇)
+⁄
+])
+−1
+,  
+(2) 
+where gs = 2 is the spin degeneracy, gv is the valley degeneracy (= 2 and 6 for the lower K-valleys and 
+the higher Q-valleys, respectively, of the conduction band in monolayer MoS2), m∗ is the effective mass, 
+h is Planck’s constant, EG is the electronic band gap (≈ 2.2 eV for monolayer MoS2, depending on its 
+dielectric environment5,14), kB is the Boltzmann constant, and T is the absolute temperature (here,  
+~300 K). Above, ψch is the channel potential, typically considered without regard to its variation in the 
+channel of 2D semiconductors (i.e., infinite centroid capacitance) but self-consistently treated in this 
+work with spatial variation of charge density. Although Cq is very small in the off-state, in the on-state a 
+large |ψch| pushes the Fermi energy into the channel conduction or valence bands, causing the bracketed 
+term in equation (2) to approach unity and saturating Cq to its degenerate value Cdq, given by12,15 
+ 
+𝐶dq = 2𝜋
+𝑞2𝑔𝑠𝑔𝑣𝑚∗
+ℎ2
+.  
+(3) 
+Considering only the lowest-energy conduction and valence bands, Cdq ≈ 70 and 200 μF/cm2 for n- and 
+p-type monolayer MoS2, respectively. Although including higher energy bands (e.g., the Q-valley along 
+the Τ-line16 in the monolayer MoS2 conduction bands) would enable larger Cq, even these lower-bound 
+estimates of Cdq greatly exceed Cox for any realistic EOT, leading most studies to neglect Csc.  
+However, previous experimental studies on monolayer semiconductors, including MoS2, MoSe2, WSe2, 
+and black phosphorus, have reported values of Cq and Csc that are much smaller than their respective Cdq 
+when the Fermi energy EF is pushed beyond the band edges.17-19 Although these smaller-than-anticipated 
+capacitances could be attributed to extrinsic contributions (like defects), recent theoretical work has 
+shown that for monolayer MoS2, other components of Csc could limit it to values much smaller than Cdq.20 
+However, the contribution of non-uniform carrier distributions across a 2D semiconductor’s thickness, 
+as well as the impact that this reduced Csc will have on CG, remain largely unexplored. 
+
+3 
+ 
+  
+ 
+Figure 1. (a) Capacitance network model of a 2D transistor. C and V represent capacitances and potentials, 
+subscripts G, S, D, and B denote quantities associated with the gate, source, drain, and the bottom insulator/sub-
+strate, respectively. ψch is the channel potential, which can vary across the thickness of the channel. The inset 
+shows CG as the series capacitance of Cox and Csc with an intermediate surface potential ψsurf. (b) Schematic of 
+the MoS2-based MOS capacitor considered in this work, along with boundary conditions applied when solving 
+equations (4) and (5).  
+ 
+In this work, we address these gaps in knowledge by self-consistently solving carrier statistics equations 
+with the electrostatic potential distribution across a monolayer MoS2-based MOS capacitor, as shown in 
+Figure 1b. We consider the spatial variation of electrostatic potential V(z) [where z is the cross-plane 
+coordinate labeled in Figure 1b], the volumetric charge density ρ(z), and the local density of states 
+(LDOS)21,22 across the monolayer thickness, and we write                             
+ 
+𝜌(𝑧) = ∓𝑞∫ LDOS(𝐸, 𝑧) [
+1
+2 ∓
+1
+2 ± 𝑓(𝐸)] d𝐸  
+(4) 
+where E is the energy, f(E) is the Fermi-Dirac distribution, upper (lower) signs are for electrons (holes), 
+and the channel carrier density nch is obtained by integrating |ρ(z)|/q. Applying a gate voltage changes 
+ρ(z) and nch by modulating the local electrostatic potential V(z), pushing EF from mid-gap towards the 
+conduction (valence) bands and populating the channel with electrons (holes). 
+Here, we assume that the intrinsic semiconductor EF is at the mid-gap energy, allowing us to equate EF 
+with qV(z) when computing f(E) in equation (4), where qV(z) is also referenced to mid-gap. As both ρ(z) 
+and V(z) are unknowns, we solve equation (4) self-consistently with Poisson’s equation, 
+ 
+𝜕
+𝜕𝑧 [𝜖(𝑧) 𝜕𝑉
+𝜕𝑧⁡] =⁡−𝜌(𝑧), 
+(5) 
+where ϵ(z) is the permittivity. We discretize V(z) and ρ(z) along a one-dimensional grid, including the 
+gate voltage boundary condition [V(z) = VG] at the top of the gate insulator and a Neumann boundary 
+condition [∂V(z)/∂z = 0] at the opposite side of the bottom insulator in the structure shown in Figure 1b. 
+We model ϵ(z) as a step-like function that transitions from ϵox to the MoS2 permittivity17 4ϵ0 (where ϵ0 is 
+the permittivity of vacuum) at z = -tch/2, then to the permittivity of SiO2 (3.9ϵ0) at z = tch/2, where tch = 
+(a)
+(b)
+0
+z
+
+4 
+ 
+  
+0.615 nm is the MoS2 monolayer thickness. We note that this dielectric profile is approximate; many 
+different estimates for the out-of-plane dielectric constant of monolayer MoS2 have been reported,4,17,23,24 
+and it is unclear how ϵ(z) varies spatially at the insulator/MoS2 interface. Furthermore, it is uncertain if 
+ϵ(z) is mostly constant across the thickness of the MoS2 monolayer or if, like graphene,25 ϵ(z) is a function 
+of position within the monolayer. Once these factors are known, they can be incorporated into our model 
+by substituting the appropriate dielectric profile into equation (5). 
+We extract the LDOS of monolayer MoS2 using density functional theory (DFT) with Quantum ES-
+PRESSO software.26 All calculations are performed on a 151 × 151 × 1 k-point grid using projector-
+augmented wave pseudopotentials with kinetic energy cutoffs of 60 Ry for wave functions and 480 Ry 
+for charge densities and potentials. After computing the LDOS for a primitive cell in three dimensions, 
+we average the LDOS across the in-plane directions to represent it only as functions of E and z. Then, to 
+ensure that the LDOS at a specific energy will always sum to the magnitude of the DOS at that same 
+energy, we express it as 
+ 
+LDOS(𝐸, 𝑧) = 𝐿(𝐸, 𝑧)DOS(𝐸), 
+(6) 
+where L(E,z) is the spatial distribution of the LDOS,21 which we normalize at each energy level En: 
+ 
+𝐿(𝐸 = 𝐸𝑛, 𝑧) =
+LDOS(𝐸=𝐸𝑛,𝑧)
+∫
+LDOS(𝐸=𝐸𝑛,𝑧)d𝑧
+∞
+−∞
+. 
+(7) 
+Figures 2a,b show L(E,z) in the conduction and valence bands, respectively. At E ≤ EC + 0.25 eV (where 
+EC is the conduction band minimum), the LDOS is confined close to the center of the MoS2 monolayer 
+(i.e., near the Mo atoms) with sharp, narrow peaks appearing just to the left and right of the main central 
+peak. These sharp satellite peaks arise from the spatial distribution of the 4𝑑𝑧2 orbital of Mo, which has 
+been shown to dominate the DOS at these energies in previous studies.27,28 Furthermore, we find that 
+broad peaks centered close to the S atoms appear in the LDOS at E > EC + 0.25 eV. As shown in the 
+projected DOS (pDOS) in Figure 2c, the S atoms begin to contribute to the DOS in the conduction bands 
+at ~0.26 eV above the conduction band minimum, corresponding to the Q-K valley separation ΔEQK from 
+our DFT simulations. Projected band structures have also shown that S atoms contribute weakly to the 
+electronic structure of monolayer MoS2 at the conduction band minimum, but they contribute noticeably 
+to the Q-valley.27,29 We therefore attribute these peaks to contributions to the DOS from the S atoms. 
+Similar laterally positioned peaks are present at all values of E ≤ EV (where EV is the valence band 
+maximum) we consider in Figure 2b, which is also consistent with the pDOS in the valence bands: as 
+shown in Figure 2d, S atoms contribute to the DOS in the valence bands at all considered energies. 
+Similarly, projected band structures have shown that both Mo and S atoms contribute significantly to the 
+valence bands of monolayer MoS2.27,29 
+We note that the exact ΔEQK for monolayer MoS2 in vacuum is not precisely known30 and that the band 
+structure of monolayer MoS2 can vary depending on strain31 and its surrounding dielectric environment.14 
+For example, the experimental ΔEQK for monolayer MoS2 encased in quartz and WS2 is ΔEQK ≈ 0.11 
+
+5 
+ 
+  
+eV,32 and simulated values range between 0.071 and 0.270 eV, depending on the approach used.30 To 
+accommodate this uncertainty in the value of ΔEQK, we investigate its effect on Csc and nch in Section S1 
+of the Supporting Information. We also note from Figures 2a,b that the LDOS extends slightly beyond 
+tch = 0.615 nm, which occurs because DFT simulations of 2D MoS2 assume that the semiconductor is 
+surrounded by vacuum; in reality, a semiconductor’s LDOS cannot so easily penetrate into an insulator.33 
+However, we shortly demonstrate that this non-ideality should not significantly affect our results. 
+ 
+Figure 2. Normalized spatial distribution of the local density of states L(E,z) across monolayer MoS2 at (a) E ≥ 
+EC, and (b) E ≤ EV, where the z coordinates of Mo and S atoms align with the location of atoms at the top of the 
+figures (size of atoms are not to scale). Projected density of states (pDOS) for (c) conduction bands and (d) 
+valence bands of monolayer MoS2, where the contributions of all orbitals from each individual atom are summed 
+together. Note that the contributions from only one S atom are shown in (c) and (d); a.u., arbitrary units. 
+ 
+In Figures 3a,b, we calculate and plot Csc of monolayer MoS2 on linear and logarithmic y-axes, respec-
+tively, by self-consistently solving equations (4) and (5) under an applied gate bias, with the correspond-
+ing nch values plotted in Figure 3c. For now, we set the permittivity of the gate insulator to an extremely 
+large value (Cox → ∞), so that ψsurf = VG, allowing us to study the intrinsic capacitance of monolayer 
+MoS2 by neglecting the potential drop across the gate insulator. We will shortly relax this assumption 
+and study monolayer MoS2-based capacitors with finite EOTs.  
+At |ψsurf| − EG/(2q) < 0.25 V, the capacitance of p-type MoS2 exceeds that of n-type MoS2, which is due 
+to the DOS near the valence band edge being larger than the DOS near the conduction band edge (Figures 
+2c,d). However, as ψsurf is pushed farther into the conduction bands, the slopes of both Csc and nch increase 
+sharply for n-type MoS2. This increase is due to the step-like increase in the DOS at the Q-valley, noting 
+that thermal broadening in equation (4) allows the DOS from the Q-valley to also contribute when EF is 
+a few kBT below the edge of the Q-valley. We note that this effect has been experimentally observed in 
+MoSe2 and WSe2 monolayers, which have similar electronic structures to that of MoS2 (including a Q-
+valley above the conduction band edge). Thus, the shape of the electron density in our Figure 3c resem-
+bles similar experimental curves for MoSe2 and WSe2 monolayers.18 
+(a)
+(b)
+(c)
+EC
+∆E  = 0.05 eV
+(d)
+ΔEQK ≈ 0.26 eV
+EV
+EV - 0.35 eV
+EC + 0.35 eV
+S
+Mo
+S
+S
+Mo
+S
+Mo
+S
+S
+Mo
+∆E  = 0.05 eV
+
+6 
+ 
+  
+  
+Figure 3. Semiconductor capacitance Csc as a function of |ψsurf| − EG/(2q) plotted on (a) linear and (b) logarithmic 
+axes. For comparison, we also plot quantum capacitance Cq as a function of the channel potential ψch [calculated 
+using equation (2) with gvm* = 1.01m0 and 2.96m0 for electrons and holes,4 respectively] in (b). (c) Charge 
+carrier density nch for n-type and p-type monolayer MoS2. The location of the Q-valley for n-type monolayer 
+MoS2 is marked in (a) and (c) with a vertical dashed line. Note, carrier densities over ~2 × 1013 cm-2 are not 
+accessible in practice with conventional dielectrics (e.g. HfO2), only with solid or liquid electrolytes. 
+ 
+We also compare our computed Csc values to the conventional Cq for both n- and p-type monolayer MoS2, 
+which we have plotted alongside Csc in Figure 3b. Although our calculated Csc closely matches Cq at 
+small gate voltage (i.e., non-degenerate surface potentials), we find that at high gate voltages (i.e., de-
+generate potentials), our computed Csc values are substantially lower than the traditionally calculated  
+Cq = Cdq = 70 μF/cm2 for n-type and 200 μF/cm2 for p-type monolayer MoS2, respectively. To understand 
+why Csc matches Cq only for non-degenerate potentials, we first plot the charge distributions and poten-
+tial across the thickness of n-type (p-type) MoS2 when ψsurf is 0.3 V below (above) the conduction (va-
+lence) band edge in Figures 4a,b. For this non-degenerate case, the charge distributions are nearly sym-
+metric across the channel, closely matching the LDOS distributions shown in Figures 2a,b. This result is 
+consistent with the potential profile shown in Figure 4b: there is nearly no potential drop across the 
+monolayer MoS2 thickness, allowing electronic states to contribute to the carrier density equally effi-
+ciently, regardless of their location in the channel. Hence, Csc ≈ Cq in this regime. 
+Next, to understand why Csc < Cq at degenerate potentials, we plot the charge distributions and potential 
+across the thickness of monolayer n-type (p-type) MoS2 when ψsurf is 0.3 V above (below) the conduction 
+(valence) band edge in Figures 4c,d. We find that for this degenerate case, the charge distributions are 
+heavily asymmetric and skewed towards the gate electrode. This asymmetry can be explained from the 
+potential drops in Figure 4d, which shows that the local electrostatic potential is highest near the gate 
+electrode and rapidly decays across the monolayer channel. We recall from Figures 2a,b that most avail-
+able states are near the center of the channel in this region of relatively low potential. Therefore, many 
+(a)
+p
+n
+n : ψsurf = EC/q
+p : ψsurf = EV/q
+Q
+Csc
+(Our 
+model)
+n : ψsurf = EC/q
+p : ψsurf = EV/q
+p
+n
+Cq
+(Eq. 2)
+p-type Cdq
+n-type Cdq
+(b)
+n
+p
+n : ψsurf = EC/q
+p : ψsurf = EV/q
+Q
+(c)
+
+7 
+ 
+  
+states are unable to efficiently contribute to nch, which is why Csc < Cdq even at high ψsurf. As a result, the 
+charge density is askew across the thickness of the MoS2 monolayer, with the S atoms opposite to the 
+gate contributing little to the channel charge. We conclude that at degenerate surface potentials, the 
+shapes of the LDOS and potential profile play pivotal roles in dictating Csc for 2D semiconductors. 
+ 
+Figure 4. (a) Distributions of charge density and (b) potential profile across the MoS2 monolayer with ψsurf inside 
+the band gap, 0.3 eV below the conduction band edge (for electrons) and 0.3 eV above the valence band edge 
+(for holes). (c) Distributions of charge density and (d) potential profile across the MoS2 monolayer with ψsurf of 
+0.3 eV above the conduction band edge (for electrons) and 0.3 eV below the valence band edge (for holes). 
+Dashed lines indicate boundaries with the gate electrode and bottom insulator. Note, the carrier densities in (c) 
+are much greater than in (a). In (c), the carrier distribution also extends slightly outside tch by < 10 % of total nch, 
+meaning that the charge centroid is closer to the gate18 and that Csc is slightly overestimated in this study. Finally, 
+we note that the potential drops are different for n- and p-type devices in Figure 4d because the capacitance of n-
+type monolayer MoS2 is smaller than that of p-type MoS2 for the ψsurf we consider here.34   
+ 
+We next consider the impact of Csc in practical MOS devices based on monolayer MoS2. Figures 5a,b 
+plot CG and nch for MOS capacitors as in Figure 1b with EOTs of 0.5, 1, and 2.5 nm (ϵox = 20ϵ0).  
+For comparison, we additionally plot Cox = 3.9ϵ0/EOT, as well as the classical carrier density  
+𝑛ch
+classical = (𝑉G − 𝑉T)𝐶ox/𝑞 (for electrons; the bracketed term is negated for holes), where the threshold 
+voltage is VT = ± EG/(2q) for n- and p-type devices, assuming the same gate metal. At EOT = 2.5 nm, CG 
+saturates close to Cox and 𝑛ch ≈ 𝑛ch
+classical, but the observed CG and nch deviate significantly from Cox and 
+𝑛ch
+classical, respectively, at EOTs of 0.5 and 1 nm. For example, at EOT = 0.5 nm, the CG of n-type MoS2 
+increases from 4.38 μF/cm2 at 0.5 V to 5.35 μF/cm2 at 1 V, remaining between 63% and 78% of Cox. 
+However, we note that CG is sensitive to small variations in VG and to the position of the Q-valley for n-
+type MoS2, as captured in Figure 5a and discussed in Supporting Information Section S1. 
+n
+p
+(a)
+(b)
+(c)
+(d)
+tch
+V(z) =ψch(z)
+p
+n
+n
+p
+n
+Gate
+Bottom  insulator
+p
+ψsurf in band gap 
+ψsurf degenerate
+Gate
+Bottom  insulator
+
+8 
+ 
+  
+ 
+Figure 5. (a) Gate capacitance CG and (b) charge carrier density nch for n-type and p-type monolayer MoS2 as 
+functions of VG at EOT = 0.5, 1, and 2.5 nm. Solid red (blue) lines represent n-type (p-type) MoS2. Dashed lines 
+mark the oxide capacitance Cox and the conventionally calculated carrier density 𝑛ch
+classical, highlighting the im-
+portance of quantum corrections in the limit of ultrathin EOT. Small red arrows mark approximately where the 
+Q-valley of n-type MoS2 begins to contribute. (c) Calculated ratio of 𝑛ch/𝑛ch
+classical for n-type MoS2 and (d) p-
+type MoS2. The true charge density is lower than the classical estimate in the limit of high VG when transistors 
+are strongly turned on. However, 𝑛ch
+classical approaches zero and underestimates the true charge density near 
+threshold. 
+  
+Similarly, we find that the classical equation overestimates nch in 2D MOS capacitors with small EOTs, 
+as shown in Figures 5c,d for n- and p-type MoS2, respectively. At EOTs of 0.5 and 1 nm, the channel 
+carrier density nch is as small as 65% and 79% (79% and 89%) of 𝑛ch
+classical for n-type (p-type) MoS2 in 
+the VG range considered here. We note that contributions from the Q-valley are visible for the n-type 
+device with EOT = 0.5 nm, leading to an increase in CG and nch when the voltage is sufficiently high. 
+This effect has also been observed experimentally in ion-gated MoSe2 and WSe2 monolayers, which have 
+similar band structures to that of monolayer MoS2.18 
+We note that 𝑛ch
+classical = 𝐶ox(𝑉G − 𝑉T)/𝑞 should not be applied near or below VT, because this expres-
+sion neglects subthreshold charge.6 Instead, nch may be approximated in both the off- and on-states by 
+taking CG as the series combination of Cq and Cox,11,12 and then integrating the result (up to the relevant 
+VG) to find the carrier density. However, as we demonstrate in Section S2 of the Supporting Information, 
+correcting for Cq in this manner still significantly overestimates both CG and nch in the on-state for 2D 
+channels with low EOT, highlighting the importance of including both quantum and centroid effects 
+when modeling these devices. 
+Finally, we assess how Csc limits CG for monolayer MoS2 compared to other semiconductors, including 
+silicon. Although decreasing tch improves the channel electrostatics, tch < 5 nm causes surface scattering 
+to limit silicon carrier mobilities.2,35,36 At the limit tch ≈ 5 nm, a previous study37 has shown that the Csc 
+of silicon limits a dual-gated silicon FET with an EOT of 0.5 nm to CG ≈ 7 μF/cm2 at an overdrive  
+VOV = VG – VT = 1 V. In direct comparison, our calculations show that a similar dual-gated structure with 
+n-type monolayer MoS2 offers CG ≈ 10.9 μF/cm2 (10.6 μF/cm2 for p-type), over 50% greater than that of 
+(a)
+(b)
+(c)
+1 nm
+0.5 nm
+n-type
+n
+p
+p
+Cox
+n
+EOT 
+(nm)
+0.5
+1
+2.5
+EOT 
+(nm)
+0.5
+1
+2.5
+(d)
+EOT= 2.5 nm
+1 nm
+0.5 nm
+p-type
+EOT= 2.5 nm
+
+9 
+ 
+  
+silicon. The MoS2 advantage persists even when the silicon thickness is reduced37 to 2.5 nm, which yields 
+CG ≈ 8.1 μF/cm2. We refer the reader to Section S2 of the Supporting Information for a description of 
+how we calculated CG and nch for dual-gated devices and for full CG and nch vs. VG curves. 
+The Csc of monolayer MoS2 compares even more favorably to III-V semiconductors, whose low DOS 
+are known to limit their Cq. A previous study38 has shown that Cq limits dual-gated InGaAs MOS capac-
+itors with EOT = 1 nm to CG < 1.6 μF/cm2 at channel thickness tch = 25 nm; as tch decreases, this CG 
+worsens because the DOS shrinks due to quantum confinement effects.38 Using the same approach as 
+above, we find that a dual-gated monolayer n-type MoS2 capacitor with EOT = 1 nm offers CG ≈ 5.55 
+μF/cm2 (or 6.00 μF/cm2 for p-type) at VOV = 1 V, over three times higher than InGaAs. The results from 
+Figure 5a also indicate that single-gated monolayer MoS2 capacitors with EOT of 2.5 nm offer higher 
+CG than those reported for single-gated In0.7Ga0.3As and InAs capacitors with similar or lower EOTs.5 
+In conclusion, we have shown that variations in carrier density (i.e., the centroid capacitance), potential, 
+and density of states across the thickness of monolayer MoS2 severely limits its on-state Csc to values 
+well below its degenerate quantum capacitance. As a result, gate capacitance estimates made by classical 
+equations must be corrected when evaluating the carrier density and mobility of devices with small EOTs 
+below ~2 nm. Nevertheless, we find that in strong inversion, the CG of dual-gated n-type monolayer 
+MoS2 capacitors is over 50% higher than dual-gated silicon MOS capacitors at EOT = 0.5 nm and over 
+three times higher than InGaAs capacitors at EOT = 1 nm. The monolayer MoS2 capacitance advantage 
+is higher at lower EOTs, ultimately indicating that strong current and transconductance may be achieved 
+in such 2D transistors if their channel mobility and contact resistance continue to be improved. 
+ 
+ASSOCIATED CONTENT 
+Supporting Information: Description and variation of the Q-K energy valley separation; comparison to 
+quantum capacitance approximation; methodology and results for dual-gated calculations. 
+Author contributions: R.K.A.B. and E.P. conceived the idea and wrote the manuscript. R.K.A.B. car-
+ried out all calculations.  
+Notes: The authors declare no competing financial interests. 
+ACKNOWLEDGMENTS 
+R.K.A.B. acknowledges support from the Stanford Graduate Fellowship (SGF) and the NSERC PGS-D 
+programs. The authors also acknowledge partial support from the Stanford SystemX Alliance and from 
+ASCENT, one of six centers in JUMP, an SRC program sponsored by DARPA. 
+ 
+ 
+
+10 
+ 
+  
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+is not observed here because we apply a Neumann boundary condition at the far side of the bottom 
+insulator. However, we have verified that the potentials across the channel of Figure 4b,d become nearly 
+identical to those obtained when we instead ground the far side of the bottom insulator (note that electrical 
+ground corresponds to the mid-gap voltage) and allow the bottom insulator thickness to approach infinity.     
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+(2015). DOI: 10.1109/TED.2015.2480380 
+ 
+ 
+ 
+
+1 
+ 
+  
+Supporting Information 
+ 
+How do Quantum Effects Influence the Capacitance and Carrier Density of 
+Monolayer MoS2 Transistors? 
+Robert K. A. Bennett1 and Eric Pop1,* 
+1Department of Electrical Engineering, Stanford University, Stanford, California 94305, U.S.A. 
+*Contact: epop@stanford.edu 
+ 
+S1. Effect of Q-K Energy Valley Separation  
+Our density functional theory (DFT) calculations yield an energy difference ΔEQK ≈ 0.26 eV between 
+the Q- and K-valleys in the conduction bands of monolayer MoS2, as labeled in Figure S1a (we note 
+this Q-valley is sometimes called Τ or Λ). However, the computed value of ΔEQK for monolayer MoS2 
+is highly dependent on the input parameters used in DFT (e.g., exchange-correlation functionals and 
+pseudopotentials),1 where the settings that yield the actual ΔEQK of monolayer MoS2 in vacuum are 
+presently unclear. The band structure of monolayer MoS2 also varies with the surrounding dielectric 
+environment,2 further complicating the question of which ΔEQK is relevant for a given physical system.  
+To accommodate this uncertainty in the “correct” ΔEQK, we repeat calculations of the semiconductor 
+capacitance Csc and carrier density nch for n-type MoS2 with ΔEQK = 0.13 eV and compare these results 
+to those obtained using ΔEQK = 0.26 eV in the main text. As p-type MoS2 does not have an associated 
+ΔEQK or similar low-energy peaks that contribute to its density of states (DOS) in the range of energies 
+of interest, the p-type results would be unchanged and are not repeated here.  
+  
+Figure S1. (a) Band structure of monolayer MoS2 near the conduction and valence band edges obtained from DFT, 
+where the energy separation ΔEQK is labeled between the Q- and K-valleys. Energies (E) are not to scale (e.g., the 
+band gap is reduced to highlight details of the conduction band). Note that we neglect spin-orbit coupling in this 
+work, although for monolayer MoS2, including spin-orbit coupling only negligibly influences the value of ΔEQK 
+obtained from DFT.1 (b) Computed semiconductor capacitance Csc and (c) carrier density nch for n-type monolayer 
+MoS2 capacitors with Q-K energy separations ΔEQK = 0.13 and 0.26 eV. 
+(a)
+ΔEQK = 0.26 eV
+ΔEQK = 0.13 eV
+ΔEQK = 0.13 eV
+ΔEQK = 0.26 eV
+(b)
+(c)
+ΔEQK
+K
+Q
+
+2 
+ 
+  
+To obtain a local DOS (LDOS) profile with ΔEQK = 0.13 eV, we take the LDOS used in the main text 
+with ΔEQK = 0.26 eV and splice together the LDOS at energies E < EC + 0.13 eV and E > EC + 0.26 eV 
+to create a continuous LDOS profile with ΔEQK = 0.13 eV. We then compute Csc and nch with this 
+LDOS using the same methodology as described in the main text.  
+As shown in Figures S1b and S1c, Csc and nch are the same for both values of ΔEQK we consider at low 
+ψsurf. However, as ψsurf increases, the states near the Q-valley contribute at lower energies for the LDOS 
+profile with ΔEQK = 0.13 eV, resulting in an earlier onset for the second linear region of Csc, thereby 
+increasing nch. Although this lower ΔEQK shifts this linear region of the Csc curve to the left, it does not 
+significantly affect the maximum value of Csc ≈ 40 μF/cm2 in the range of ψsurf we consider. 
+Next, we repeat the calculations of CG and nch presented in Figures 4a,b of the main text at EOTs of 
+0.5, 1, and 2.5 nm using ΔEQK = 0.13 and 0.26 eV. As shown in Figures S2a and S2b, the value of 
+ΔEQK used affects neither Csc nor nch at EOT = 2.5 nm since CG is dominated by the oxide capacitance 
+Cox at sufficiently large EOTs. At EOT = 0.5 and 1 nm, however, we find that the higher Csc at ΔEQK = 
+0.13 eV allows CG and nch to grow closer to the classical limit compared to ΔEQK = 0.26 eV. This result 
+signifies that smaller ΔEQK provides further advantage of monolayer MoS2 over Si and III-V channels 
+from the point of view of quantum capacitance and channel carrier density at a given overdrive voltage  
+VOV = VG – VT. In practice, note that ΔEQK is controlled by the strain applied3 and may be controlled by 
+the environmental dielectric as well. 
+ 
+Figure S2. (a) Gate capacitance CG and (b) carrier density nch for n-type MoS2 with Q-K energy separations 
+ΔEQK = 0.13 and 0.26 eV at EOTs of 0.5, 1, and 2.5 nm. Dotted green (solid red) lines represent the LDOS 
+profile obtained using ΔEQK = 0.13 (0.26) eV. Dashed lines mark the oxide capacitance 𝐶ox = 3.9𝜖0/EOT  and 
+the conventionally calculated carrier density 𝑛ch
+classical = 𝐶ox(𝑉G − 𝑉T)/𝑞 , where the threshold voltage is  
+VT = EG/(2q). 
+ 
+S2. Comparison to Quantum Capacitance Approximation 
+As discussed in the main text, the classical approximation of gate capacitance, CG = Cox (where Cox is 
+the 
+oxide 
+capacitance) 
+and 
+the 
+classical 
+approximation 
+for 
+charge 
+carrier 
+density,  
+𝑛ch
+classical = 𝐶ox(𝑉G − 𝑉T)/𝑞, are well-known to be inaccurate near or below the threshold voltage VT. 
+Instead, CG is typically modeled in 2D semiconductors as the series combination of the quantum 
+capacitance Cq and the oxide capacitance Cox, yielding CG-1 ≈ Cox-1 + Cq-1. Since Cq is a function of the 
+semiconductor’s surface potential, when using this equation, CG must be solved iteratively such that the 
+ΔEQK = 0.26 eV
+ΔEQK = 0.13 eV
+Cox
+nch
+(classical)
+ΔEQK = 0.13 eV
+ΔEQK = 0.26 eV
+(a)
+(b)
+EOT 
+(nm)
+0.5
+1
+2.5
+EOT 
+(nm)
+0.5
+1
+2.5
+
+3 
+ 
+  
+voltage drop across the oxide and semiconductor are self-consistent with Cq. Then, nch may be 
+approximated at any VG by integrating this result to obtain the carrier density, 
+ 
+ 
+ 
+ 
+𝑛ch
+Cq−corrected ≈
+1
+𝑞 ∫
+[𝐶q
+−1(𝑉G
+′) + 𝐶ox
+−1]
+−1d𝑉G
+′.
+𝑉G
+−𝐸G/2𝑞
+   
+      (S1) 
+As shown in Figure S3 below, the approximation CG-1 ≈ Cox-1 + Cq-1 matches the rigorously calculated 
+CG presented in the main text in the subthreshold region (here |VG| < EG/2q). However, this 
+approximation considerably overestimates the CG of devices with EOT ≤ 1 nm at larger overdrive 
+voltages. This finding is consistent with our previous result from Figure 3b in the main text, which 
+shows that Cq similarly overestimates the semiconductor’s capacitance in the on-state. 
+ 
+Figure S3: Gate capacitance CG for n-type and p-type monolayer MoS2 as functions of VG at EOT = 0.5, 1, and 
+2.5 nm. Solid red (blue) lines represent the CG of n-type (p-type) MoS2 calculated using the full ab initio 
+approach described in the main text that includes charge centroid effects. Dash-dotted red (blue) lines are the 
+approximation CG-1 ≈ Cox-1 + Cq-1 (where Cq is calculated using the self-consistent approach described above) 
+which does not include centroid effects, and black dashed lines mark the oxide capacitance, Cox. The 
+discrepancies between solid lines and approximations highlight the importance of quantum and charge centroid 
+effects in the on-state, which are most important at small EOTs. 
+ 
+Similarly, to compare our rigorously computed nch in the main text to the carrier density corrected with 
+only the quantum (not centroid) capacitance, we plot our calculated nch from the main text alongside 
+Equation S1 in Figure S4, below. For a more thorough comparison, we also include our original 
+𝑛ch
+classical on the same plot. At an EOT of 2.5 nm, Equation S1 accurately approximates our rigorously 
+calculated nch in both the off- and on-states. However, at EOT = 0.5 and 1 nm, Equation S1 does not 
+correctly predict the charge in the on-state significantly better than the classical approximation 
+𝑛ch
+classical. Again, this result can be understood based on Figure 3b in the main text, where we show that 
+our rigorously calculated semiconductor capacitance Csc < Cq in the on-state. From these results, we 
+conclude that although including corrections for Cq enables good approximations of CG and nch in and 
+near the off-state, the more rigorous approach for calculating these quantities (i.e., including spatial 
+variations in the density of states, potential, and volumetric charge density) presented in the main text 
+should be used to understand 2D semiconductor devices with sub-1 nm EOT in the on-state. 
+EOT 
+(nm)
+0.5
+1
+2.5
+n
+p
+Cox
+p-type (Cox
+-1 + Cq
+-1)-1
+n-type (Cox
+-1 + Cq
+-1)-1
+This work
+
+4 
+ 
+  
+ 
+Figure S4: Charge carrier density nch for n- and p-type monolayer MoS2 as functions of VG at (a) EOT 
+= 2.5 nm, (b) EOT = 1 nm, (c) EOT = 0.5 nm. Solid red (blue) lines represent nch of n- (p-) type MoS2 
+calculated using the full ab initio approach described in the main text, which includes charge centroid 
+effects. Dashed red (blue) lines represent the quantum capacitance-corrected carrier density  
+𝑛ch
+Cq−corrected (Equation S1, using the self-consistent approach described above) that does not include 
+centroid effects, and black dashed lines mark the classically calculated charge carrier density, 
+𝑛ch
+classical = 𝐶ox(𝑉G − 𝑉T)/𝑞. The discrepancies between solid lines and approximations highlight the 
+importance of quantum and charge centroid effects in the on-state, which are most important  
+at small EOTs. 
+ 
+S3. Monolayer MoS2 Dual-Gated Capacitors 
+We calculate nch and CG of dual-gated monolayer MoS2 capacitors by solving equations (4) and (5) in 
+the main text self-consistently, just as we do when computing these quantities for single-gated devices. 
+Here we use the device schematic shown in Figure S5a, which is similar to the single-gated device in 
+Figure 1b except that the relative permittivities of both the top and bottom insulator are set to 20 and 
+the thicknesses of the top and bottom insulators are identical. We update the bottom boundary 
+condition [previously ∂V(z)/∂z = 0 at this boundary for single-gated devices] to V(z) = VG, where VG is 
+the gate voltage applied at both the top and bottom electrodes. We plot CG and nch at EOT = 0.5, 1, and 
+2.5 nm for both n-type and p-type monolayer MoS2 for this dual-gated device in Figures S5b,c. 
+ 
+ 
+ 
+p-type nch
+Cq-corrected
+n-type nch
+Cq-corrected
+p-type nch
+Cq-corrected
+n-type nch
+Cq-corrected
+n
+p
+(a)
+EOT = 2.5 nm
+p-type nch
+Cq-corrected
+n-type nch
+Cq-corrected
+(b)
+n
+p
+EOT = 1 nm
+(c)
+n
+p
+EOT = 0.5 nm
+
+5 
+ 
+  
+ 
+Figure S5. (a) Schematic of a monolayer MoS2 dual-gated MOS capacitor with boundary conditions applied 
+when solving equations (4) and (5). (b) Gate capacitance CG and (c) carrier density nch for these dual-gated 
+devices. Solid red (blue) lines represent n-type (p-type) MoS2. Dashed lines mark the total dual-gated oxide 
+capacitance 𝐶ox = 2 ∗ (3.9𝜖0/EOT) (where the prefactor of 2 accounts for both gates) and the conventionally 
+calculated carrier density 𝑛ch
+classical = 𝐶ox(𝑉G − 𝑉T)/𝑞, where the threshold voltage is VT = ± EG/(2q) for n- and 
+p-type devices, assuming the same gate metal. 
+ 
+SUPPORTING REFERENCES: 
+1 
+Gaddemane, G., Gopalan, S., Van de Put, M. L. & Fischetti, M. V. Limitations of ab initio 
+methods to predict the electronic-transport properties of two-dimensional semiconductors: the 
+computational example of 2H-phase transition metal dichalcogenides. J Comput Electron 20, 
+49-59 (2021). DOI: 10.1007/s10825-020-01526-1 
+2 
+Ryou, J., Kim, Y.-S., Kc, S., & Cho, K. Monolayer MoS2 Bandgap Modulation by Dielectric 
+Environments and Tunable Bandgap Transistors. Sci Rep 6, 29184 (2016). DOI: 
+10.1038/srep29184 
+3 
+Hosseini, M., Elahi, M., Pourfath, M., Esseni, & D. Strain induced mobility modulation in 
+single-layer MoS2. J Phys D: Appl. Phys 48, 375104 (2015). DOI: 10.1088/0022-
+3727/48/37/375104 
+n
+p
+Cox
+EOT 
+(nm)
+0.5
+1
+2.5
+(b)
+p
+n
+EOT 
+(nm)
+0.5
+2.5
+1
+(c)
+(a)
+
diff --git a/j9E1T4oBgHgl3EQf0QVw/content/tmp_files/load_file.txt b/j9E1T4oBgHgl3EQf0QVw/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0c84d0ff8de5b3e6b85d443541183cbe7a4ead2e
--- /dev/null
+++ b/j9E1T4oBgHgl3EQf0QVw/content/tmp_files/load_file.txt
@@ -0,0 +1,685 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf,len=684
+page_content='1  How do Quantum Effects Influence the Capacitance and Carrier Density of Monolayer MoS2 Transistors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Robert K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Bennett1 and Eric Pop1,* 1Department of Electrical Engineering, Stanford University, Stanford, California 94305, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' *Contact: epop@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='edu  ABSTRACT: When transistor gate insulators have nanometer-scale equivalent oxide thickness (EOT), the gate capacitance (CG) becomes smaller than the oxide capacitance (Cox) due to the quantum capacitance and charge centroid capacitance of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Here, we study the capacitance of monolayer MoS2 as a proto- typical two-dimensional (2D) channel while considering spatial variations in the potential, charge density, and density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm EOT, the monolayer MoS2 capacitance is smaller than its quantum capaci- tance, limiting the single-gated CG of an n-type channel to between 63% and 78% of Cox for gate overdrive voltages between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 and 1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Despite these limitations, for dual-gated devices, the on-state CG of monolayer MoS2 is 50% greater than that of silicon at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm EOT and more than three times that of InGaAs at 1 nm EOT, indicating that 2D semiconductors are promising for nanoscale devices at future technology nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  KEYWORDS: Semiconductor capacitance, quantum capacitance, centroid capacitance, electrostatics, field- effect transistor, 2D semiconductor  Two-dimensional (2D) semiconductors have emerged over the last decade as promising candidates for channel materials in sub-10-nm metal-oxide-semiconductor field-effect transistors (MOSFETs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1,2 Using 2D monolayer semiconductors in such transistors is appealing from an electrostatic perspective because their ultrathin channel (< 1 nm) reduces the impact of lateral fringing fields while increasing the 2D semiconductor’s out-of-plane capacitance Csc (sometimes called the inversion layer capacitance in bulk semiconductor transistors in the on-state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Although conventional bulk semiconductors (like silicon) suf- fer from mobility degradation as their channel thickness is reduced to a few nanometers, 2D semicon- ductors maintain good carrier mobilities even at their monolayer limit, allowing them to simultaneously offer excellent electrostatic control and good on-state conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3,4 In a field-effect transistor, the total gate capacitance CG = q∂nch/∂VG [where q is the elementary charge, nch is the number of charge carriers (electrons or holes) per unit area, and VG is the gate voltage] is given by the series capacitance of Csc with the gate insulator capacitance, denoted here as Cox (acknowledging that gate insulators may have nitrides or other components),5,6 as shown in Figure 1a:  1  𝐶G =  1  𝐶sc +  1  𝐶ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  (1)  2  The semiconductor channel’s contribution to CG is negligible when Csc ≫ Cox, at which point CG ≈ Cox = ϵox/tox, where ϵox and tox are the insulator’s permittivity and thickness, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' If Csc is comparable to Cox, however, then Csc can limit CG, thereby limiting the maximum carrier densities achievable in the FET on-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 For example, we demonstrate in this work that for monolayer MoS2, Csc is negligible when the gate insulator’s equivalent oxide thickness (EOT) is ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm, although the precise EOT at which Csc becomes negligible varies between semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5,7-9 The Csc has two main components: the centroid capacitance (due to the penetration of the charge centroid into the semiconductor channel5,10) and the quantum capacitance Cq (due to Fermi level movement with respect to the energy bands in a semiconductor channel with finite density of states9,11-13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' For 2D semi- conductors, the centroid capacitance has often been taken as infinite (implicitly assuming Csc ≈ Cq), and Cq is evaluated as12  𝐶q = 2𝜋 𝑞2𝑔𝑠𝑔𝑣𝑚∗ ℎ2 (1 + exp\u2061[𝐸G (2𝑘B𝑇) ⁄ ] 2cosh[𝑞𝜓𝑐h (𝑘B𝑇) ⁄ ]) −1 , (2) where gs = 2 is the spin degeneracy, gv is the valley degeneracy (= 2 and 6 for the lower K-valleys and the higher Q-valleys, respectively, of the conduction band in monolayer MoS2), m∗ is the effective mass, h is Planck’s constant, EG is the electronic band gap (≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='2 eV for monolayer MoS2, depending on its dielectric environment5,14), kB is the Boltzmann constant, and T is the absolute temperature (here, ~300 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Above, ψch is the channel potential, typically considered without regard to its variation in the channel of 2D semiconductors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', infinite centroid capacitance) but self-consistently treated in this work with spatial variation of charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Although Cq is very small in the off-state, in the on-state a large |ψch| pushes the Fermi energy into the channel conduction or valence bands, causing the bracketed term in equation (2) to approach unity and saturating Cq to its degenerate value Cdq, given by12,15  𝐶dq = 2𝜋 𝑞2𝑔𝑠𝑔𝑣𝑚∗ ℎ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (3) Considering only the lowest-energy conduction and valence bands, Cdq ≈ 70 and 200 μF/cm2 for n- and p-type monolayer MoS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Although including higher energy bands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', the Q-valley along the Τ-line16 in the monolayer MoS2 conduction bands) would enable larger Cq, even these lower-bound estimates of Cdq greatly exceed Cox for any realistic EOT, leading most studies to neglect Csc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, previous experimental studies on monolayer semiconductors, including MoS2, MoSe2, WSe2, and black phosphorus, have reported values of Cq and Csc that are much smaller than their respective Cdq when the Fermi energy EF is pushed beyond the band edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='17-19 Although these smaller-than-anticipated capacitances could be attributed to extrinsic contributions (like defects), recent theoretical work has shown that for monolayer MoS2, other components of Csc could limit it to values much smaller than Cdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='20 However, the contribution of non-uniform carrier distributions across a 2D semiconductor’s thickness, as well as the impact that this reduced Csc will have on CG, remain largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (a) Capacitance network model of a 2D transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' C and V represent capacitances and potentials, subscripts G, S, D, and B denote quantities associated with the gate, source, drain, and the bottom insulator/sub- strate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' ψch is the channel potential, which can vary across the thickness of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The inset shows CG as the series capacitance of Cox and Csc with an intermediate surface potential ψsurf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (b) Schematic of the MoS2-based MOS capacitor considered in this work, along with boundary conditions applied when solving equations (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  In this work, we address these gaps in knowledge by self-consistently solving carrier statistics equations with the electrostatic potential distribution across a monolayer MoS2-based MOS capacitor, as shown in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We consider the spatial variation of electrostatic potential V(z) [where z is the cross-plane coordinate labeled in Figure 1b], the volumetric charge density ρ(z), and the local density of states (LDOS)21,22 across the monolayer thickness, and we write  𝜌(𝑧) = ∓𝑞∫ LDOS(𝐸, 𝑧) [ 1 2 ∓ 1 2 ± 𝑓(𝐸)] d𝐸 (4) where E is the energy, f(E) is the Fermi-Dirac distribution, upper (lower) signs are for electrons (holes), and the channel carrier density nch is obtained by integrating |ρ(z)|/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Applying a gate voltage changes ρ(z) and nch by modulating the local electrostatic potential V(z), pushing EF from mid-gap towards the conduction (valence) bands and populating the channel with electrons (holes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Here, we assume that the intrinsic semiconductor EF is at the mid-gap energy, allowing us to equate EF with qV(z) when computing f(E) in equation (4), where qV(z) is also referenced to mid-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' As both ρ(z) and V(z) are unknowns, we solve equation (4) self-consistently with Poisson’s equation,  𝜕 𝜕𝑧 [𝜖(𝑧) 𝜕𝑉 𝜕𝑧\u2061] =\u2061−𝜌(𝑧), (5) where ϵ(z) is the permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We discretize V(z) and ρ(z) along a one-dimensional grid, including the gate voltage boundary condition [V(z) = VG] at the top of the gate insulator and a Neumann boundary condition [∂V(z)/∂z = 0] at the opposite side of the bottom insulator in the structure shown in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We model ϵ(z) as a step-like function that transitions from ϵox to the MoS2 permittivity17 4ϵ0 (where ϵ0 is the permittivity of vacuum) at z = -tch/2, then to the permittivity of SiO2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='9ϵ0) at z = tch/2, where tch = (a) (b) 0 z  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='615 nm is the MoS2 monolayer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We note that this dielectric profile is approximate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' many different estimates for the out-of-plane dielectric constant of monolayer MoS2 have been reported,4,17,23,24 and it is unclear how ϵ(z) varies spatially at the insulator/MoS2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Furthermore, it is uncertain if ϵ(z) is mostly constant across the thickness of the MoS2 monolayer or if, like graphene,25 ϵ(z) is a function of position within the monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Once these factors are known, they can be incorporated into our model by substituting the appropriate dielectric profile into equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We extract the LDOS of monolayer MoS2 using density functional theory (DFT) with Quantum ES- PRESSO software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 All calculations are performed on a 151 × 151 × 1 k-point grid using projector- augmented wave pseudopotentials with kinetic energy cutoffs of 60 Ry for wave functions and 480 Ry for charge densities and potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' After computing the LDOS for a primitive cell in three dimensions, we average the LDOS across the in-plane directions to represent it only as functions of E and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Then, to ensure that the LDOS at a specific energy will always sum to the magnitude of the DOS at that same energy, we express it as  LDOS(𝐸, 𝑧) = 𝐿(𝐸, 𝑧)DOS(𝐸), (6) where L(E,z) is the spatial distribution of the LDOS,21 which we normalize at each energy level En:  𝐿(𝐸 = 𝐸𝑛, 𝑧) = LDOS(𝐸=𝐸𝑛,𝑧) ∫ LDOS(𝐸=𝐸𝑛,𝑧)d𝑧 ∞ −∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (7) Figures 2a,b show L(E,z) in the conduction and valence bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' At E ≤ EC + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='25 eV (where EC is the conduction band minimum), the LDOS is confined close to the center of the MoS2 monolayer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', near the Mo atoms) with sharp, narrow peaks appearing just to the left and right of the main central peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' These sharp satellite peaks arise from the spatial distribution of the 4𝑑𝑧2 orbital of Mo, which has been shown to dominate the DOS at these energies in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='27,28 Furthermore, we find that broad peaks centered close to the S atoms appear in the LDOS at E > EC + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' As shown in the projected DOS (pDOS) in Figure 2c, the S atoms begin to contribute to the DOS in the conduction bands at ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV above the conduction band minimum, corresponding to the Q-K valley separation ΔEQK from our DFT simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Projected band structures have also shown that S atoms contribute weakly to the electronic structure of monolayer MoS2 at the conduction band minimum, but they contribute noticeably to the Q-valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='27,29 We therefore attribute these peaks to contributions to the DOS from the S atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Similar laterally positioned peaks are present at all values of E ≤ EV (where EV is the valence band maximum) we consider in Figure 2b, which is also consistent with the pDOS in the valence bands: as shown in Figure 2d, S atoms contribute to the DOS in the valence bands at all considered energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Similarly, projected band structures have shown that both Mo and S atoms contribute significantly to the valence bands of monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='27,29 We note that the exact ΔEQK for monolayer MoS2 in vacuum is not precisely known30 and that the band structure of monolayer MoS2 can vary depending on strain31 and its surrounding dielectric environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='14 For example, the experimental ΔEQK for monolayer MoS2 encased in quartz and WS2 is ΔEQK ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='11  5  eV,32 and simulated values range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='071 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='270 eV, depending on the approach used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='30 To accommodate this uncertainty in the value of ΔEQK, we investigate its effect on Csc and nch in Section S1 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We also note from Figures 2a,b that the LDOS extends slightly beyond tch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='615 nm, which occurs because DFT simulations of 2D MoS2 assume that the semiconductor is surrounded by vacuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' in reality, a semiconductor’s LDOS cannot so easily penetrate into an insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='33 However, we shortly demonstrate that this non-ideality should not significantly affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Normalized spatial distribution of the local density of states L(E,z) across monolayer MoS2 at (a) E ≥ EC, and (b) E ≤ EV, where the z coordinates of Mo and S atoms align with the location of atoms at the top of the figures (size of atoms are not to scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Projected density of states (pDOS) for (c) conduction bands and (d) valence bands of monolayer MoS2, where the contributions of all orbitals from each individual atom are summed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Note that the contributions from only one S atom are shown in (c) and (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', arbitrary units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  In Figures 3a,b, we calculate and plot Csc of monolayer MoS2 on linear and logarithmic y-axes, respec- tively, by self-consistently solving equations (4) and (5) under an applied gate bias, with the correspond- ing nch values plotted in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' For now, we set the permittivity of the gate insulator to an extremely large value (Cox → ∞), so that ψsurf = VG, allowing us to study the intrinsic capacitance of monolayer MoS2 by neglecting the potential drop across the gate insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We will shortly relax this assumption and study monolayer MoS2-based capacitors with finite EOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' At |ψsurf| − EG/(2q) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='25 V, the capacitance of p-type MoS2 exceeds that of n-type MoS2, which is due to the DOS near the valence band edge being larger than the DOS near the conduction band edge (Figures 2c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, as ψsurf is pushed farther into the conduction bands, the slopes of both Csc and nch increase sharply for n-type MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' This increase is due to the step-like increase in the DOS at the Q-valley, noting that thermal broadening in equation (4) allows the DOS from the Q-valley to also contribute when EF is a few kBT below the edge of the Q-valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We note that this effect has been experimentally observed in MoSe2 and WSe2 monolayers, which have similar electronic structures to that of MoS2 (including a Q- valley above the conduction band edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Thus, the shape of the electron density in our Figure 3c resem- bles similar experimental curves for MoSe2 and WSe2 monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='18 (a) (b) (c) EC ∆E  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='05 eV (d) ΔEQK ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV EV EV - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='35 eV EC + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='35 eV S Mo S S Mo S Mo S S Mo ∆E  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='05 eV  6  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Semiconductor capacitance Csc as a function of |ψsurf| − EG/(2q) plotted on (a) linear and (b) logarithmic axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' For comparison, we also plot quantum capacitance Cq as a function of the channel potential ψch [calculated using equation (2) with gvm* = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='01m0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='96m0 for electrons and holes,4 respectively] in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (c) Charge carrier density nch for n-type and p-type monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The location of the Q-valley for n-type monolayer MoS2 is marked in (a) and (c) with a vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Note, carrier densities over ~2 × 1013 cm-2 are not accessible in practice with conventional dielectrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' HfO2), only with solid or liquid electrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  We also compare our computed Csc values to the conventional Cq for both n- and p-type monolayer MoS2, which we have plotted alongside Csc in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Although our calculated Csc closely matches Cq at small gate voltage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', non-degenerate surface potentials), we find that at high gate voltages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', de- generate potentials), our computed Csc values are substantially lower than the traditionally calculated Cq = Cdq = 70 μF/cm2 for n-type and 200 μF/cm2 for p-type monolayer MoS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' To understand why Csc matches Cq only for non-degenerate potentials, we first plot the charge distributions and poten- tial across the thickness of n-type (p-type) MoS2 when ψsurf is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3 V below (above) the conduction (va- lence) band edge in Figures 4a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' For this non-degenerate case, the charge distributions are nearly sym- metric across the channel, closely matching the LDOS distributions shown in Figures 2a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' This result is consistent with the potential profile shown in Figure 4b: there is nearly no potential drop across the monolayer MoS2 thickness, allowing electronic states to contribute to the carrier density equally effi- ciently, regardless of their location in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Hence, Csc ≈ Cq in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Next, to understand why Csc < Cq at degenerate potentials, we plot the charge distributions and potential across the thickness of monolayer n-type (p-type) MoS2 when ψsurf is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3 V above (below) the conduction (valence) band edge in Figures 4c,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  We find that for this degenerate case, the charge distributions are heavily asymmetric and skewed towards the gate electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' This asymmetry can be explained from the potential drops in Figure 4d, which shows that the local electrostatic potential is highest near the gate electrode and rapidly decays across the monolayer channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We recall from Figures 2a,b that most avail- able states are near the center of the channel in this region of relatively low potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Therefore, many (a) p n n : ψsurf = EC/q p : ψsurf = EV/q Q Csc (Our model) n : ψsurf = EC/q p : ψsurf = EV/q p n Cq (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' 2) p-type Cdq n-type Cdq (b) n p n : ψsurf = EC/q p : ψsurf = EV/q Q (c)  7  states are unable to efficiently contribute to nch, which is why Csc < Cdq even at high ψsurf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' As a result, the charge density is askew across the thickness of the MoS2 monolayer, with the S atoms opposite to the gate contributing little to the channel charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We conclude that at degenerate surface potentials, the shapes of the LDOS and potential profile play pivotal roles in dictating Csc for 2D semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (a) Distributions of charge density and (b) potential profile across the MoS2 monolayer with ψsurf inside the band gap, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3 eV below the conduction band edge (for electrons) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3 eV above the valence band edge (for holes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (c) Distributions of charge density and (d) potential profile across the MoS2 monolayer with ψsurf of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3 eV above the conduction band edge (for electrons) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3 eV below the valence band edge (for holes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Dashed lines indicate boundaries with the gate electrode and bottom insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Note, the carrier densities in (c) are much greater than in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' In (c), the carrier distribution also extends slightly outside tch by < 10 % of total nch, meaning that the charge centroid is closer to the gate18 and that Csc is slightly overestimated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Finally, we note that the potential drops are different for n- and p-type devices in Figure 4d because the capacitance of n- type monolayer MoS2 is smaller than that of p-type MoS2 for the ψsurf we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='34  We next consider the impact of Csc in practical MOS devices based on monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Figures 5a,b plot CG and nch for MOS capacitors as in Figure 1b with EOTs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm (ϵox = 20ϵ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' For comparison, we additionally plot Cox = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='9ϵ0/EOT, as well as the classical carrier density 𝑛ch classical = (𝑉G − 𝑉T)𝐶ox/𝑞 (for electrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' the bracketed term is negated for holes), where the threshold voltage is VT = ± EG/(2q) for n- and p-type devices, assuming the same gate metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' At EOT = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm, CG saturates close to Cox and 𝑛ch ≈ 𝑛ch classical, but the observed CG and nch deviate significantly from Cox and 𝑛ch classical, respectively, at EOTs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 and 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' For example, at EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm, the CG of n-type MoS2 increases from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='38 μF/cm2 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 V to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='35 μF/cm2 at 1 V, remaining between 63% and 78% of Cox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, we note that CG is sensitive to small variations in VG and to the position of the Q-valley for n- type MoS2, as captured in Figure 5a and discussed in Supporting Information Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' n p (a) (b) (c) (d) tch V(z) =ψch(z) p n n p n Gate Bottom  insulator p ψsurf in band gap ψsurf degenerate Gate Bottom  insulator  8  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (a) Gate capacitance CG and (b) charge carrier density nch for n-type and p-type monolayer MoS2 as functions of VG at EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Solid red (blue) lines represent n-type (p-type) MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Dashed lines mark the oxide capacitance Cox and the conventionally calculated carrier density 𝑛ch classical, highlighting the im- portance of quantum corrections in the limit of ultrathin EOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Small red arrows mark approximately where the Q-valley of n-type MoS2 begins to contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (c) Calculated ratio of 𝑛ch/𝑛ch classical for n-type MoS2 and (d) p- type MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The true charge density is lower than the classical estimate in the limit of high VG when transistors are strongly turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, 𝑛ch classical approaches zero and underestimates the true charge density near threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Similarly, we find that the classical equation overestimates nch in 2D MOS capacitors with small EOTs, as shown in Figures 5c,d for n- and p-type MoS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' At EOTs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 and 1 nm, the channel carrier density nch is as small as 65% and 79% (79% and 89%) of 𝑛ch classical for n-type (p-type) MoS2 in the VG range considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We note that contributions from the Q-valley are visible for the n-type device with EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm, leading to an increase in CG and nch when the voltage is sufficiently high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' This effect has also been observed experimentally in ion-gated MoSe2 and WSe2 monolayers, which have similar band structures to that of monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='18 We note that 𝑛ch classical = 𝐶ox(𝑉G − 𝑉T)/𝑞 should not be applied near or below VT, because this expres- sion neglects subthreshold charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='6 Instead, nch may be approximated in both the off- and on-states by taking CG as the series combination of Cq and Cox,11,12 and then integrating the result (up to the relevant VG) to find the carrier density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, as we demonstrate in Section S2 of the Supporting Information, correcting for Cq in this manner still significantly overestimates both CG and nch in the on-state for 2D channels with low EOT, highlighting the importance of including both quantum and centroid effects when modeling these devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Finally, we assess how Csc limits CG for monolayer MoS2 compared to other semiconductors, including silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Although decreasing tch improves the channel electrostatics, tch < 5 nm causes surface scattering to limit silicon carrier mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='2,35,36 At the limit tch ≈ 5 nm, a previous study37 has shown that the Csc of silicon limits a dual-gated silicon FET with an EOT of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm to CG ≈ 7 μF/cm2 at an overdrive VOV = VG – VT = 1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' In direct comparison, our calculations show that a similar dual-gated structure with n-type monolayer MoS2 offers CG ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='9 μF/cm2 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='6 μF/cm2 for p-type), over 50% greater than that of (a) (b) (c) 1 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm n-type n p p Cox n EOT (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 EOT (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 (d) EOT= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm 1 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm p-type EOT= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm  9  silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The MoS2 advantage persists even when the silicon thickness is reduced37 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm, which yields CG ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1 μF/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We refer the reader to Section S2 of the Supporting Information for a description of how we calculated CG and nch for dual-gated devices and for full CG and nch vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' VG curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The Csc of monolayer MoS2 compares even more favorably to III-V semiconductors, whose low DOS are known to limit their Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' A previous study38 has shown that Cq limits dual-gated InGaAs MOS capac- itors with EOT = 1 nm to CG < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='6 μF/cm2 at channel thickness tch = 25 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' as tch decreases, this CG worsens because the DOS shrinks due to quantum confinement effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='38 Using the same approach as above, we find that a dual-gated monolayer n-type MoS2 capacitor with EOT = 1 nm offers CG ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='55 μF/cm2 (or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='00 μF/cm2 for p-type) at VOV = 1 V, over three times higher than InGaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The results from Figure 5a also indicate that single-gated monolayer MoS2 capacitors with EOT of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm offer higher CG than those reported for single-gated In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='7Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3As and InAs capacitors with similar or lower EOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 In conclusion, we have shown that variations in carrier density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', the centroid capacitance), potential, and density of states across the thickness of monolayer MoS2 severely limits its on-state Csc to values well below its degenerate quantum capacitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  As a result, gate capacitance estimates made by classical equations must be corrected when evaluating the carrier density and mobility of devices with small EOTs below ~2 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Nevertheless, we find that in strong inversion, the CG of dual-gated n-type monolayer MoS2 capacitors is over 50% higher than dual-gated silicon MOS capacitors at EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm and over three times higher than InGaAs capacitors at EOT = 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The monolayer MoS2 capacitance advantage is higher at lower EOTs, ultimately indicating that strong current and transconductance may be achieved in such 2D transistors if their channel mobility and contact resistance continue to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  ASSOCIATED CONTENT Supporting Information: Description and variation of the Q-K energy valley separation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' comparison to quantum capacitance approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' methodology and results for dual-gated calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Author contributions: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' conceived the idea and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' car- ried out all calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Notes: The authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' ACKNOWLEDGMENTS R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
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+page_content=' Strain-Enhanced Mobility of Monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Nano Lett 22, 8052–8059 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='nanolett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='2c01707 32 Hill, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Rigosi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Rim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Flynn, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
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+page_content=' Nano Lett 16, 4831-4837 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='nanolett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='6b01007 33 Fiore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Klinkert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Ducry, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Backman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' & Luisier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Influence of the hBN Dielectric Layers on the Quantum Transport Properties of MoS2 Transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Mater 15 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='3390/ma15031062 34 In a physical device, the remainder of the potential would be dropped across the bottom insulator, which is not observed here because we apply a Neumann boundary condition at the far side of the bottom insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, we have verified that the potentials across the channel of Figure 4b,d become nearly identical to those obtained when we instead ground the far side of the bottom insulator (note that electrical ground corresponds to the mid-gap voltage) and allow the bottom insulator thickness to approach infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  35 Uchida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Experimental study on carrier transport mechanism in ultrathin-body SOI nand p- MOSFETs with SOI thickness less than 5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Digest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Int Electron Devices Meeting, 47-50 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1109/IEDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1175776 36 English, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Shine, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Dorgan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Saraswat, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' & Pop, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Improved Contacts to MoS2 Transistors by Ultra-High Vacuum Metal Deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Nano Lett 16, 3824-3830 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='nanolett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='6b01309 37 Khan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Ashraf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' & Khosru, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Effects of wave function penetration on gate capacitance modeling of nanoscale double gate MOSFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' 2007 IEEE Conference on Electron Devices aEnd Solid- State Circuits, 137-140 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1109/EDSSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='4450081 38 Yadav, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Capacitance Modeling in III–V FinFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' IEEE  Trans Electron Devices 62, 3892-3897 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1109/TED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='2480380  1  Supporting Information  How do Quantum Effects Influence the Capacitance and Carrier Density of Monolayer MoS2 Transistors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Robert K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Bennett1 and Eric Pop1,* 1Department of Electrical Engineering, Stanford University, Stanford, California 94305, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' *Contact: epop@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='edu  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Effect of Q-K Energy Valley Separation Our density functional theory (DFT) calculations yield an energy difference ΔEQK ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV between the Q- and K-valleys in the conduction bands of monolayer MoS2, as labeled in Figure S1a (we note this Q-valley is sometimes called Τ or Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, the computed value of ΔEQK for monolayer MoS2 is highly dependent on the input parameters used in DFT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', exchange-correlation functionals and pseudopotentials),1 where the settings that yield the actual ΔEQK of monolayer MoS2 in vacuum are presently unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The band structure of monolayer MoS2 also varies with the surrounding dielectric environment,2 further complicating the question of which ΔEQK is relevant for a given physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' To accommodate this uncertainty in the “correct” ΔEQK, we repeat calculations of the semiconductor capacitance Csc and carrier density nch for n-type MoS2 with ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV and compare these results to those obtained using ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' As p-type MoS2 does not have an associated ΔEQK or similar low-energy peaks that contribute to its density of states (DOS) in the range of energies of interest, the p-type results would be unchanged and are not repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (a) Band structure of monolayer MoS2 near the conduction and valence band edges obtained from DFT, where the energy separation ΔEQK is labeled between the Q- and K-valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Energies (E) are not to scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', the band gap is reduced to highlight details of the conduction band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Note that we neglect spin-orbit coupling in this work, although for monolayer MoS2, including spin-orbit coupling only negligibly influences the value of ΔEQK obtained from DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1 (b) Computed semiconductor capacitance Csc and (c) carrier density nch for n-type monolayer MoS2 capacitors with Q-K energy separations ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (a) ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV (b) (c) ΔEQK K Q  2  To obtain a local DOS (LDOS) profile with ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV, we take the LDOS used in the main text with ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV and splice together the LDOS at energies E < EC + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV and E > EC + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV to create a continuous LDOS profile with ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We then compute Csc and nch with this LDOS using the same methodology as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' As shown in Figures S1b and S1c, Csc and nch are the same for both values of ΔEQK we consider at low ψsurf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, as ψsurf increases, the states near the Q-valley contribute at lower energies for the LDOS profile with ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV, resulting in an earlier onset for the second linear region of Csc, thereby increasing nch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Although this lower ΔEQK shifts this linear region of the Csc curve to the left, it does not significantly affect the maximum value of Csc ≈ 40 μF/cm2 in the range of ψsurf we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Next, we repeat the calculations of CG and nch presented in Figures 4a,b of the main text at EOTs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm using ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' As shown in Figures S2a and S2b, the value of ΔEQK used affects neither Csc nor nch at EOT = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm since CG is dominated by the oxide capacitance Cox at sufficiently large EOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' At EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 and 1 nm, however, we find that the higher Csc at ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV allows CG and nch to grow closer to the classical limit compared to ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  This result signifies that smaller ΔEQK provides further advantage of monolayer MoS2 over Si and III-V channels from the point of view of quantum capacitance and channel carrier density at a given overdrive voltage VOV = VG – VT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' In practice, note that ΔEQK is controlled by the strain applied3 and may be controlled by the environmental dielectric as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (a) Gate capacitance CG and (b) carrier density nch for n-type MoS2 with Q-K energy separations ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV at EOTs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Dotted green (solid red) lines represent the LDOS profile obtained using ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26) eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Dashed lines mark the oxide capacitance 𝐶ox = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='9𝜖0/EOT  and the conventionally calculated carrier density 𝑛ch classical = 𝐶ox(𝑉G − 𝑉T)/𝑞 , where the threshold voltage is VT = EG/(2q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Comparison to Quantum Capacitance Approximation As discussed in the main text, the classical approximation of gate capacitance, CG = Cox (where Cox is the oxide capacitance) and the classical approximation for charge carrier density, 𝑛ch classical = 𝐶ox(𝑉G − 𝑉T)/𝑞, are well-known to be inaccurate near or below the threshold voltage VT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Instead, CG is typically modeled in 2D semiconductors as the series combination of the quantum capacitance Cq and the oxide capacitance Cox, yielding CG-1 ≈ Cox-1 + Cq-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Since Cq is a function of the semiconductor’s surface potential, when using this equation, CG must be solved iteratively such that the ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV Cox nch (classical) ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='13 eV ΔEQK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='26 eV (a) (b) EOT (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 EOT (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5  3  voltage drop across the oxide and semiconductor are self-consistent with Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Then, nch may be approximated at any VG by integrating this result to obtain the carrier density,  𝑛ch  Cq−corrected ≈  1  𝑞 ∫  [𝐶q  −1(𝑉G  ′) + 𝐶ox  −1]  −1d𝑉G  ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  𝑉G  −𝐸G/2𝑞  (S1) As shown in Figure S3 below, the approximation CG-1 ≈ Cox-1 + Cq-1 matches the rigorously calculated CG presented in the main text in the subthreshold region (here |VG| < EG/2q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, this approximation considerably overestimates the CG of devices with EOT ≤ 1 nm at larger overdrive voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' This finding is consistent with our previous result from Figure 3b in the main text, which shows that Cq similarly overestimates the semiconductor’s capacitance in the on-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Figure S3: Gate capacitance CG for n-type and p-type monolayer MoS2 as functions of VG at EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Solid red (blue) lines represent the CG of n-type (p-type) MoS2 calculated using the full ab initio approach described in the main text that includes charge centroid effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Dash-dotted red (blue) lines are the approximation CG-1 ≈ Cox-1 + Cq-1 (where Cq is calculated using the self-consistent approach described above) which does not include centroid effects, and black dashed lines mark the oxide capacitance, Cox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The discrepancies between solid lines and approximations highlight the importance of quantum and charge centroid effects in the on-state, which are most important at small EOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  Similarly, to compare our rigorously computed nch in the main text to the carrier density corrected with only the quantum (not centroid) capacitance, we plot our calculated nch from the main text alongside Equation S1 in Figure S4, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' For a more thorough comparison, we also include our original 𝑛ch classical on the same plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' At an EOT of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm, Equation S1 accurately approximates our rigorously calculated nch in both the off- and on-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' However, at EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 and 1 nm, Equation S1 does not correctly predict the charge in the on-state significantly better than the classical approximation 𝑛ch classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Again, this result can be understood based on Figure 3b in the main text, where we show that our rigorously calculated semiconductor capacitance Csc < Cq in the on-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' From these results, we conclude that although including corrections for Cq enables good approximations of CG and nch in and near the off-state, the more rigorous approach for calculating these quantities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', including spatial variations in the density of states, potential, and volumetric charge density) presented in the main text should be used to understand 2D semiconductor devices with sub-1 nm EOT in the on-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' EOT (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 n p Cox p-type (Cox -1 + Cq -1)-1 n-type (Cox -1 + Cq -1)-1 This work  4  Figure S4: Charge carrier density nch for n- and p-type monolayer MoS2 as functions of VG at (a) EOT = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm, (b) EOT = 1 nm, (c) EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Solid red (blue) lines represent nch of n- (p-) type MoS2 calculated using the full ab initio approach described in the main text, which includes charge centroid effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Dashed red (blue) lines represent the quantum capacitance-corrected carrier density 𝑛ch Cq−corrected (Equation S1, using the self-consistent approach described above) that does not include centroid effects, and black dashed lines mark the classically calculated charge carrier density, 𝑛ch classical = 𝐶ox(𝑉G − 𝑉T)/𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' The discrepancies between solid lines and approximations highlight the importance of quantum and charge centroid effects in the on-state, which are most important at small EOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Monolayer MoS2 Dual-Gated Capacitors We calculate nch and CG of dual-gated monolayer MoS2 capacitors by solving equations (4) and (5) in the main text self-consistently, just as we do when computing these quantities for single-gated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Here we use the device schematic shown in Figure S5a, which is similar to the single-gated device in Figure 1b except that the relative permittivities of both the top and bottom insulator are set to 20 and the thicknesses of the top and bottom insulators are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We update the bottom boundary condition [previously ∂V(z)/∂z = 0 at this boundary for single-gated devices] to V(z) = VG, where VG is the gate voltage applied at both the top and bottom electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' We plot CG and nch at EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm for both n-type and p-type monolayer MoS2 for this dual-gated device in Figures S5b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  p type nch  Cq corrected  n type nch  Cq corrected  p type nch  Cq corrected  n type nch  Cq corrected  n  p  (a)  EOT = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm  p type nch  Cq corrected  n type nch  Cq corrected  (b)  n  p  EOT = 1 nm  (c)  n  p  EOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 nm  5  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (a) Schematic of a monolayer MoS2 dual-gated MOS capacitor with boundary conditions applied when solving equations (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' (b) Gate capacitance CG and (c) carrier density nch for these dual-gated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Solid red (blue) lines represent n-type (p-type) MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Dashed lines mark the total dual-gated oxide capacitance 𝐶ox = 2 ∗ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='9𝜖0/EOT) (where the prefactor of 2 accounts for both gates) and the conventionally calculated carrier density 𝑛ch classical = 𝐶ox(𝑉G − 𝑉T)/𝑞, where the threshold voltage is VT = ± EG/(2q) for n- and p-type devices, assuming the same gate metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='  SUPPORTING REFERENCES: 1 Gaddemane, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Gopalan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Van de Put, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' & Fischetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Limitations of ab initio methods to predict the electronic-transport properties of two-dimensional semiconductors: the computational example of 2H-phase transition metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' J Comput Electron 20, 49-59 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1007/s10825-020-01526-1 2 Ryou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Kc, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', & Cho, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Monolayer MoS2 Bandgap Modulation by Dielectric Environments and Tunable Bandgap Transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Sci Rep 6, 29184 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1038/srep29184 3 Hosseini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Elahi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Pourfath, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=', Esseni, & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Strain induced mobility modulation in single-layer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' J Phys D: Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' Phys 48, 375104 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='1088/0022- 3727/48/37/375104 n p Cox EOT (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 (b) p n EOT (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
+page_content='5 1 (c) (a)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf'}
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+Schrödinger Equation Driven by the Square of a Gaussian Field:
+Instanton Analysis in the Large Amplification Limit
+Philippe Mounaix1, ∗
+1CPHT, CNRS, École polytechnique,
+Institut Polytechnique de Paris, 91120 Palaiseau, France.
+(Dated: January 31, 2023)
+Abstract
+We study the tail of p(U), the probability distribution of U = |ψ(0, L)|2, for large U, ψ(x, z) being
+the solution to ∂zψ(x, z) −
+i
+2m∇2
+⊥ψ(x, z) = g|S(x, z)|2ψ(x, z), where S(x, z) is a complex Gaussian
+random field, z and x respectively are the axial and transverse coordinates, with 0 ≤ z ≤ L,
+and both m ̸= 0 and g > 0 are real parameters.
+We perform the first instanton analysis of
+the corresponding Martin-Siggia-Rose action, from which it is found that the realizations of S
+concentrate onto long filamentary instantons, as U → +∞.
+The tail of p(U) is deduced from
+the statistics of the instantons. The critical value of g above which ⟨U⟩ diverges is checked to
+coincide with the one obtained by the completely different approach developed in Mounaix et al.
+2006 Commun. Math. Phys. 264 741 (and Erratum 2008 Commun. Math. Phys. 280 281).
+Analytical predictions are supported by numerical simulations that clearly show the statistical bias
+of S towards the instanton, at large U. For the biased realizations of S, the high maxima — or
+‘hot spots’ — of |S(x, z)|2 tend to cluster in the instanton region.
+Keywords: stochastic partial differential equations, instanton analysis, extreme event statistics, laser-plasma
+interactions
+∗Electronic address: philippe.mounaix@polytechnique.edu
+1
+arXiv:2301.13000v1  [cond-mat.stat-mech]  30 Jan 2023
+
+I.
+INTRODUCTION
+In the second part of their seminal paper on the breakdown of linear instability in stim-
+ulated Brillouin scattering [1], Rose and DuBois investigated the following equation for the
+complex amplitude ψ(x, z) of the scattered light electric field
+�
+�
+�
+∂zψ(x, z) −
+i
+2m∇2
+⊥ψ(x, z) = g|S(x, z)|2ψ(x, z),
+0 ≤ z ≤ L, x ∈ Λ ⊂ Rd, and ψ(x, 0) = 1.
+(1)
+In Eq. (1), z and x respectively denote the axial and transverse coordinates in a plasma of
+length L and cross-sectional domain Λ (often a torus like, e.g., in mathematics oriented work
+and/or numerical simulations using spectral methods). The boundary condition at z = 0 is
+taken to be a constant for simplicity and m ̸= 0 is a real parameter introduced for conve-
+nience. In Ref. [1], the coupling constant g > 0 is proportional to the average laser intensity
+and the complex amplitude of the laser electric field S(x, z) is a homogeneous Gaussian
+random field with zero mean and normalized intensity ⟨|S(x, z)|2⟩ = 1. For our purposes,
+we can be less restrictive and take S(x, z) transversally homogeneous with normalization
+L−1 � L
+0 ⟨|S(x, z)|2⟩ dz = 1. From now on, we accept the idealizations inherent in the deriva-
+tion of Eq. (1), setting aside the question of its validity as a realistic model (which varies
+from one physical problem to the other). As a stochastic PDE, the diffraction-amplification
+problem (1) is a Schrödinger equation driven by the square of a Gaussian field.
+Using heuristic arguments and numerical simulations, Rose and DuBois found that the
+expected value of the scattered energy density, ⟨|ψ(x0, L)|2⟩, at some given x0 ∈ Λ diverges
+for every L > 0 when g is greater than some critical value, gc(L), yet to be determined.
+Here, the average ⟨|ψ|2⟩ is taken over the realizations of the Gaussian field S. Physically,
+this divergence was interpreted in [1] as indicating the breakdown of the linear model (1) and
+the onset of a saturated nonlinear regime in high overintensities, or hot spots, of |S(x, z)|2.
+We will shortly come back to the role of the hot spots in the divergence of ⟨|ψ(x0, L)|2⟩. Note
+that in the limit referred to in [1] as the independent hot spot model, this divergence was
+pointed out by Akhmanov et al. 20 years before [2]. The problem was then analyzed in [3–5]
+from a more rigorous mathematical point of view, establishing the numerical results of Ref.
+[1] on much firmer ground and giving the exact expression of the critical coupling gc(L). In
+the following, we will take x0 = 0 without loss of generality (by statistical invariance under
+x-translation) and we will write U = |ψ(0, L)|2.
+2
+
+Whether or not ⟨U⟩ diverges depends on the upper tail of p(U) — the probability distri-
+bution function (PDF) of U —, which is determined by the realizations of S(x, z) yielding
+a large U, with U and S(x, z) related to each other through Eq. (1). It is then natural to
+ask what these particular realizations of S(x, z) are like, from which probability distribution
+they are drawn, and if the corresponding tail of p(U) does give the correct value of gc(L).
+Answering these questions is the subject of this paper.
+To put our work into perspective, it is interesting to recall how the existence of gc(L) has
+been interpreted in laser-plasma physics literature since Ref. [1]. The interpretation relies
+on the implicit assumption that the realizations of |S(x, z)|2 giving rise to a large U in the
+tail of p(U) and the generic ones for which U is in the bulk of p(U) are alike, in the sense
+of being made up of local, statistically independent, overintensities, or hot spots, separated
+from each other by a few correlation lengths of S(x, z). Hot spot contribution to the ampli-
+fication of |ψ|2 can then be computed by using the remarkable result that intense hot spots
+have a non-random profile depending on the correlation function of S(x, z) and being the
+same for each hot spot [6, 7]. Thus, intense hot spots are entirely characterized by their
+random intensity which turns out to be exponentially distributed (for large intensity and
+to within slow, algebraic, corrections) [6, 8]. For g large enough, intense hot spots become
+statistically significant as the exponentially large amplification they produce outbalances
+their exponentially small scarcity, leading to the divergence of ⟨U⟩. The smallest value of g
+at which this divergence occurs defines the critical coupling gc(L) and for g > gc(L) physics
+could be expected to be dominated by intense hot spots. Unfortunately, this interpretation
+fails to give the correct value of gc(L) for L greater than a hot spot length [9, 10]. The
+assumption of high intensity, statistically independent hot spots giving the dominant con-
+tribution to the amplification in the large U limit must be revisited. Large values of U are
+produced by rare realizations of S(x, z) that have no reason a priori to look like generic
+realizations with no other structures than uncorrelated, local hot spots randomly scattered
+in Λ × [0, L]. It may or may not be so: the answer will come out of the calculations.
+In the simpler diffraction-free case where m−1 = 0 in Eq. (1), the problem reduces to a
+mere 1D amplification along z with ψ(L) = exp
+�
+g
+� L
+0 |S(z)|2dz
+�
+. A large value of |ψ(L)|2
+corresponds to a large value of
+� L
+0 |S(z)|2dz. Thus, the realizations of S(z) that form the
+tail of p(U) are the ones with a large L2-norm. These realizations were studied thoroughly
+in [10–12]. Let C(z, z′) = ⟨S(z)S(z′)∗⟩ and define the covariance operator TC acting on
+3
+
+f(z) ∈ L2([0, L]) by
+(TCf)(z) =
+� L
+0
+C(z, z′) f(z′) dz′,
+(2)
+with 0 ≤ z ≤ L. Write µ1 > 0 the largest eigenvalue of TC with degeneracy d1. It was
+proved in [10, 11] that the realizations of S(z) with a large L2-norm concentrate onto the
+fundamental eigenspace of TC, i.e., the eigenspace associated with the largest eigenvalue µ1.
+More specifically, writing {φ1, · · · , φd1} an orthonormal basis of the fundamental eigenspace
+of TC, one has
+S(z) ∼ √η
+d1
+�
+i=ν
+aνφν(z)
+(∥S∥2 → +∞),
+(3)
+with η ∼ ∥S∥2
+2, where ∥ · ∥2 denotes the L2-norm over [0, L]. The aνs are complex numbers
+normalized to �d1
+ν=1 |aν|2 = 1. The probability distribution of η has the gamma-distribution
+tail p(η) ∼ ηd1−1e−η/µ1 for large η, and the aνs define a random 2d1-dimensional (real) unit
+vector a with coordinates Re(aν) and Im(aν) (1 ≤ ν ≤ d1) the direction of which is uniformly
+distributed over the unit (2d1 − 1)-sphere. From Eq. (3) it is clear that the realizations of
+S(z) with a large L2-norm are less random than the Gaussian field S(z) itself. It only takes
+2d1 random quantities to characterize these realizations entirely: η and the direction of a.
+For instance, if µ1 is not degenerate (d1 = 1), Eq. (3) yields
+S(z)
+∥S∥2
+∼ eiθφ1(z)
+(∥S∥2 → +∞),
+(4)
+where θ is a random phase uniformly distributed over [0, 2π) and |S(z)|/∥S∥2 ∼ |φ1(z)| is
+non-random, which means that the profile of S(z) is purely deterministic in this case. Note
+that Eq. (4) rules out any description in terms of localized hot spots when L is large, as
+φ1(z) typically is a one-bump delocalized mode spreading over the whole domain 0 ≤ z ≤ L
+(see [10] for details). As will be seen further on, the randomness reduction of S when the
+amplification is large occurs in the m−1 ̸= 0 case too.
+From U = |ψ(L)|2 = exp(2g∥S∥2
+2) and η ∼ ∥S∥2
+2 as ∥S∥2 → +∞, one gets η ∼ (2g)−1 ln U
+as U → +∞. The tail of p(U) is then readily obtained from p(η) ∼ ηd1−1e−η/µ1 and the
+change of variables from η to U. One finds, in logarithmic form,
+ln p(U) ∼ ln
+� 1
+U p
+�
+η = 1
+2g ln U
+��
+(5)
+= −
+�
+1 +
+1
+2µ1g
+�
+ln U + (d1 − 1) ln ln U + O(1)
+(U → +∞),
+4
+
+from which it follows that p(U) has a leading algebraic tail ∝ U −ζ (modulated by logarithmic
+corrections in the amplitude) with exponent ζ = (1 + 1/2µ1g) depending continuously on
+the parameters of the model. Injecting this result into ⟨U⟩ =
+� +∞
+1
+Up(U) dU, one finds that
+the critical coupling in the diffraction-free case is given by gc(L) = 1/2µ1 (where µ1 depends
+on L) [10]. Note that it is also possible to determine the tail of p(U) exactly from the full
+Gaussian statistics of S without using η ∼ ∥S∥2
+2, which makes it possible to estimate the
+contribution of the subleading corrections to Eq. (3). Skipping the details, one finds that
+these corrections do not contribute to ln p(U) by terms greater than O(1) as U → +∞.
+To conclude this brief overview of diffraction-free results, let us mention the interesting
+connection between the concentration onto the fundamental eigenspace of TC in Eq. (3) and
+the Bose-Einstein condensation of S(z) in the ‘thermodynamic’ limit defined by L → +∞
+and ∥S∥2
+2 → +∞ with fixed ∥S∥2
+2/L, see [12].
+Note also that more general concentra-
+tion properties can be found in the limit where the large L2-norm is replaced with a large
+quadratic or linear form, see [13, 14].
+By contrast, much less is known in the general case with diffraction where m−1 ̸= 0 in
+Eq. (1). The only results so far are (i) the numerical ones in the second part of [1] and (ii)
+the analytical calculation of the critical coupling performed in [5] where it is proved that the
+critical coupling without diffraction cannot be less than the one with diffraction, the latter
+being given by
+gc(L) =
+1
+2 supx(·)∈B(0,L) µ1[x(·)],
+(6)
+and the former by 1/2µ1[x(·) ≡ 0]. In Eq. (6), B(0, L) denotes the set of all the continuous
+paths in Λ satisfying x(L) = 0 and µ1[x(·)] is the largest eigenvalue of the covariance operator
+Tx(·) defined by Eq. (2) with C(z, z′) = ⟨S(x(z), z)S(x(z′), z′)∗⟩ (see also Eq. (43)).
+The
+question
+then
+arises
+whether
+the
+presence
+of
+the
+non-local
+quantity
+supx(·)∈B(0,L) µ1[x(·)] in the expression for gc(L) — a presence which does not appear in
+hot spot models — is the signature of a corresponding non-local structure in the realizations
+of S(x, z) giving rise to a large amplification. To answer this question we need to find a way
+to identify such realizations. The corresponding tail of p(U) will then be tested in return
+by checking that the critical coupling it yields coincides with the one in Eq. (6). The calcu-
+lations in [10, 11] are of no help as being specific to the diffraction-free case. To determine
+S(x, z) when U is large and get the tail of p(U) in the presence of diffraction we need a
+5
+
+different approach.
+A possible line of attack is through the functional integral formalism introduced by
+Janssen [15], DeDominicis [16, 17], and Phythian [18] (see also [19, 20]).
+This method
+provides a formal description of classical statistical dynamics in terms of functional integrals
+analogous to Feynman’s action-integral formalism of quantum theory. Applying the method
+to the stochastic equation (1), one finds that p(U) can be formally written as the functional
+integral
+p(U) ∝
+�
+ϕ(x,0)=b
+δ(U − |ϕ(0, L)|2) eA D2ϕ D2 ˜ϕ D2S,
+(7)
+where ϕ and ˜ϕ are complex Martin-Siggia-Rose conjugate fields [21], D2 ≡ DRe(·)DIm(·),
+and A ≡ A(ϕ, ˜ϕ, S) is an ‘action’ depending on ϕ, ˜ϕ, and S, often referred to as the Martin-
+Siggia-Rose (MSR) action in the literature. When U is large, the functional integral on
+the right-hand side of Eq. (7) is determined by the field configuration — called the ‘leading
+instanton’ — corresponding to the highest saddle-point of the action A. Note that, usually,
+S is integrated out in Eq. (7), leading to
+p(U) ∝
+�
+ϕ(x,0)=b
+δ(U − |ϕ(0, L)|2) eL D2ϕ D2 ˜ϕ,
+(8)
+where the action L ≡ L(ϕ, ˜ϕ) is given by
+eL(ϕ, ˜ϕ) ∝
+�
+eA(ϕ, ˜ϕ,S) D2S.
+(9)
+The expressions of p(U) in Eqs. (7) and (8) are equivalent. Either can be used and the large
+U behavior of p(U) can equally be obtained from an instanton analysis of Eq. (8), instead of
+Eq. (7). As we want to determine the most probable realizations of S(x, z) when U is large,
+it is natural to keep S explicit in the problem and to use the functional representation (7).
+When U is large, the fluctuations of the fields around the instanton are small and can be
+integrated out in Eq. (7) as standard Gaussian fluctuations, yielding the leading asymptotic
+behavior
+ln p(U) ∼ ln
+�
+δ(U − |ϕinst(0, L)|2)
+�
+Sinst
+(U → +∞),
+(10)
+where the subscript ‘inst’ stands for leading instanton and ⟨·⟩Sinst denotes the average over
+the realizations of Sinst. In the diffraction-free case, it will be checked in Sec. II that Sinst
+coincides with the right-hand side of Eq. (3) and that the tail of p(U) in Eq. (10) is the same
+as the one in Eq. (5).
+6
+
+The first attempt to obtain PDF tails from the highest saddle-point of the action in a
+functional integral representation was made by Giles for Navier-Stokes turbulence in [22].
+Unfortunately, the perturbative approach followed in this work was doomed to fail as in-
+stantons are nonperturbative objects by nature. The proper, nonperturbative, instanton
+analysis of intermittency in fluid turbulence was performed by Falkovich et al. in [23]. See
+also the instanton analyses of PDF tails in forced Burgers turbulence in [24, 25]. Here, we
+give the first instanton analysis of the stochastic amplifier (1) in the large amplification limit
+of interest in the overcritical regime g > gc(L). Our strategy is two-step:
+• write the MSR action A(ϕ, ˜ϕ, S) for the stochastic amplifier (1) and find the corre-
+sponding leading instanton Sinst. The realizations of Sinst define the driver onto which
+the realizations of S concentrate in the large U limit, which generalizes the diffraction-
+free result in Eq. (3) to the case with diffraction;
+• use the instanton ϕinst and Sinst obtained at the first step on the right-hand side of
+Eq. (10) to get the tail of p(U).
+Before entering the details of the calculations, it is useful to give a brief summary of the
+main new results obtained in this paper.
+⋄ We show that in the large U limit, the realizations of S concentrate onto large-scale
+filamentary instantons running along specific non-random paths in B(0, L). These
+paths are the ones maximizing the largest eigenvalue µ1[x(·)] of the covariance operator
+Tx(·) defined in Eq. (43). In the case of a ‘single-filament instanton’ (see Sec. IV), we
+prove that
+S(x, z) ∼
+1
+µmax
+� L
+0
+C(x − xinst(z′), z, z′)
+� d1
+�
+ν=1
+cνφν(z′)
+�
+dz′
+(U → +∞),
+(11)
+where C(x − x′, z, z′) = ⟨S(x, z)S(x′, z′)∗⟩, xinst(·) maximizes µ1[x(·)], and µmax ≡
+µ1[xinst(·)] with degeneracy d1 and orthonormal eigenfunctions φν. The cνs are i.i.d.
+complex Gaussian random variables with ⟨cν⟩ = ⟨c2
+ν⟩ = 0 and ⟨|cν|2⟩ = µmax. The
+instanton on the right-hand side of Eq. (11) — which lives within a long thin tube,
+or filament, running along xinst(·) (see the end of Sec. IV A) — is less random than
+the generic realizations of S for which U is in the bulk of p(U): it only takes the d1
+7
+
+(complex) random variables cν to characterize the instanton entirely. If µmax is not
+degenerate (d1 = 1), Eq. (11) reduces to
+S(x, z)
+∥S∥2
+∼ Aei arg(c1)
+� L
+0
+C(x − xinst(z′), z, z′) φ1(z′) dz′
+(U → +∞),
+where A > 0 is a constant, and the profile of S(x, z) defined by |S(x, z)|/∥S∥2 is
+asymptotically non-random as U → +∞.
+⋄ We determine the tail of p(U) for large U from the statistics of the instanton on the
+right-hand side of Eq. (11). We find that p(U) has a leading algebraic tail ∝ U −ζ with
+exponent ζ = (1 + 1/2µmaxg), modulated by a slow varying amplitude (slower than
+algebraic). Injecting this result into ⟨U⟩ =
+� +∞
+1
+Up(U) dU, we find that ⟨U⟩ diverges
+for all g > 1/2µmax. The critical coupling is thus given by gc(L) = 1/2µmax (where µmax
+depends on L), in agreement with Eq. (6). We can then explain the intriguing presence
+of the non-local quantity µmax in the expression of gc(L) as a direct consequence of
+the fact that the realizations of S causing the divergence of ⟨U⟩ are realizations of the
+non-local instanton (11), (rather than of localized hot spots, as is widely assumed).
+⋄ Finally, the emergence of the instanton in the realizations of S as U increases is
+observed in numerical simulations (see Sec. V). For large but finite U in the sampled
+range, the emerging large-scale instanton coexists with fluctuation induced small-scale
+hot spots. The presence of the instanton causes the hot spots to cluster in the instanton
+region instead of being uniformly scattered in Λ × [0, L], and the level of |S(x, z)|2
+between the hot spots remains significantly higher than it would be in the absence of
+instanton (see Figs. 6, 8(a), and 9).
+The paper is organized as follows. In Section II, we test the functional approach by
+revisiting the diffraction-free problem where the results are already known. In Section III,
+we write the instanton equations for the full problem with diffraction in the case of one
+transverse dimension (d = 1) and we specify the class of S we consider. Section IV is devoted
+to the solution of the instanton equations in the case of ‘single-filament’ instantons. The
+corresponding tail of p(U) is determined. In Section V, we verify our analytical predictions
+via numerical simulations. Finally, we discuss our results and their implications, especially
+in laser-matter interaction physics, and we give potential perspectives in Section VI. Some
+technical material is relegated to the appendices.
+8
+
+II.
+AMPLIFICATION WITHOUT DIFFRACTION REVISITED
+As a warm-up to the full problem (1), we test the functional approach on the simpler
+problem without diffraction and see how the results in Eqs. (3) and (5) can also be obtained
+from an instanton analysis of the appropriate MSR action.
+A.
+The MSR action A(ϕ, ˜ϕ, S)
+In the diffraction-free limit, m−1 = 0, the equation (1) reduces to the 1D stochastic
+amplifier (for fixed x, not written)
+�
+�
+�
+dzψ(z) − g|S(z)|2ψ(z) = 0,
+0 ≤ z ≤ L and ψ(0) = 1.
+(12)
+Let F[ψ(·)] be a functional of ψ(z) solution to Eq. (12). From the general formalism devel-
+oped in [15–20] it can be shown that F[ψ(·)] admits the functional integral representation
+F[ψ(·)] =
+�
+ϕ(0)=b
+F[ϕ(·)] e
+i
+2 (⟨ ˜ϕ|dz−g|S|2|ϕ⟩+c. c.) D2ϕ D2 ˜ϕ,
+(13)
+with Dirac’s bracket notation ⟨f|O|h⟩ =
+� L
+0 f(z)∗(Oh)(z) dz. Note that since ψ(0) is real,
+ψ(z) is also real for all z and a representation with real ϕ and ˜ϕ would have been sufficient.
+In Eq. (13) we have kept complex ϕ and ˜ϕ in anticipation of the generalization to the case
+with diffraction where ψ is a complex field. Now, using (13) with F[ψ(·)] = δ(U − |ψ(L)|2)
+in p(U) = ⟨δ(U − |ψ(L)|2)⟩S, where ⟨·⟩S denotes the average over the realizations of S, and
+writing TC the covariance operator of S (see Eq. (2)), one gets
+p(U) =
+�
+ϕ(0)=b
+�
+δ(U − |ϕ(L)|2) e
+i
+2 (⟨ ˜ϕ|dz−g|S|2|ϕ⟩+c. c.)�
+S D2ϕ D2 ˜ϕ
+∝
+�
+ϕ(0)=b
+δ(U − |ϕ(L)|2) e
+i
+2 (⟨ ˜ϕ|dz−g|S|2|ϕ⟩+c. c.)−⟨S|T −1
+C |S⟩ D2ϕ D2 ˜ϕ D2S.
+(14)
+The functional integral representation of p(U) in Eq. (14) is of the same form as the one in
+Eq. (7) with the MSR action
+A(ϕ, ˜ϕ, S) = i
+2
+��
+˜ϕ
+��dz − g|S|2�� ϕ
+�
++ c. c.
+�
+−
+�
+S
+��T −1
+C
+�� S
+�
+= i
+2
+�� L
+0
+˜ϕ∗(z)
+�
+dzϕ(z) − g|S(z)|2ϕ(z)
+�
+dz + c. c.
+�
+−
+� L
+0
+S∗(z)(T −1
+C S)(z) dz.
+(15)
+9
+
+B.
+Leading instanton and tail of p(U)
+The leading instanton which determines the large U behavior of p(U) in Eq. (14) is a
+stationary point of A(ϕ, ˜ϕ, S) under the restriction |ϕ(L)|2 = U. According to the usual
+procedure of Lagrange multipliers [26], it can be found as a stationary point of the action
+A′(ϕ, ˜ϕ, S) = A(ϕ, ˜ϕ, S) + λ|ϕ(L)|2 without restriction, where λ is a Lagrange multiplier.
+Write δA′(ϕ, ˜ϕ, S) the variation of A′(ϕ, ˜ϕ, S) under variations of the fields and their com-
+plex conjugates treated as independent variables, with endpoints ϕ(0) = ϕ∗(0) = 1, and
+˜ϕ(L+) = ˜ϕ∗(L+) = 0 (by causality principle. See, e.g., Ref. [23]). The stationarity condition
+δA′(ϕ, ˜ϕ, S) = 0 leads to the equations
+dzϕ(z) − g|S(z)|2ϕ(z) = 0 with ϕ(0) = 1,
+dz ˜ϕ(z) + g|S(z)|2 ˜ϕ(z) = −2iλϕ(L)δ(z − L) with ˜ϕ(L+) = 0,
+(16)
+dz ˜ϕ∗(z) + g|S(z)|2 ˜ϕ∗(z) = −2iλϕ∗(L)δ(z − L) with ˜ϕ∗(L+) = 0,
+and
+(T −1
+C S)(z) = −ig
+2 [ ˜ϕ∗(z)ϕ(z) + ˜ϕ(z)ϕ∗(z)] S(z),
+(17)
+or, equivalently,
+dzϕ(z) − g|S(z)|2ϕ(z) = 0 with ϕ(0) = 1,
+dz ˜ϕ(z) + g|S(z)|2 ˜ϕ(z) = 0 with ˜ϕ(L) = 2iλϕ(L),
+(18)
+dz ˜ϕ∗(z) + g|S(z)|2 ˜ϕ∗(z) = 0 with ˜ϕ∗(L) = 2iλϕ∗(L),
+and
+[TC ( ˜ϕ∗ϕ + ˜ϕϕ∗) S] (z) = 2i
+g S(z).
+(19)
+Note that if Re(λ) ̸= 0, ˜ϕ∗(z) as given by the third equation (18) is different from the complex
+conjugate of the solution to the second equation (18) for ˜ϕ(z) (i.e. ˜ϕ∗(z) ̸= ˜ϕ(z)∗). This
+seemingly strange result is a consequence of treating the fields and their complex conjugates
+as independent when varying the action.
+For the auxiliary (unphysical) field ˜ϕ, having
+˜ϕ∗(z) ̸= ˜ϕ(z)∗ in the instanton solution is not forbidden a priori, unlike the physical fields ϕ
+and S for which instantons are observable realizations with ϕ∗(z) = ϕ(z)∗ and S∗(z) = S(z)∗
+for all 0 ≤ z ≤ L.
+10
+
+The equations (18) are readily solved. One gets,
+ϕ(z) = eg
+� z
+0 |S(z′)|2dz′,
+˜ϕ(z) = 2iλϕ(L)eg
+� L
+z |S(z′)|2dz′,
+(20)
+˜ϕ∗(z) = 2iλϕ∗(L)eg
+� L
+z |S(z′)|2dz′,
+and ˜ϕ∗(z)ϕ(z) = ˜ϕ(z)ϕ∗(z) = 4iλ|ϕ(L)|2 = 4iλU, independent of z. Injecting this solution
+onto the left-hand side of Eq. (19), one obtains the eigenvalue equation
+(TCS)(z) =
+1
+2λgU S(z).
+(21)
+It follows immediately from Eq. (21) that an instanton solution for S is an eigenfunction of its
+covariance operator TC, which fixes the value of λ for each instanton, namely λ = 1/(2µngU),
+where µ1 > µ2 > · · · > 0 are the eigenvalues of TC. The leading instanton Sinst corresponds to
+the largest eigenvalue µ1. Writing d1 the degeneracy of µ1 and {φ1, · · · , φd1} an orthonormal
+basis of the fundamental eigenspace of TC, i.e., the eigenspace associated with µ1, one gets
+Sinst(z) =
+d1
+�
+ν=1
+cνφν(z),
+(22)
+where cν =
+� L
+0 Sinst(z)φν(z)∗dz is a complex number. The other components of the leading
+instanton, ϕinst, ˜ϕinst, and ˜ϕ∗
+inst, are the instanton solution in Eq. (20) with S = Sinst and
+λ = 1/(2µ1gU). In the following, we will only need the expression of ϕinst(L),
+ϕinst(L) = exp
+�
+g∥Sinst∥2
+2
+�
+,
+(23)
+where ∥ · ∥2 denotes the L2-norm over [0, L].
+Integrating out the fluctuations of the fields around the leading instanton in Eq. (14) and
+using the expressions of Sinst and ϕinst in Eqs. (22) and (23), one obtains the diffraction-free
+version of the general asymptotic expression (10),
+ln p(U) ∼ ln
+�
+· · ·
+�
+δ
+�
+U − exp
+�
+2g
+d1
+�
+ν=1
+|cν|2
+��
+d1
+�
+ν=1
+exp
+�
+−|cν|2
+µ1
+� d2cν
+πµ1
+= ln
+1
+Γ(d1)µ1
+� +∞
+0
+δ
+�
+U − e2gη� � η
+µ1
+�d1−1
+e−η/µ1dη
+(U → +∞),
+(24)
+where we have made the change of variable �d1
+ν=1 |cν|2 = η. It can be seen in Eq. (24) that the
+cνs in Eq. (22) are independent (complex) Gaussian random variables with ⟨cν⟩ = ⟨c2
+ν⟩ = 0
+11
+
+and ⟨|cν|2⟩ = µ1. Thus, writing cν = aν√η in Eq. (22), one gets
+Sinst(z) = √η
+d1
+�
+ν=1
+aνφν(z),
+(25)
+where η is a gamma-distributed random variable with p(η) = [Γ(d1)µ1]−1(η/µ1)d1−1e−η/µ1,
+and the aνs define a random 2d1-dimensional (real) unit vector a with coordinates Re(aν)
+and Im(aν) the direction of which is uniformly distributed over the unit (2d1−1)-sphere. For
+large U (hence large η), the realizations of S which contribute to the tail of p(U) concentrate
+onto the leading instanton, S(z) ∼ Sinst(z) (η → +∞), and one recovers the result of [10, 11]
+recalled in Eq. (3). It remains to perform the integral over η in Eq. (24), which can be done
+without difficulty. One obtains the asymptotic behavior given in Eq. (5),
+ln p(U) = −
+�
+1 +
+1
+2µ1g
+�
+ln U + (d1 − 1) ln ln U + O(1)
+(U → +∞),
+as expected.
+III.
+AMPLIFICATION WITH DIFFRACTION: GENERAL SETTING
+The approach followed in the previous section to deal with the diffraction-free case is
+completely different from the one in [10, 11].
+Having checked that both give the same
+results, we can now move on to the next step and use the instanton analysis to deal with
+the full problem with diffraction.
+A.
+MSR action and instanton equations
+We consider the transversally one-dimensional (d = 1) version of Eq. (1),
+�
+�
+�
+∂zψ(x, z) −
+i
+2m∂2
+x2ψ(x, z) = g|S(x, z)|2ψ(x, z),
+0 ≤ z ≤ L, x ∈ Λ ⊂ R, and ψ(x, 0) = 1,
+(26)
+where we take for Λ the circle of length ℓ (and radius ℓ/2π). The random field S is homo-
+geneous along x with normalization L−1 � L
+0 ⟨|S(x, z)|2⟩ dz = 1. (The generalization to more
+than one transverse dimension is straightforward.) Our goal is to determine the realizations
+of S and the tail of p(U) in the large U limit for U = |ψ(0, L)|2, with ψ(x, z) solution to
+Eq. (26). Write
+Dz, x2 ≡ ∂z −
+i
+2m∂2
+x2,
+(27)
+12
+
+and TC the covariance operator of S defined by
+(TCf)(x, z) =
+�
+Λ
+� L
+0
+C(x − x′, z, z′) f(x′, z′) dz′ dx′,
+f(x, z) ∈ L2(Λ × [0, L]),
+(28)
+with C(x − x′, z, z′) = ⟨S(x, z)S(x′, z′)∗⟩. The counterpart of Eq. (14) in the problem with
+diffraction reads
+p(U) ∝
+�
+ϕ(x,0)=b
+δ
+�
+U − |ϕ(0, L)|2�
+e
+i
+2 (⟨ ˜ϕ|Dz, x2−g|S|2|ϕ⟩+c. c.)−⟨S|T −1
+C |S⟩ D2ϕ D2 ˜ϕ D2S,
+(29)
+which is of the same form as in Eq. (7) with MSR action
+A(ϕ, ˜ϕ, S) = i
+2
+��
+˜ϕ
+��Dz, x2 − g|S|2�� ϕ
+�
++ c. c.
+�
+−
+�
+S
+��T −1
+C
+�� S
+�
+= i
+2
+��
+Λ
+� L
+0
+˜ϕ∗(x, z)
+�
+Dz, x2ϕ(x, z) − g|S(x, z)|2ϕ(x, z)
+�
+dz dx + c. c.
+�
+(30)
+−
+�
+Λ
+� L
+0
+S∗(x, z)(T −1
+C S)(x, z) dz dx.
+The derivation of the instanton equations from the action in Eq. (30) follows exactly the
+same line as in the diffraction-free case in Sec. II B. Varying A(ϕ, ˜ϕ, S) with the Lagrange
+multiplier term λ|ϕ(0, L)|2 and setting the variation to zero, one obtains the equations
+�
+Dz, x2 − g|S(x, z)|2�
+ϕ(x, z) = 0 with ϕ(x, 0) = 1,
+�
+Dz, x2 + g|S(x, z)|2�
+˜ϕ(x, z) = −2iλϕ(0, L)δ(x)δ(z − L) with ˜ϕ(x, L+) = 0,
+(31)
+�
+D∗
+z, x2 + g|S(x, z)|2�
+˜ϕ∗(x, z) = −2iλϕ∗(0, L)δ(x)δ(z − L) with ˜ϕ∗(x, L+) = 0,
+and
+(T −1
+C S)(x, z) = −ig
+2 [ ˜ϕ∗(x, z)ϕ(x, z) + ˜ϕ(x, z)ϕ∗(x, z)] S(x, z),
+(32)
+or, equivalently,
+�
+Dz, x2 − g|S(x, z)|2�
+ϕ(x, z) = 0 with ϕ(x, 0) = 1,
+�
+Dz, x2 + g|S(x, z)|2�
+˜ϕ(x, z) = 0 with ˜ϕ(x, L) = 2iλϕ(0, L)δ(x),
+(33)
+�
+D∗
+z, x2 + g|S(x, z)|2�
+˜ϕ∗(x, z) = 0 with ˜ϕ∗(x, L) = 2iλϕ∗(0, L)δ(x),
+and
+[TC ( ˜ϕ∗ϕ + ˜ϕϕ∗) S] (x, z) = 2i
+g S(x, z).
+(34)
+The equations (33) are readily solved in terms of Feynman-Kac propagator,
+K(x2, z2; x1, z1) =
+� x(z2)=x2
+x(z1)=x1
+e
+� z2
+z1 [ im
+2 ˙x(τ)2+g|S(x(τ),τ)|2] dτDx,
+(35)
+13
+
+with z2 > z1, where the path-integral is over the set of all the continuous paths in Λ satisfying
+x(z1) = x1 and x(z2) = x2. One gets
+ϕ(x, z) =
+�
+Λ
+K(x, z; y, 0) dy,
+˜ϕ(x, z) = 2iλϕ(0, L) K(0, L; x, z)∗,
+(36)
+˜ϕ∗(x, z) = 2iλϕ∗(0, L) K(0, L; x, z).
+Like in the diffraction-free case, one has ˜ϕ∗(x, z) ̸= ˜ϕ(x, z)∗ if Re(λ) ̸= 0 (see the discussion
+below Eq. (19)). Using the expressions (36) on the left-hand side of Eq. (34), one obtains
+ϕ∗(0, L) G1(x, z) + ϕ(0, L) G2(x, z) = 1
+λg S(x, z),
+(37)
+with
+G1(x, z) =
+� L
+0
+�
+Λ
+�
+Λ
+K(0, L; x′, z′)K(x′, z′; ξ, 0)
+×C(x − x′, z, z′) S(x′, z′) dx′ dξ dz′,
+(38)
+and
+G2(x, z) =
+� L
+0
+�
+Λ
+�
+Λ
+K(0, L; x′, z′)∗K(x′, z′; ξ, 0)∗
+×C(x − x′, z, z′) S(x′, z′) dx′ dξ dz′.
+(39)
+In the large U limit, S concentrates onto the leading instanton, Sinst, solution to Eq. (37),
+and ϕ(x, z) concentrates onto ϕinst(x, z) given by the Feynman-Kac path-integral for ϕ(x, z)
+in Eq. (36) with S = Sinst. The key to solving Eq. (37) follows from the fact that, in this
+limit, the path-integrals in Eqs (38) and (39) are dominated by the contribution of the
+paths with the largest amplification, the contribution of the other paths being subdominant.
+These dominant trajectories run in the vicinity of ‘ridge paths’ along which |Sinst(x, z)|2 is
+at a global maximum (for every given z). Assuming that the ridge paths are all continuous
+(to be checked a posteriori, once Sinst is known), Eq. (37) simplifies and for a class of S that
+we will now specify, it can be solved explicitly.
+B.
+Specification of S(x, z)
+We assume that S(x, z) can be expressed as a finite random Fourier sum,
+S(x, z) =
+�
+(n,j)∈I
+s(n,j)
+�σ(n,j)
+ℓ
+e2iπnx/ℓΦ(n,j)(z),
+(40)
+14
+
+where I is a finite subset of Z × N. The s(n,j)s are complex Gaussian random variables
+with ⟨s(n,j)⟩ = ⟨s(n,j)s(m,k)⟩ = 0 and ⟨s(n,j)s∗
+(m,k)⟩ = δnmδjk, the σ(n,j)s are positive constants
+normalized to �
+(n,j)∈I σ(n,j) = Lℓ, and, for fixed n, the Φ(n,j)s are orthonormal continuous
+functions of 0 ≤ z ≤ L. Using Eq. (40) in C(x − x′, z, z′) = ⟨S(x, z)S(x′, z′)∗⟩ one gets
+C(x − x′, z, z′) =
+�
+(n,j)∈I
+σ(n,j)
+ℓ
+e2iπn(x−x′)/ℓΦ(n,j)(z)Φ(n,j)(z′)∗,
+(41)
+from which it follows that σ(n,j) and e2iπnx/ℓΦ(n,j)(z)/
+√
+ℓ are the eigenvalues and orthonormal
+eigenfunctions of the covariance operator TC defined in Eq. (28).
+Equation (40) generalizes models of spatially smoothed laser beams in which laser light
+is represented by a superposition of monochromatic beamlets the amplitudes of which are
+independent random variables [6]. For a large number of beamlets these random variables
+can be taken as Gaussian and the laser electric field takes on the form (40) in which the
+sum over j reduces to j = 1 with Φ(n,1)(z) = (1/
+√
+L) exp[iα(2πn/ℓ)2z)] where α is a (real)
+constant. Moreover, every centered Gaussian field with a continuous correlation function
+has an expansion of the form (40), possibly with an infinite sum [27]. Combining this result
+with the practically unavoidable existence of some natural cut-off making the sum finite
+(like, e.g., in numerical simulations), one can safely expects the expression in Eq. (40) to be
+quite generic, at least from a practical point of view.
+Let B(0, L) denote the set of all the continuous paths in Λ satisfying x(L) = 0 and define
+M[x(·)] the |I| × |I| positive definite matrix with components
+M(n,j)(m,k)[x(·)] =
+√σ(n,j)σ(m,k)
+ℓ
+� L
+0
+e2iπ(m−n)x(z)/ℓΦ(n,j)(z)∗Φ(m,k)(z) dz,
+(42)
+in which x(·) ∈ B(0, L). Write µ1[x(·)] > 0 the largest eigenvalue of M[x(·)]. We consider
+cases fulfilling the following two assumptions:
+(i) all the paths maximizing µ1[x(·)] are in B(0, L);
+(ii) there is a finite number of paths in B(0, L) maximizing µ1[x(·)].
+It is proved in [5] that the eigenvalues of M[x(·)] are equal to the ones of Tx(·) defined by
+(Tx(·)f)(z) =
+� L
+0
+C(x(z) − x(z′), z, z′) f(z′) dz′.
+f(z) ∈ L2([0, L]).
+(43)
+15
+
+It follows in particular that µ1[x(·)] is invariant under the path transformations leaving
+C(x(z) − x(z′), z, z′) unchanged and that the image of a path maximizing µ1[x(·)] by such a
+transformation is also a path maximizing µ1[x(·)].
+Assumption (i) is a central feature of the class of S we consider in this paper, together
+with the random Fourier representation in Eq. (40).
+We don’t know whether Eq. (37)
+could be solved analytically in the large U limit without this assumption. The technical
+restriction (ii) will be used in Sec. IV B . Assumptions (i) and (ii) are fulfilled in most cases
+of practical interest. Lifting (ii) raises tricky technical problems yet to be solved; this will
+be the subject of a future work. Finally, for notational convenience we define
+µmax =
+sup
+x(·)∈B(0,L)
+µ1[x(·)],
+(44)
+the supremum being reached in B(0, L), by Assumption (i).
+IV.
+SINGLE-FILAMENT INSTANTON AND TAIL OF p(U)
+In the following, we consider the simplest case where for each realization of Sinst there is
+only one ridge path of |Sinst(x, z)|2 in B(0, L), denoted by xinst(·), dominating the large U
+limit of the Feynman-Kac integrals in Eqs (38) and (39). Note that xinst(·) may be different
+from one realization of Sinst to the other. The set of all the realizations of Sinst having the
+same xinst(·) defines a random field, denoted by Sxinst(·)
+inst
+, referred to in the following as a
+‘single-filament instanton’ (the reason for this name will appear more clearly at the end of
+Sec. IV A). We will write ϕxinst(·)
+inst
+(x, z) the Feynman-Kac path-integral for ϕ(x, z) in Eq. (36)
+with S = Sxinst(·)
+inst
+.
+Single-filament instantons are not the only possible solutions to Eq. (37). Multi-filament
+instantons are also possible if realizations of |Sinst(x, z)|2 have more than one ridge path. The
+conditions for single- or multi-filament instantons are specified below Eq. (49) as well as at
+the end of Appendix A. The study of multi-filament instantons being excessively intricate,
+we restrict ourselves to single-filament instantons for the sake of clarity and readability.
+A.
+Leading instanton
+Assume that xinst(·) is continuous (to be checked a posteriori). As finite sums of continu-
+ous functions, both S(x, z) and C(x − x′, z, z′) are continuous functions of their arguments.
+16
+
+It follows in particular that for fixed x, z, and z′, the product C(x − x′, z, z′) S(x′, z′) on the
+right-hand side of Eqs (38) and (39) is a continuous function of x′. Integrating over ξ and
+x′ at fixed z′ and using the fact that, in the large U limit, only the vicinity of x′ = xinst(z′)
+contributes, one gets the large U behavior of G1,2(x, z),
+G1(x, z) ∼
+�
+Λ
+K(0, L; ξ, 0) dξ
+� L
+0
+C(x − xinst(z′), z, z′) Sxinst(·)
+inst
+(xinst(z′), z′) dz′
+= ϕxinst(·)
+inst
+(0, L)
+� L
+0
+C(x − xinst(z′), z, z′) Sxinst(·)
+inst
+(xinst(z′), z′) dz′
+(U → +∞), (45)
+and
+G2(x, z) ∼
+�
+Λ
+K(0, L; ξ, 0)∗ dξ
+� L
+0
+C(x − xinst(z′), z, z′) Sxinst(·)
+inst
+(xinst(z′), z′) dz′
+= ϕ(0, L)xinst(·) ∗
+inst
+� L
+0
+C(x − xinst(z′), z, z′) Sxinst(·)
+inst
+(xinst(z′), z′) dz′
+(U → +∞). (46)
+Injecting these expressions onto the left-hand side of Eq. (37), one obtains the instanton
+equation
+� L
+0
+C(x − xinst(z′), z, z′) Sxinst(·)
+inst
+(xinst(z′), z′) dz′ =
+1
+2λgU Sxinst(·)
+inst
+(x, z),
+(47)
+where we have used the equality |ϕxinst(·)
+inst
+(0, L)|2 = U imposed by the delta function on the
+right-hand side of Eq. (29).
+Equation (47) can be solved in two different ways, depending on wether or not the Fourier
+decompositions (40) and (41) for Sxinst(·)
+inst
+and C(x−xinst(z′), z, z′) are used in Eq. (47). Using
+these decompositions, one gets the eigenvalue equation
+�
+(m,k)∈I
+M(n,j)(m,k)[xinst(·)] s(m,k) =
+1
+2λgU s(n,j),
+(48)
+which fixes λ at λ = 1/(2µn[xinst(·)]gU), where µ1[xinst(·)] > µ2[xinst(·)] > · · · > 0 are the
+eigenvalues of M[xinst(·)]. The leading instanton Sxinst(·)
+inst
+corresponds to the largest eigenvalue
+µ1[xinst(·)] with xinst(·) maximizing µ1[x(·)]. The fact that xinst(·) exists and is continuous
+is ensured by the assumption (i). It is worth noticing that xinst(·) is a non-random path.
+Thus, for every path xinst(·) ∈ B(0, L) maximizing µ1[x(·)], there is a leading instanton
+Sxinst(·)
+inst
+(x, z) =
+�
+(n,j)∈I
+s(n,j)
+�σ(n,j)
+ℓ
+e2iπnx/ℓΦ(n,j)(z),
+(49)
+17
+
+where s (with components s(n,j)) is an eigenvector of M[xinst(·)] associated with the largest
+eigenvalue µ1[xinst(·)] = µmax. It is checked in Appendix A that xinst(·) is indeed a ridge
+path of |Sxinst(·)
+inst
+(x, z)|2, as it should be.
+The calculation in Appendix A also specifies under what condition on S the leading
+instanton in Eq. (49) is a single-filament instanton: the fundamental eigenspace of M[xinst(·)]
+and the one of M[x(·)] for every other path maximizing µ1[x(·)], if any, must be essentially
+disjoint1. In particular, if the fundamental eigenspaces of M[x(·)] for all the different paths
+maximizing µ1[x(·)] are essentially disjoint, all the instantons are single-filament instantons.
+This is the case considered in this paper. Conversely, if the fundamental eigenspaces of
+M[x(·)] for different paths maximizing µ1[x(·)] have a non trivial intersection, then multi-
+filament instantons come into play as possible solutions to Eq. (37).
+We now solve the equation (47) without using the Fourier decompositions (40) and (41).
+Taking x = xinst(z) in Eq. (47), one finds that Sxinst(·)
+inst
+(xinst(z), z) is an eigenfunction of
+Txinst(·) with eigenvalue 1/2λgU, which fixes λ at λ = 1/(2µn[xinst(·)]gU), where µ1[xinst(·)] >
+µ2[xinst(·)] > · · · > 0 are the eigenvalues of Txinst(·). (Recall that Tx(·) and M[x(·)] have the
+same eigenvalues with the same multiplicities [5]). Again, Sxinst(·)
+inst
+corresponds to the largest
+eigenvalue µ1[xinst(·)] with xinst(·) maximizing µ1[x(·)], i.e., µ1[xinst(·)] = µmax. Writing d1
+the degeneracy of µ1[xinst(·)] and {φ1, · · · , φd1} an orthonormal basis of the fundamental
+eigenspace of Txinst(·), one has
+Sxinst(·)
+inst
+(xinst(z), z) =
+d1
+�
+ν=1
+cνφν(z),
+(50)
+where cν =
+� L
+0 Sxinst(·)
+inst
+(xinst(z), z)φν(z)∗dz is a complex number. Injecting (50) into Eq. (47),
+one obtains
+Sxinst(·)
+inst
+(x, z) =
+1
+µmax
+� L
+0
+C(x − xinst(z′), z, z′)
+� d1
+�
+ν=1
+cνφν(z′)
+�
+dz′.
+(51)
+• Equivalence of the Fourier and convolution representations of Sxinst(·)
+inst
+The fact that the expressions of Sxinst(·)
+inst
+(x, z) in Eqs. (49) and (51) are equivalent is proved
+1 ‘essentially disjoint’ and ‘trivial intersection’ mean that the intersection reduces to the zero vector.
+18
+
+in Appendix B, with s(n,j) in Eq. (49) and cν in Eq. (51) related to each other by
+s(n,j) =
+1
+õmax
+d1
+�
+ν=1
+cνe(ν)
+(n,j) and cν = √µmax
+�
+(n,j)∈I
+s(n,j)e(ν) ∗
+(n,j),
+(52)
+where {e(1), · · · , e(d1)} is an orthonormal basis of the fundamental eigenspace of M[xinst(·)]
+the vectors of which are defined by their components
+e(ν)
+(n,j) =
+� σ(n,j)
+ℓ µmax
+� L
+0
+e−2iπnxinst(z′)/ℓΦ(n,j)(z′)∗φν(z′) dz′.
+(53)
+Note that by Eq. (49) for Sxinst(·)
+inst
+(xinst(z), z) and the definition of e(ν)
+(n,j) in Eq. (53), the
+expression of cν in Eq. (52) coincides with cν =
+� L
+0 Sxinst(·)
+inst
+(xinst(z), z)φν(z)∗dz, as it should
+be (see below Eq. (50)).
+• Statistical properties of S in the large U limit
+The statistical properties of the cνs and s(n,j)s are readily obtained from the ones of the
+s(n,j)s in Eq. (40). Since the s(n,j)s are i.i.d. standard complex Gaussian random variables
+with ⟨s(n,j)⟩ = ⟨s2
+(n,j)⟩ = 0 and ⟨|s(n,j)|2⟩ = 1, the orthogonal projection of the |I|-dimensional
+(complex) vector s with coordinates s(n,j) onto any given direction is also a standard complex
+Gaussian random variable statistically independent of the projections onto the orthogonal
+directions.
+Thus, projecting s onto the direction of the vector e(ν) defined in Eq. (53)
+(with given 1 ≤ ν ≤ d1) and writing cν = √µmax s · e(ν) ∗, one finds that the cνs are
+i.i.d.
+(complex) Gaussian random variables with ⟨cν⟩ = ⟨c2
+ν⟩ = 0 and ⟨|cν|2⟩ = µmax.
+The statistical properties of the s(n,j)s are different from the ones of the s(n,j)s because s
+is restricted to the d1-dimensional fundamental eigenspace of M[xinst(·)] (with d1 ≪ |I|,
+typically), which induces correlations between the s(n,j)s. From the first Eq. (52) and the
+statistical properties of the cνs, one finds that the s(n,j)s are correlated complex Gaussian
+random variables with ⟨s(n,j)⟩ = ⟨s(n,j)s(m,k)⟩ = 0 and ⟨s(n,j)s∗
+(m,k)⟩ = �d1
+ν=1 e(ν)
+(n,j)e(ν) ∗
+(m,k).
+From Eq. (51) and S ∼ Sxinst(·)
+inst
+(U → +∞), it is clear that the realizations of S most likely
+to produce a large value of U are less random than the unconditioned field S itself. Besides
+the (non-random) ridge path xinst(·), it only takes the 2d1 real Gaussian random variables
+Re(cν) and Im(cν) to characterize these realizations entirely. For instance, if µ1[xinst(·)] is
+not degenerate (d1 = 1), Eq. (51) and S ∼ Sxinst(·)
+inst
+yield
+S(x, z)
+∥S∥2
+∼ Aeiθ
+� L
+0
+C(x − xinst(z′), z, z′) φ1(z′) dz′
+(U → +∞),
+(54)
+19
+
+where ∥ · ∥2 denotes the L2-norm over Λ × [0, L], A is a positive constant, and θ is a
+random phase uniformly distributed over [0, 2π). From Eq. (54) it follows immediately that
+|S(x, z)|/∥S∥2 is non-random, which means that the profile of S(x, z) is purely deterministic
+in this case. This result generalizes the diffraction-free deterministic profile of S in Eq. (4)
+when diffraction is taken into account.
+• Typical shape of Sxinst(·)
+inst
+Although the Fourier representation of Sxinst(·)
+inst
+in Eq. (49) is very useful to deal with
+technical points like in Appendix A, it is far from clear as to the structure of Sxinst(·)
+inst
+in real
+space. By contrast, it is easier to figure out the shape of Sxinst(·)
+inst
+(x, z) from the convolution
+representation in Eq. (51), knowing the correlation function C(x − x′, z, z′). As a simple
+illustration, take, e.g., C(x−x′, z, z′) = f[(x−x′)/xc, (z−z′)/zc], where xc and zc respectively
+denote transverse and axial correlation lengths, f(x, z) being negligibly small outside the
+domain defined by both |x| ≤ 1 and |z| ≤ 1. Assuming xc ≪ ℓ, zc ≪ L and a ‘gentle’
+ridge path with | ˙xinst(z)| ≲ ℓ/L for all 0 ≤ z ≤ L, it is not difficult to show from Eq. (51)
+that Sxinst(·)
+inst
+lives within a thin tube, or filament, of radius ρ ≲ xc + ℓzc/L ≪ ℓ along the
+path xinst(·). This is the reason for the name ‘single-filament instanton’ given to Sxinst(·)
+inst
+.
+Numerical simulations confirm the elongated profile of Sxinst(·)
+inst
+(see Fig. 1 in Sec. V).
+B.
+Tail of p(U)
+Write Ninst the number of single-filament instantons (Ninst is the number of paths maxi-
+mizing µ1[x(·)], which is finite by assumption (ii)). As mentioned below Eq. (49), we consider
+cases where the fundamental eigenspaces of M[x(·)] for all the different paths maximizing
+µ1[x(·)] are essentially disjoint. It means that the instantons — that are all single-filament
+instantons — are mutually exclusive realizations of S.
+As a result, the total instanton
+contribution to the tail of p(U) in Eq. (29) is the sum of the contributions of the Ninst single-
+filament instantons. Let π(i)(U) denotes the contribution of the ith single-filament instanton.
+It will be seen below that the leading term of ln π(i)(U) in the large U limit does not depend
+on i. Writing ln f(U) this term and π(i)(U) = f(U)A(i)(U) with ln A(i)(U) = o[ln f(U)] as
+20
+
+U → +∞, one has
+ln p(U) ∼ ln
+Ninst
+�
+i=1
+π(i)(U) = ln
+Ninst
+�
+i=1
+f(U)A(i)(U)
+= ln f(U) + ln
+Ninst
+�
+i=1
+A(i)(U)
+(55)
+= ln f(U) + o[ln f(U)]
+(U → +∞).
+Thus, at leading order, ln p(U) ∼ ln f(U) where ln f(U) is the leading term of ln π(i)(U) for
+all 1 ≤ i ≤ Ninst. Let [LT](U→+∞)(·) denote the leading term of the asymptotic expansion
+of (·) as U → +∞. Picking a i and integrating out the fluctuations of the fields around the
+corresponding ith single-filament instanton (with ridge-path xinst(·)) in Eq. (29), one obtains
+ln f(U) = [LT](U→+∞) ln
+�
+δ(U − |ϕxinst(·)
+inst
+(0, L)|2)
+�
+Sxinst(·)
+inst
+(56)
+= [LT](U→+∞) ln
+��
+· · ·
+�
+δ
+�
+U − |ϕxinst(·)
+inst
+(0, L)|2�
+d1
+�
+ν=1
+exp
+�
+− |cν|2
+µmax
+�
+d2cν
+πµmax
+�
+.
+To go further we need the behavior of |ϕxinst(·)
+inst
+(0, L)|2 as a function of the cνs in the large
+U limit. First, we make the change of variables η = �d1
+ν=1 |cν|2 and cν = aν√η, where η is
+a gamma-distributed random variable with p(η) = [Γ(d1)µmax]−1(η/µmax)d1−1e−η/µmax, and
+the aνs define a random 2d1-dimensional (real) unit vector a with coordinates Re(aν) and
+Im(aν) the direction of which is uniformly distributed over the unit (2d1 − 1)-sphere. For
+finite L and ℓ, a large U implies a large η. Writing
+|ϕxinst(·)
+inst
+(0, L)|2 = A(η, a) e2gη,
+(57)
+without loss of generality, on the right-hand side of Eq. (56) (with variables η and a), one
+obtains
+ln f(U) = [LT](U→+∞) ln
+�� +∞
+0
+δ
+�
+U − A(η, a) e2gη�
+p(η) dη
+�
+a
+= [LT](U→+∞) ln
+�� +∞
+0
+δ (η − η(U, a))
+|2g + ∂η ln A(η, a)| U p(η) dη
+�
+a
+(58)
+= [LT](U→+∞) ln
+K(d1)
+U 1+1/2µmaxg
+�η(U, a) A(η(U, a), a)1/2µmaxg
+|2g + ∂η ln A(η(U, a), a)|
+�
+a
+where ⟨·⟩a denotes the average over the direction of a, K(d1) = [Γ(d1)µd1
+max]−1, and η(U, a)
+is the solution to A(η, a) e2gη = U. It is shown in Appendix C that
+limη→+∞
+1
+η ln A(η, a) = 0,
+limη→+∞ ∂η ln A(η, a) = 0,
+(59)
+21
+
+for every direction of a. If d1 = 1, A(η, a) ≡ A(η) does not depend on a and the limits
+in Eq. (59) are trivially uniform with respect to the direction of a. If d1 > 1, it is not
+unreasonable to expect that uniform convergence also applies to non-pathological instantons,
+like in Eq. (51) (with variables η and a), where no particular direction of a stands out
+significantly from the others, which could jeopardize uniform convergence in its vicinity. We
+therefore assume that the limits in Eq. (59) are uniform in a also for d1 > 1. Proving this
+conjecture is another problem that we are unable to solve at the present time. Under this
+assumption, it follows from Eq. (59) that at leading order in the large U limit, η(U, a) ∼
+(2g)−1 ln U and |2g + ∂η ln A(η(U, a), a)| ∼ 2g uniformly in a. Uniform asymptotics makes
+it possible to interchange asymptotics and average over the direction of a on the right-hand
+side of Eq. (58), which yields
+ln f(U) = [LT](U→+∞) ln
+�K(d1)
+4g2
+ln U
+U 1+1/2µmaxg ⟨A((2g)−1 ln U, a)1/2µmaxg⟩a
+�
+= −
+�
+1 +
+1
+2µmaxg
+�
+ln U,
+(60)
+where we have used ln⟨A((2g)−1 ln U, a)1/2µmaxg⟩a = o(ln U) (see the end of appendix C).
+Note that the expression of ln f(U) in Eq. (60) is independent of i, as announced. Using
+Eq. (60) on the right-hand side of Eq. (55), one finally obtains the tail of p(U) as
+ln p(U) = −
+�
+1 +
+1
+2µmaxg
+�
+ln U + o(ln U)
+(U → +∞),
+(61)
+from which it follows that p(U) has a leading algebraic tail ∝ U −ζ modulated by a slow
+varying amplitude (slower than algebraic) with exponent ζ = (1 + 1/2µmaxg). Injecting this
+result into ⟨U⟩ =
+� +∞
+1
+Up(U) dU, one finds that the critical coupling in the case of single-
+filament instantons is given by gc(L) = 1/2µmax (where µmax depends on L), in agreement
+with the general result of Ref. [5] recalled in Eq. (6).
+V.
+NUMERICAL RESULTS
+In this section, we report on numerical simulations we have performed to test our ana-
+lytical results. First of all, we emphasize on the fact that all our predictions are asymptotic
+results the convergence of which is notoriously slow. The very large values of U correspond-
+ing to the instanton dominated regime are too rarely sampled for our analytical results to
+be observed numerically. What can be observed, however, as a good indication of their
+22
+
+validity is the emergence of a statistical bias of S towards the instanton as the amplification
+increases within the sampled range.
+For definiteness, we have taken
+S(x, z) =
+50
+�
+n=−50
+sn
+√ςn exp i
+�
+2πn
+ℓ x +
+�2πn
+ℓ
+�2 z
+2
+�
+,
+(62)
+where the sns are complex Gaussian random variables with ⟨sn⟩ = ⟨snsm⟩ = 0 and ⟨sns∗
+m⟩ =
+δnm, and the spectral density ςn is normalized to �50
+n=−50 ςn = 1. Equation (62) — which is
+of the form (40) in which the sum over j reduces to j = 1 and ςn = σ(n,1)/Lℓ — is reminiscent
+of models of spatially smoothed laser beams [6], where S is a solution to the paraxial wave
+equation
+∂zS(x, z) + i
+2∂2
+x2S(x, z) = 0,
+(63)
+here with boundary condition S(x, 0) = �50
+n=−50 sn√ςn exp(2iπnx/ℓ). To ensure that the
+space average ℓ−1 �
+Λ S(x, z) dx is zero for all z and every realization of S, as expected for
+the electric field of a smoothed laser beam, the mode at n = 0 is excluded by taking ς0 = 0.
+Here we show the results for the Gaussian spectrum
+ςn̸=0 ∝ exp
+�
+−
+�πn
+ℓ
+�2�
+.
+(64)
+(Other widely used spectra, like top-hat and Cauchy spectra, give similar results.)
+For each realization of S on a cylinder of length L = 10 and circumference ℓ = 20, we have
+solved Eq. (1) by using a symmetrized z-split method [28] which propagates the diffraction
+term, (i/2m)∂2
+x2ψ(x, z), in Fourier space and the amplification term, g|S(x, z)|2ψ(x, z), in
+real space.
+We have taken m = 0.7, and g = 0.5.
+To get a better statistics of large
+amplification values, we have considered Umax = |ψ(xmax, L)|2 instead of U = |ψ(0, L)|2,
+where xmax is the value of x maximizing |ψ(x, L)|2 (i.e., the location of the highest peak
+of |ψ(x, L)|2). As Umax increases, S concentrates onto the leading instanton(s) arriving at
+x(L) ≃ xmax.
+By statistical invariance under x-translation, the leading instanton(s) arriving at x(L) = y
+is simply given by Sy
+inst(x, z) = Sinst(x − y, z), where Sinst(x, z) is the leading instanton(s)
+arriving at x(L) = 0. We have determined Sinst from Eqs. (49), (62), and (64), with numer-
+ically computed eigenvector(s) s. We have found a unique single-filament instanton with
+xinst(·) ≡ 0, µmax = 4.34984 and d1 = 1. The critical coupling is gc(L) = 1/(2µmax) ≃
+23
+
+0.11495 and g = 0.5 is in the above critical regime with g/gc(L) ≃ 4.35 > 1. Figure 1 shows
+the contour plots of |Sinst|2 and ‘hot spot profile’ |C|2 [6, 8].
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+z
+x
+(a)
+0
+0.5
+1
+-4
+-2
+0
+2
+4
+-10
+-5
+0
+5
+10
+z - L/2
+(b)
+FIG. 1: (a) Contour plot of |Sinst(x, z)|2 normalized to maxΛ×[0,L] |Sinst(x, z)|2 = 1. (b) Contour
+plot of the ‘hot spot profile’ |C(x, z)|2.
+Define ˆSy
+inst = Sy
+inst/∥Sy
+inst∥2,Λ×[0,L] and ˆS = S/∥S∥2,Λ×[0,L] where ∥ · ∥2,Λ×[0,L] is the L2-
+norm on Λ × [0, L]. Write Sy
+∥ =
+�
+ˆSy
+inst, ˆS
+�
+ˆSy
+inst the component of ˆS along Sy
+inst. We have
+measured the difference between S and the instanton through the minimized L2-distance
+D ≡ d2
+�
+ˆS, Symin
+∥
+�
+= min
+y∈Λ ∥ ˆS − Sy
+∥∥2,Λ×[0,L]
+=
+�
+1 − max
+y∈Λ |( ˆS, ˆSy
+inst)|2,
+(65)
+where ymin is the value of y minimizing ∥ ˆS − Sy
+∥∥2,Λ×[0,L]. The smaller D, the closer S to the
+instanton arriving at x(L) = ymin. The fact that ymin can be different from xmax is due to
+the fluctuations of S away from the instanton, the relative amplitude of which is measured
+by D. For D smaller than average, ymin ≃ xmax with a relatively small dispersion of the
+data points about ymin = xmax, as can be seen in Fig. 5. Using the Fourier representations
+(62) for both S and Sy
+Inst on the right-hand side of Eq. (65), one gets
+D =
+�
+1 − maxy∈Λ | �
+n ςnˆsnˆs∗
+ne2iπny/ℓ|2
+(�
+n ςn|sn|2) (�
+n ςn|sn|2) ,
+(66)
+which is the expression we have used in the simulations.
+24
+
+We drew 105 independent realizations of S denoted in the following by {S}. Figure 2
+shows the probability distribution of D estimated from {S} and the realizations in {S} with
+Umax above the 90th and 99th percentiles. The last two are conditional probabilities knowing
+that Umax ≥ 5 1012 and Umax ≥ 8.6 1017, respectively. One can see a clear tendency of D
+to decrease with increasing Umax: the subsamples of {S} conditioned on a large Umax are
+statistically biased toward the instanton compared with the unconditioned sample {S} itself.
+This numerical result for large but finite Umax is consistent with the predicted concentration
+of S onto the instanton for U → +∞.
+p ()
+(a)
+(b)
+(c)
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+2
+4
+6
+8
+
+FIG. 2: Probability distribution of D estimated from (a) {S}, (b) the realizations in {S} with Umax
+above the 90th percentile, and (c) the realizations in {S} with Umax above the 99th percentile.
+p (log10Umax)
+(a)
+(b)
+(c)
+8
+18
+28
+0.02
+0.06
+0.1
+0.14
+log10Umax
+FIG. 3: Probability distribution of log10 Umax estimated from (a) {S}, (b) {S}10%, and (c) {S}1%.
+25
+
+To study this bias phenomenon in more detail, we have used the two samples {S}10%
+and {S}1% respectively defined as the realizations in {S} with D below the 10th and 1st
+percentiles.
+These samples correspond to D ≤ 0.8 for {S}10% and D ≤ 0.7 for {S}1%.
+Figure 3 shows the probability distribution of log10 Umax estimated from (a) {S}, (b) {S}10%,
+and (c) {S}1%. The last two are conditional probabilities knowing that D ≤ 0.8 and D ≤ 0.7,
+respectively. In this figure, the statistical bias of S, already observed in Fig. 2, appears as
+the clear tendency of Umax to increase with decreasing D.
+The concentration of S onto the instanton implies that for all ε > 0 and 0 < a < b,
+one has lima→+∞ Prob.(D ≤ ε| a ≤ Umax < b) = 1. Thus, for a large enough it is not
+unreasonable to expect Prob.(D ≤ ε| a ≤ Umax < b) to increase with increasing a, which
+should be possible to check numerically. Prob.(D ≤ ε| a ≤ Umax < b) can be estimated by
+the percentage of realizations of S with D ≤ ε among the realizations with a ≤ Umax < b. In
+Figure 4 we show the results for ε = 0.8 and 0.7 (i.e., S in {S}10% and {S}1%, respectively),
+a = 10n, and b = 10n+2, with n an integer. It can be seen that both curves increase with
+increasing Umax, as expected. Note, e.g., that 30% of the realizations with Umax ≃ 1028 are
+in {S}1% (i.e., have D ≤ 0.7), when {S}1% represents only 1% of all the realizations in {S}:
+the emergence of a statistical bias of S with increasing amplification is clearly visible.
+relative histogram of Umax
+108
+1018
+1028
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+{S}10 %
+{S}1 %
+Umax
+FIG. 4: Percentage of realizations of S in {S}10% (orange, upper curve) and {S}1% (green, lower
+curve) for Umax in [10n, 10n+2), with n varying from 0 to 28. As guides to the eyes, the solid lines
+are nonlinear fits of the corresponding data points. Dashed lines are continuations of these fits to
+higher Umax (disregarding the data points in this domain)
+26
+
+Figure 5 shows a scatter plot of xmax and ymin for the 103 realizations in {S} with the
+largest Umax (viz., Umax ≥ 8.6 1017). Red circles and gray squares correspond to realizations
+with D ≤ 0.7 and D > 0.7, respectively (i.e., realizations in {S}1% and {S} \ {S}1%). It can
+be checked that the dispersion of the data points about ymin = xmax is indeed smaller for
+smaller D, as announced below Eq. (65). Note that due to the periodic boundary condition
+in Λ, the distance between xmax and ymin is min(|xmax −ymin|, ℓ−|xmax −ymin|) and the data
+points in the left-upper and right-lower corners are actually close to the diagonal.
+ymin
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+-10
+-5
+0
+5
+10
+-10
+-5
+0
+5
+10
+xmax
+FIG. 5: Scatter plot of xmax and ymin for the realizations in {S} with Umax ≥ 8.6 1017. Red circles
+and gray squares are realizations in {S}1% (D ≤ 0.7) and {S} \ {S}1% (D > 0.7), respectively.
+We have compared the realizations in {S}1% and {S}\{S}1% near the edge of the sampled
+domain of Umax, where {S}1% becomes statistically significant according to the results in
+Fig. 4. We have considered realizations with Umax ≥ 1027. There are 15 such realizations in
+{S} among which 5 in {S}1% (D ≤ 0.7) and 10 in {S} \ {S}1% (D > 0.7). In Figures 6 and
+7 we show two pairs of typical realizations picked in {S}1% and {S} \ {S}1%, respectively
+(technical details are given in the captions). For each realization, the theoretical instanton
+arriving at x(L) = ymin is indicated by a dashed contour, solution to |Sinst(x−ymin, z)|2 = 0.75
+with |Sinst|2 as in Fig. 1. Intense localized hot spots similar to the theoretical one in Fig. 1(b)
+are clearly visible in both figures. In Fig. 6 (D ≤ 0.7), hot spots occur inside the dashed
+line, in the instanton region. Note also that the level of |S(x, z)|2 is significantly higher than
+average throughout the instanton region (∼ 6, while ⟨|S(x, z)|2⟩ = 1), which seems difficult
+to explain by generic fluctuations (i.e. independent, small-scale hot spots). On the other
+hand, in Fig. 7 (D > 0.7), hot spots occur anywhere and the levels of |S(x, z)|2 inside and
+27
+
+outside the instanton region are quite comparable (hot spots excluded).
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+z
+x - xmax
+(a)
+fraction of max
+0
+0.5
+1
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+z
+(b)
+FIG. 6: Contour plots of |S(x, z)|2 for two realizations in {S}1% (D ≤ 0.7) with Umax ≥ 1027.
+The dashed contours indicate the theoretical instanton arriving at x(L) = ymin for the considered
+realization. (a): Umax = 4.3 1029, D = 0.68, and maxΛ×[0,L] |S(x, z)|2 = 14.83. (b): Umax =
+1.15 1027, D = 0.537, and maxΛ×[0,L] |S(x, z)|2 = 13.
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+z
+x - xmax
+(a)
+fraction of max
+0
+0.5
+1
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+z
+(b)
+FIG. 7: Plots similar to the ones shown in Figure 6 for two realizations in {S} \ {S}1% (D > 0.7).
+(a): Umax = 6.92 1029, D = 0.8, and maxΛ×[0,L] |S(x, z)|2 = 17.79. (b): Umax = 4.95 1027, D = 0.82,
+and maxΛ×[0,L] |S(x, z)|2 = 17.12.
+28
+
+The robustness of these observations from one realization to the other can be tested
+through the sample mean of |S(x + xmax, z)|2 in which the realizations are translated to
+align the maxima of |ψ(x, L)|2 with each other at the same position (here, x = 0).
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+z
+x
+(a)
+fraction of max
+0
+0.5
+1
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+z
+(b)
+FIG. 8: Contour plots of the sample mean of |S(x + xmax, z)|2 for the realizations in (a) {S}1% and
+(b) {S} \ {S}1%, with Umax ≥ 1027. The dashed contours indicate the region where the sample
+mean of |Sinst(x + xmax − ymin, z)|2 is greater than 75% of its maximum value.
+In Figure 8, we show the results for the same 15 realizations with Umax ≥ 1027 as above.
+The region of the sample mean of |Sinst(x+xmax−ymin, z)|2 is indicated by a dashed contour.
+The red spot seen in both figures (a) and (b) at x = 0 and z ≃ L is a cumulative effect
+reflecting the tendency of |S(x, z)|2 to have a bump — not necessarily high — right behind
+the maximum of |ψ(x, L)|2. Such recurring bumps can be observed in the four realizations
+shown in Figs. 6 and 7: plots are never blue at x = xmax and z ≃ L. Figure 8(a) shows the
+result for the realizations in {S}1%. In substance, it confirms the observations already made
+about the Fig. 6; namely, the observed level of |S(x, z)|2 inside the dashed contour is the
+superposition of an average elevation of the level (the emerging instanton) and fluctuations of
+comparable amplitude. The presence of such an average elevation inside the dashed contour
+results in a larger probability that high maxima of |S(x, z)|2 occur inside the instanton
+region. It is a pure statistical effect similar to the well known enhancement of correlations
+of peaks in Gaussian fields [29, 30], the large-scale instanton playing the same role as the
+‘signal’ and ‘background field’ in [29] and [30], respectively. As a consequence, one observes
+29
+
+(i) a tendency for the hot spots to cluster in the instanton region and (ii) a level of |S(x, z)|2
+between the hot spots significantly higher than the average level outside the instanton region.
+These two points (i) and (ii) are strong indications of the presence of the instanton in the
+realizations of {S}1%. Hot spot clustering is illustrated in Fig. 9 for the same realizations as
+in Fig. 6. By contrast, no particular structure is observed in Fig. 8(b) for the realizations
+in {S} \ {S}1% (except the red spot at x = 0 and z ≃ L). It means that neither instanton
+nor clustering of hot spots are significant in those realizations.
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+z
+x - xmax
+(a)
+0
+2
+4
+6
+8
+10-10
+-5
+0
+5
+10
+z
+(b)
+FIG. 9: Positions of the local maxima of |S(x, z)|2 higher than 75% of the global maximum (blue
+circles) for the same realizations as in Fig. 6. High maxima cluster in the instanton region indicated
+by the dashed contour. (See caption of Fig. 6 for details.)
+Combining numerical results with analytical predictions, one can now infer what realiza-
+tions of S(x, z) contribute to p(U). As long as the value of U is in the bulk of p(U), an
+overwhelming majority of the realizations of S(x, z) contributing to p(U) are generic real-
+izations with hot spots uniformly scattered in Λ × [0, L] and D close to its typical value at
+D ≃ 0.86. The situation changes gradually as U increases into the tail of p(U). Namely, the
+larger U the larger the percentage of atypical realizations with D smaller than, say, its first
+percentile — like the ones in Fig. 6 — to the detriment of generic realizations — like the ones
+in Fig. 7. In those atypical realizations, the hot spots cluster in the instanton region instead
+of being uniformly scattered in Λ × [0, L] and the level of |S(x, z)|2 between the hot spots
+remains abnormally high (see Figs. 6, 8(a), and 9). Letting U → +∞, the percentage of
+atypical realizations goes up to 100% while D and the relative fluctuations-to-instanton am-
+30
+
+plitude decrease to zero with probability one. In this limit, the tail of p(U) is asymptotically
+dominated by the instanton which determines the the critical coupling gc(L).
+VI.
+DISCUSSION AND PERSPECTIVES
+In this paper, we have studied the large amplification limit of a linear amplifier driven by
+the square of a Gaussian random field. We have considered the same model as in Refs. [1]
+and [5] in which the propagation is that of a free Schrödinger equation. By performing the
+first instanton analysis of the corresponding MSR action, we have identified the realizations
+of the Gaussian field most likely to produce a large amplification.
+We have found that
+when U = |ψ(0, L)|2 gets large, for ψ solution to Eq. (1) with S defined in Sec. III B,
+the realizations of S concentrate onto large-scale filamentary instantons running along the
+path(s) maximizing the largest eigenvalue of the covariance operator defined in Eq. (43).
+This result explains the otherwise mysterious presence of this maximized eigenvalue in the
+expression of gc(L) found in [5] (see Eq. (6)). We have then derived the tail of p(U) for large U
+from the instanton contribution and checked that the resulting critical coupling does coincide
+with the one in Ref. [5]. From this analysis, it follows in particular that the realizations of
+S causing the divergence of ⟨|ψ(0, L)|2⟩ for g > gc(L) are long filamentary structures (the
+instantons) rather than localized hot-spots, as assumed in hot-spot models [1]. This result
+extends the conclusions of Ref. [10] to the full problem (1) with diffraction.
+Our analytical predictions are supported by numerical simulations that clearly show a
+statistical bias of S towards the instanton, as U increases. The larger U in the sampled
+range, the larger the fraction of atypical realizations of S in which a large-scale instanton
+coexists with fluctuation induced localized hot spots. (See [10] for a quantitative comparison
+of hot spot and instanton contributions to the amplification in the diffraction-free case.) In
+those atypical realizations, hot spots are not uniformly distributed in Λ × [0, L] but tend
+to cluster in the instanton region. For the parameters used in our simulations we failed to
+sample values of U large enough that the fluctuations of S away from the instanton could be
+neglected. Hot spot clustering and nonlinear evolution of the coupled hot spots/instanton
+system are interesting subjects that would deserve to be dealt with in more depth, especially
+in laser-plasma interaction physics.
+The work presented here is only a first step toward a comprehensive study of Eq. (1) in
+31
+
+the large amplification limit. There are various directions along which investigations could
+be pushed further. Obviously, trying to lift all or part of the assumptions made in Sec. III B
+appears as a natural next step, especially the technical restriction (ii), as well as proving the
+conjecture that the limits in Eq. (59) are uniform also for degenerate µmax (d1 > 1). The
+second natural next step is to investigate the possibility of multi-filament instantons.
+Another challenging line of research is the study of a possible intermittency of |ψ(x, L)|2
+and its connection with our results, as we will now explain. Typically, a unique realization
+of S is available in a given experimental environment, like, e.g., in a laser-plasma interaction
+experiment. In this case, ⟨|ψ(0, L)|2⟩ is replaced with the space average |Λ|−1 �
+Λ |ψ(x, L)|2 dx
+for a generic realization of S. As rare events, instantons are very unlikely to contribute to
+the latter quantity unless |Λ| is large and the space average is dominated by the contribution
+of scarce, intense peaks of |ψ(x, L)|2 the high amplitude of which outbalances their scarcity.
+The question is then whether such a peak-dominated behavior — called ‘intermittency’ in the
+literature on random media [31] — can be observed in the solution to Eq. (1) for g > gc(L).
+If so, our results imply that S(x, z) in the region of Λ×[0, L] upstream from a dominant peak
+of |ψ(x, L)|2 is a filament instanton arriving at the peak location. Intermittency of |ψ(x, L)|2
+is thus important as connecting our instanton analysis approach with experimental results in
+the above critical regime g > gc(L). The interested reader will find a detailed introduction
+to intermittency in random media in Ref. [31].
+Finally, it would also be interesting to investigate the small m behavior of the same
+problem. For m → 0, it can be shown that ψ(x, z) reduces to
+ψ(x, z) = exp
+� g
+|Λ| ∥S∥2
+2,Λ×[0,L]
+�
+.
+(67)
+Thus, in this limit, the realizations of S giving rise to a large U are the ones with a large
+L2-norm, which are known to concentrate onto the fundamental eigenspace of the covariance
+operator TC defined in Eq. (28), as U → +∞ [10, 11]. If S is given by the random Fourier
+sum (40) with, e.g., σ(n,j) < σ(0,0) for all (n, j) ̸= (0, 0), the fundamental eigenspace of TC
+reduces to the functions ∝ Φ(0,0)(z) independent of x, and the realizations of S in the large
+U limit are completely delocalized in Λ, in striking contrast to the filamentary instantons we
+have found for a fixed m ̸= 0. This simple example indicates that the two limits U → +∞
+and m → 0 do not commute, which raises the natural question of how precisely the crossover
+occurs between the limits ‘U → +∞ then m → 0’ and ‘m → 0 then U → +∞’. Answering
+32
+
+this question will elucidate the intriguing transition suggested by the above example, from
+filamentary to delocalized instantons, as m goes to zero.
+In conclusion, it may be noted that the number of highly non-trivial questions raised
+by the seemingly simple linear problem (1) is quite remarkable.
+Following on from the
+work presented here, we hope that those questions will motivate interesting research in both
+statistical physics and laser-matter interaction physics where the linear amplifier model (1)
+first appeared.
+Acknowledgments
+The author warmly thanks Satya N Majumdar, Denis Pesme, and Grégory Schehr for
+their interest and valuable advice about the manuscript. He also thanks Harvey A Rose and
+Joel L Lebowitz for the inspiring discussions he had with them on related subjects.
+Appendix A: Paths maximizing µ1[x(·)] and ridge paths of |Sinst(x, z)|2
+In this appendix we show that xinst(·) is a ridge path of |Sxinst(·)
+inst
+(x, z)|2. From Eqs. (42)
+and (49), one gets
+� L
+0
+|Sxinst(·)
+inst
+(xinst(z), z)|2 dz = s†M[xinst(·)]s = µmax∥s∥2,
+(A1)
+where s is in the fundamental eigenspace of M[xinst(·)], and
+� L
+0
+|Sxinst(·)
+inst
+(x(z), z)|2 dz = s†M[x(·)]s ≤ µ1[x(·)]∥s∥2
+= µ1[x(·)]
+µmax
+� L
+0
+|Sxinst(·)
+inst
+(xinst(z), z)|2 dz ≤
+� L
+0
+|Sxinst(·)
+inst
+(xinst(z), z)|2 dz,
+(A2)
+yielding
+sup
+x(·)∈B(0,L)
+� L
+0
+|Sxinst(·)
+inst
+(x(z), z)|2 dz =
+� L
+0
+|Sxinst(·)
+inst
+(xinst(z), z)|2 dz.
+(A3)
+Equation (A3) means that in the path-integral for ϕxinst(·)
+inst
+(0, L), xinst(·) is a path along which
+the amplification is maximum. Now, assume that there is A ⊂ [0, L] with |A| ≡
+� L
+0 1z∈Adz >
+0 such that for all z ∈ A, there is x ∈ Λ with |Sxinst(·)
+inst
+(x, z)|2 > |Sxinst(·)
+inst
+(xinst(z), z)|2. It follows
+immediately that
+� L
+0
+sup
+x∈Λ
+|Sxinst(·)
+inst
+(x, z)|2dz >
+� L
+0
+|Sxinst(·)
+inst
+(xinst(z), z)|2 dz,
+(A4)
+33
+
+and from Eqs. (A3) and (A4) one should have
+� L
+0
+sup
+x∈Λ
+|Sxinst(·)
+inst
+(x, z)|2dz >
+sup
+x(·)∈B(0,L)
+� L
+0
+|Sxinst(·)
+inst
+(x(z), z)|2 dz,
+(A5)
+in contradiction with the Lemma A1 in Ref. [5] according to which one must have an equality.
+Thus, there is no such A and since every given realization of Sxinst(·)
+inst
+(x, z) in Eq. (49) is
+a continuous function of x and z, one has |Sxinst(·)
+inst
+(x, z)|2 ≤ |Sxinst(·)
+inst
+(xinst(z), z)|2 for all
+0 ≤ z ≤ L and x ∈ Λ. This proves that for all the realizations of Sxinst(·)
+inst
+(x, z) in Eq. (49)
+with s in the fundamental eigenspace of M[xinst(·)], xinst(·) is a ridge path of |Sxinst(·)
+inst
+(x, z)|2
+along which the amplification is maximum.
+Assume that there is a s in the fundamental eigenspace of M[xinst(·)] and yinst(·) ∈
+B(0, L) with yinst(·) ̸= xinst(·) such that yinst(·) is also a ridge path of |Sxinst(·)
+inst
+(x, z)|2 along
+which the amplification is maximum. Then, yinst(·) maximizes µ1[x(·)] and s belongs to the
+fundamental eigenspace of M[yinst(·)] (otherwise, the amplification along yinst(·) would be
+less than along xinst(·)). Since s belongs to the fundamental eigenspaces of both M[xinst(·)]
+and M[yinst(·)], their intersection is necessarily non trivial. It shows that the number of
+ridge paths depends on the relative structure of the fundamental eigenspaces of M[x(·)] for
+the different paths maximizing µ1[x(·)]. If the fundamental eigenspaces of M[x(·)] for all
+the paths maximizing µ1[x(·)] are essentially disjoint, s cannot belong to more than one
+fundamental eigenspace and each realization of the instanton has only one ridge path. This
+is the case considered in the paper. On the other hand, if the fundamental eigenspaces of
+M[x(·)] for different paths maximizing µ1[x(·)] have a non trivial intersection, then for all
+the realizations with s in the intersection, the instanton has more than one ridge path. This
+case corresponds to multi-filament instantons.
+Appendix B: Equivalence of the Fourier and convolution representations of Sinst
+In this appendix, we prove the equivalence of the expressions of Sxinst(·)
+inst
+(x, z) in Eqs. (49)
+and (51). Permuting the sum and the integral on the right-hand side of Eq. (51) and using
+the Fourier decomposition (41) for C(x − xinst(z′), z, z′), one readily finds that the equation
+(51) can be rewritten as
+Sxinst(·)
+inst
+(x, z) =
+d1
+�
+ν=1
+cνΩν(x, z),
+(B1)
+34
+
+with
+Ων(x, z) =
+�
+(n,j)∈I
+e(ν)
+(n,j)
+� σ(n,j)
+ℓ µmax
+e2iπnx/ℓΦ(n,j)(z),
+(B2)
+where e(ν) is a vector defined by its components
+e(ν)
+(n,j) =
+� σ(n,j)
+ℓ µmax
+� L
+0
+e−2iπnxinst(z′)/ℓΦ(n,j)(z′)∗φν(z′) dz′.
+(B3)
+Showing that the equation (49) can also be written in the form of Eq. (B1) requires a little
+more work. From Eqs. (41), (42), and (B3) it can be checked that
+�
+M[xinst(·)]e(ν)�
+(n,j) =
+�
+(m,k)∈I
+M(n,j)(m,k)[xinst(·)] e(ν)
+(m,k)
+=
+� σ(n,j)
+ℓ µmax
+� L
+0
+e−2iπnxinst(z)/ℓΦ(n,j)(z)∗ ⟨z|Txinst(·)|φν⟩ dz
+= µmax e(ν)
+(n,j),
+(B4)
+and
+e(µ) ∗ · e(ν) =
+�
+(n,j)∈I
+e(µ) ∗
+(n,j)e(ν)
+(n,j) =
+1
+µmax
+⟨φµ|Txinst(·)|φν⟩
+= ⟨φµ|φν⟩ = δµν,
+(B5)
+which means that {e(1), · · · , e(d1)} is an orthonormal basis of the fundamental eigenspace of
+M[xinst(·)] on which s can be decomposed. Writing
+s(n,j) =
+1
+õmax
+d1
+�
+ν=1
+cνe(ν)
+(n,j) with cν = √µmax
+�
+(n,j)∈I
+s(n,j)e(ν) ∗
+(n,j),
+(B6)
+on the right-hand side of Eq. (49) and permuting the sums over (n, j) and ν, one obtains the
+same equation (B1), as expected, which proves the equivalence of Eqs. (49) and (51) with
+s(n,j) and cν related to each other by Eq. (B6).
+Appendix C: Limit of η−1 ln A(η, a) and ∂η ln A(η, a) as η → +∞
+In this appendix we derive the two limits in Eq. (59).
+We will use the convolution
+representation (51). To make the dependence of Sxinst(·)
+inst
+on c explicit we write Sxinst(·)
+inst
+(x, z) ≡
+Sxinst(·)
+inst
+(x, z, c) = √η Sxinst(·)
+inst
+(x, z, a), with η = �d1
+ν=1 |cν|2 and cν = aν√η. Deriving the
+Feynman-Kac path-integral representation of ϕxinst(·)
+inst
+(0, L),
+ϕxinst(·)
+inst
+(0, L) =
+�
+x(·)∈B(0,L)
+e
+� L
+0
+�
+im
+2 ˙x(τ)2+gη|Sxinst(·)
+inst
+(x(τ),τ,a)|2�
+dτDx,
+(C1)
+35
+
+with respect to η for fixed a and using the fact that
+� L
+0 |Sxinst(·)
+inst
+(xinst(τ), τ, a)|2dτ = 1, one
+gets
+∂ηϕxinst(·)
+inst
+(0, L) = g
+�
+x(·)∈B(0,L)
+�� L
+0
+|Sxinst(·)
+inst
+(x(τ), τ, a)|2dτ
+�
+× e
+� L
+0
+�
+im
+2 ˙x(τ)2+gη|Sxinst(·)
+inst
+(x(τ),τ,a)|2�
+dτDx
+∼ g
+�
+x(·)∈B(0,L)
+�� L
+0
+|Sxinst(·)
+inst
+(xinst(τ), τ, a)|2dτ
+�
+× e
+� L
+0
+�
+im
+2 ˙x(τ)2+gη|Sxinst(·)
+inst
+(x(τ),τ,a)|2�
+dτDx
+= g
+�
+x(·)∈B(0,L)
+e
+� L
+0
+�
+im
+2 ˙x(τ)2+gη|Sxinst(·)
+inst
+(x(τ),τ,a)|2�
+dτDx
+= gϕxinst(·)
+inst
+(0, L)
+(η → +∞),
+(C2)
+from which it follows that
+∂η ln |ϕxinst(·)
+inst
+(0, L)|2 = 2Re
+�
+∂ηϕxinst(·)
+inst
+(0, L)
+ϕxinst(·)
+inst
+(0, L)
+�
+∼ 2g
+(η → +∞).
+(C3)
+Thus, for all ε > 0 there is η0(ε, a) > 0 such that for every η ≥ η0(ε, a),
+2g(1 − ε) ≤ ∂η ln |ϕxinst(·)
+inst
+(0, L)|2 ≤ 2g(1 + ε).
+(C4)
+Writing |ϕxinst(·)
+inst
+(0, L)|2 = A(η, a) e2gη in Eq. (C4), one obtains
+−2gε ≤ ∂η ln A(η, a) ≤ 2gε,
+(C5)
+for every η ≥ η0(ε, a), and since ε can be taken arbitrarily small, Eq. (C5) reduces to
+lim
+η→+∞ ∂η ln A(η, a) = 0,
+(C6)
+which is the second limit in Eq. (59). To get the first limit, we integrate Eq. (C5) from
+η0(ε, a) to any η > η0(ε, a), which yields
+−2gεη + K+(ε, a) ≤ ln A(η, a) ≤ 2gεη + K−(ε, a),
+(C7)
+with K±(ε, a) = ln A(η0(ε, a), a) ± 2gεη0(ε, a). Note that ln A(η0(ε, a), a) exists, otherwise
+ln A(η, a) would have a vertical asymptote at η = η0(ε, a), in contradiction with Eq. (C5).
+It remains to divide Eq. (C7) by η:
+−2gε − |K+(ε, a)|
+η
+≤ 1
+η ln A(η, a) ≤ 2gε + |K−(ε, a)|
+η
+,
+(C8)
+36
+
+where we have used K+(ε, a) ≥ −|K+(ε, a)| and K−(ε, a) ≤ |K−(ε, a)|. Now, for η large
+enough, namely η ≥ max[η0(ε, a), ε−1|K+(ε, a)|, ε−1|K−(ε, a)|], Eq. (C8) gives
+−3gε ≤ 1
+η ln A(η, a) ≤ 3gε,
+(C9)
+and since ε can be taken arbitrarily small, one finally obtains
+lim
+η→+∞
+1
+η ln A(η, a) = 0,
+(C10)
+which is the first limit in Eq. (59).
+We now prove the relation ln⟨A((2g)−1 ln U, a)1/2µmaxg⟩a = o(ln U) used in Eq. (60). Mak-
+ing the conjecture that Eq. (C10) holds uniformly with respect to the direction of a (see the
+discussion below Eq. (59)), one finds that for all ε > 0 there is η0(ε) > 0 independent of a
+such that for every η ≥ η0(ε),
+exp(−εη) ≤ A(η, a) ≤ exp(εη),
+(C11)
+whence,
+exp
+�
+−
+εη
+2µmaxg
+�
+≤ A(η, a)1/2µmaxg ≤ exp
+�
+εη
+2µmaxg
+�
+,
+(C12)
+and, averaging over a then taking the logarithm,
+−
+εη
+2µmaxg ≤ ln⟨A(η, a)1/2µmaxg⟩a ≤
+εη
+2µmaxg,
+(C13)
+for every η ≥ η0(ε). Since ε can be taken arbitrarily small, Eq. (C13) reduces to
+lim
+η→+∞
+1
+η ln⟨A(η, a)1/2µmaxg⟩a = 0,
+(C14)
+which means that ln⟨A(η, a)1/2µmaxg⟩a = o(η), as announced.
+[1] Rose H A and DuBois D F 1994 Phys. Rev. Lett. 72 2883
+[2] Akhmanov S A, D’yakov Yu E and Pavlov L I 1974 Sov. Phys. JETP 39 249
+[3] Asselah A, Dai Pra P, Lebowitz J L and Mounaix Ph 2001 J. Stat. Phys. 104 1299
+[4] Mounaix Ph and Lebowitz J L 2004 J. Phys. A: Math. Gen. 37 5289
+[5] Mounaix Ph, Collet P and Lebowitz J L 2006 Commun. Math. Phys. 264 741 and 2008
+Commun. Math. Phys. 280 281
+37
+
+[6] Rose H A and DuBois D F 1993 Phys. Fluids B 5 590
+[7] Adler J 1981 The Geometry of Random Fields. (New York: Wiley)
+[8] Garnier J 1999 Phys. Plasmas 6 1601
+[9] Mounaix Ph 2001 Phys. Rev. Lett. 87 085006
+[10] Mounaix Ph and Divol L 2004 Phys. Rev. Lett. 93 185003
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+[12] Mounaix Ph, Majumdar S N and Banerjee A 2012 J. Phys. A: Math. Theor. 45 115002
+[13] Mounaix Ph 2015 J. Stat. Phys. 160 561
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+[15] Janssen H K 1976 Z. Phys. B 23 377
+[16] DeDominicis C 1976 J. Phys. (Paris) Colloq. 1 247
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+[18] Phythian R 1977 J. Phys. A 10 777
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+[21] Martin P C, Siggia E D and Rose H A 1973 Phys. Rev. A 8 423
+[22] Giles M J 1995 Phys. Fluids 7 2785
+[23] Falkovich G, Kolokolov I, Lebedev V and Migdal A 1996 Phys. Rev. E 54 4896
+[24] Gurarie V and Migdal A 1996 Phys. Rev. E 54 4908
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+John Wiley & Sons)
+[27] Adler R J and Taylor J E 2007 Random Fields and Geometry Springer Monographs in Math-
+ematics (New York: Springer)
+[28] Strang G 1968 SIAM J. Numer. Anal. 5 (3) 506
+[29] Kaiser N 1984 Ap. J. (Letters) 284 L9
+[30] Bardeen J M, Bond J R, Kaiser N and Szalay A S 1986 Ap. J. 304 15
+[31] Molchanov S A 1991 Acta. Appl. Math. 22 139 and references therein
+38
+
diff --git a/jdFPT4oBgHgl3EQfFjQr/content/tmp_files/load_file.txt b/jdFPT4oBgHgl3EQfFjQr/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1800 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf,len=1799
+page_content='Schrödinger Equation Driven by the Square of a Gaussian Field:  Instanton Analysis in the Large Amplification Limit  Philippe Mounaix1, ∗  1CPHT, CNRS, École polytechnique,  Institut Polytechnique de Paris, 91120 Palaiseau, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (Dated: January 31, 2023)  Abstract  We study the tail of p(U), the probability distribution of U = |ψ(0, L)|2, for large U, ψ(x, z) being  the solution to ∂zψ(x, z) −  i  2m∇2  ⊥ψ(x, z) = g|S(x, z)|2ψ(x, z), where S(x, z) is a complex Gaussian  random field, z and x respectively are the axial and transverse coordinates, with 0 ≤ z ≤ L,  and both m ̸= 0 and g > 0 are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We perform the first instanton analysis of  the corresponding Martin-Siggia-Rose action, from which it is found that the realizations of S  concentrate onto long filamentary instantons, as U → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The tail of p(U) is deduced from  the statistics of the instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The critical value of g above which ⟨U⟩ diverges is checked to  coincide with the one obtained by the completely different approach developed in Mounaix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  2006 Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 264 741 (and Erratum 2008 Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 280 281).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Analytical predictions are supported by numerical simulations that clearly show the statistical bias  of S towards the instanton, at large U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For the biased realizations of S, the high maxima — or  ‘hot spots’ — of |S(x, z)|2 tend to cluster in the instanton region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Keywords: stochastic partial differential equations, instanton analysis, extreme event statistics, laser-plasma  interactions  ∗Electronic address: philippe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='mounaix@polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='edu  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='13000v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='stat-mech]  30 Jan 2023  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  INTRODUCTION  In the second part of their seminal paper on the breakdown of linear instability in stim-  ulated Brillouin scattering [1], Rose and DuBois investigated the following equation for the  complex amplitude ψ(x, z) of the scattered light electric field  �  �  �  ∂zψ(x, z) −  i  2m∇2  ⊥ψ(x, z) = g|S(x, z)|2ψ(x, z),  0 ≤ z ≤ L, x ∈ Λ ⊂ Rd, and ψ(x, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (1)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1), z and x respectively denote the axial and transverse coordinates in a plasma of  length L and cross-sectional domain Λ (often a torus like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', in mathematics oriented work  and/or numerical simulations using spectral methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The boundary condition at z = 0 is  taken to be a constant for simplicity and m ̸= 0 is a real parameter introduced for conve-  nience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [1], the coupling constant g > 0 is proportional to the average laser intensity  and the complex amplitude of the laser electric field S(x, z) is a homogeneous Gaussian  random field with zero mean and normalized intensity ⟨|S(x, z)|2⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For our purposes,  we can be less restrictive and take S(x, z) transversally homogeneous with normalization  L−1 � L  0 ⟨|S(x, z)|2⟩ dz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' From now on, we accept the idealizations inherent in the deriva-  tion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1), setting aside the question of its validity as a realistic model (which varies  from one physical problem to the other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As a stochastic PDE, the diffraction-amplification  problem (1) is a Schrödinger equation driven by the square of a Gaussian field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Using heuristic arguments and numerical simulations, Rose and DuBois found that the  expected value of the scattered energy density, ⟨|ψ(x0, L)|2⟩, at some given x0 ∈ Λ diverges  for every L > 0 when g is greater than some critical value, gc(L), yet to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Here, the average ⟨|ψ|2⟩ is taken over the realizations of the Gaussian field S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Physically,  this divergence was interpreted in [1] as indicating the breakdown of the linear model (1) and  the onset of a saturated nonlinear regime in high overintensities, or hot spots, of |S(x, z)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We will shortly come back to the role of the hot spots in the divergence of ⟨|ψ(x0, L)|2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note  that in the limit referred to in [1] as the independent hot spot model, this divergence was  pointed out by Akhmanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 20 years before [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The problem was then analyzed in [3–5]  from a more rigorous mathematical point of view, establishing the numerical results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  [1] on much firmer ground and giving the exact expression of the critical coupling gc(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In  the following, we will take x0 = 0 without loss of generality (by statistical invariance under  x-translation) and we will write U = |ψ(0, L)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  2  Whether or not ⟨U⟩ diverges depends on the upper tail of p(U) — the probability distri-  bution function (PDF) of U —, which is determined by the realizations of S(x, z) yielding  a large U, with U and S(x, z) related to each other through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It is then natural to  ask what these particular realizations of S(x, z) are like, from which probability distribution  they are drawn, and if the corresponding tail of p(U) does give the correct value of gc(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Answering these questions is the subject of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  To put our work into perspective, it is interesting to recall how the existence of gc(L) has  been interpreted in laser-plasma physics literature since Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The interpretation relies  on the implicit assumption that the realizations of |S(x, z)|2 giving rise to a large U in the  tail of p(U) and the generic ones for which U is in the bulk of p(U) are alike, in the sense  of being made up of local, statistically independent, overintensities, or hot spots, separated  from each other by a few correlation lengths of S(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Hot spot contribution to the ampli-  fication of |ψ|2 can then be computed by using the remarkable result that intense hot spots  have a non-random profile depending on the correlation function of S(x, z) and being the  same for each hot spot [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Thus, intense hot spots are entirely characterized by their  random intensity which turns out to be exponentially distributed (for large intensity and  to within slow, algebraic, corrections) [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For g large enough, intense hot spots become  statistically significant as the exponentially large amplification they produce outbalances  their exponentially small scarcity, leading to the divergence of ⟨U⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The smallest value of g  at which this divergence occurs defines the critical coupling gc(L) and for g > gc(L) physics  could be expected to be dominated by intense hot spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Unfortunately, this interpretation  fails to give the correct value of gc(L) for L greater than a hot spot length [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The  assumption of high intensity, statistically independent hot spots giving the dominant con-  tribution to the amplification in the large U limit must be revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Large values of U are  produced by rare realizations of S(x, z) that have no reason a priori to look like generic  realizations with no other structures than uncorrelated, local hot spots randomly scattered  in Λ × [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It may or may not be so: the answer will come out of the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  In the simpler diffraction-free case where m−1 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1), the problem reduces to a  mere 1D amplification along z with ψ(L) = exp  �  g  � L  0 |S(z)|2dz  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' A large value of |ψ(L)|2  corresponds to a large value of  � L  0 |S(z)|2dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Thus, the realizations of S(z) that form the  tail of p(U) are the ones with a large L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' These realizations were studied thoroughly  in [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Let C(z, z′) = ⟨S(z)S(z′)∗⟩ and define the covariance operator TC acting on  3  f(z) ∈ L2([0, L]) by  (TCf)(z) =  � L  0  C(z, z′) f(z′) dz′,  (2)  with 0 ≤ z ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Write µ1 > 0 the largest eigenvalue of TC with degeneracy d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It was  proved in [10, 11] that the realizations of S(z) with a large L2-norm concentrate onto the  fundamental eigenspace of TC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', the eigenspace associated with the largest eigenvalue µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  More specifically, writing {φ1, · · · , φd1} an orthonormal basis of the fundamental eigenspace  of TC, one has  S(z) ∼ √η  d1  �  i=ν  aνφν(z)  (∥S∥2 → +∞),  (3)  with η ∼ ∥S∥2  2, where ∥ · ∥2 denotes the L2-norm over [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The aνs are complex numbers  normalized to �d1  ν=1 |aν|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The probability distribution of η has the gamma-distribution  tail p(η) ∼ ηd1−1e−η/µ1 for large η, and the aνs define a random 2d1-dimensional (real) unit  vector a with coordinates Re(aν) and Im(aν) (1 ≤ ν ≤ d1) the direction of which is uniformly  distributed over the unit (2d1 − 1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (3) it is clear that the realizations of  S(z) with a large L2-norm are less random than the Gaussian field S(z) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It only takes  2d1 random quantities to characterize these realizations entirely: η and the direction of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  For instance, if µ1 is not degenerate (d1 = 1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (3) yields  S(z)  ∥S∥2  ∼ eiθφ1(z)  (∥S∥2 → +∞),  (4)  where θ is a random phase uniformly distributed over [0, 2π) and |S(z)|/∥S∥2 ∼ |φ1(z)| is  non-random, which means that the profile of S(z) is purely deterministic in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note  that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (4) rules out any description in terms of localized hot spots when L is large, as  φ1(z) typically is a one-bump delocalized mode spreading over the whole domain 0 ≤ z ≤ L  (see [10] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As will be seen further on, the randomness reduction of S when the  amplification is large occurs in the m−1 ̸= 0 case too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  From U = |ψ(L)|2 = exp(2g∥S∥2  2) and η ∼ ∥S∥2  2 as ∥S∥2 → +∞, one gets η ∼ (2g)−1 ln U  as U → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The tail of p(U) is then readily obtained from p(η) ∼ ηd1−1e−η/µ1 and the  change of variables from η to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' One finds, in logarithmic form,  ln p(U) ∼ ln  � 1  U p  �  η = 1  2g ln U  ��  (5)  = −  �  1 +  1  2µ1g  �  ln U + (d1 − 1) ln ln U + O(1)  (U → +∞),  4  from which it follows that p(U) has a leading algebraic tail ∝ U −ζ (modulated by logarithmic  corrections in the amplitude) with exponent ζ = (1 + 1/2µ1g) depending continuously on  the parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Injecting this result into ⟨U⟩ =  � +∞  1  Up(U) dU, one finds that  the critical coupling in the diffraction-free case is given by gc(L) = 1/2µ1 (where µ1 depends  on L) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note that it is also possible to determine the tail of p(U) exactly from the full  Gaussian statistics of S without using η ∼ ∥S∥2  2, which makes it possible to estimate the  contribution of the subleading corrections to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Skipping the details, one finds that  these corrections do not contribute to ln p(U) by terms greater than O(1) as U → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  To conclude this brief overview of diffraction-free results, let us mention the interesting  connection between the concentration onto the fundamental eigenspace of TC in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (3) and  the Bose-Einstein condensation of S(z) in the ‘thermodynamic’ limit defined by L → +∞  and ∥S∥2  2 → +∞ with fixed ∥S∥2  2/L, see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Note also that more general concentra-  tion properties can be found in the limit where the large L2-norm is replaced with a large  quadratic or linear form, see [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  By contrast, much less is known in the general case with diffraction where m−1 ̸= 0 in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The only results so far are (i) the numerical ones in the second part of [1] and (ii)  the analytical calculation of the critical coupling performed in [5] where it is proved that the  critical coupling without diffraction cannot be less than the one with diffraction, the latter  being given by  gc(L) =  1  2 supx(·)∈B(0,L) µ1[x(·)],  (6)  and the former by 1/2µ1[x(·) ≡ 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (6), B(0, L) denotes the set of all the continuous  paths in Λ satisfying x(L) = 0 and µ1[x(·)] is the largest eigenvalue of the covariance operator  Tx(·) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (2) with C(z, z′) = ⟨S(x(z), z)S(x(z′), z′)∗⟩ (see also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (43)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The  question  then  arises  whether  the  presence  of  the  non-local  quantity  supx(·)∈B(0,L) µ1[x(·)] in the expression for gc(L) — a presence which does not appear in  hot spot models — is the signature of a corresponding non-local structure in the realizations  of S(x, z) giving rise to a large amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' To answer this question we need to find a way  to identify such realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The corresponding tail of p(U) will then be tested in return  by checking that the critical coupling it yields coincides with the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The calcu-  lations in [10, 11] are of no help as being specific to the diffraction-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' To determine  S(x, z) when U is large and get the tail of p(U) in the presence of diffraction we need a  5  different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  A possible line of attack is through the functional integral formalism introduced by  Janssen [15], DeDominicis [16, 17], and Phythian [18] (see also [19, 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  This method  provides a formal description of classical statistical dynamics in terms of functional integrals  analogous to Feynman’s action-integral formalism of quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Applying the method  to the stochastic equation (1), one finds that p(U) can be formally written as the functional  integral  p(U) ∝  �  ϕ(x,0)=b  δ(U − |ϕ(0, L)|2) eA D2ϕ D2 ˜ϕ D2S,  (7)  where ϕ and ˜ϕ are complex Martin-Siggia-Rose conjugate fields [21], D2 ≡ DRe(·)DIm(·),  and A ≡ A(ϕ, ˜ϕ, S) is an ‘action’ depending on ϕ, ˜ϕ, and S, often referred to as the Martin-  Siggia-Rose (MSR) action in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' When U is large, the functional integral on  the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (7) is determined by the field configuration — called the ‘leading  instanton’ — corresponding to the highest saddle-point of the action A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note that, usually,  S is integrated out in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (7), leading to  p(U) ∝  �  ϕ(x,0)=b  δ(U − |ϕ(0, L)|2) eL D2ϕ D2 ˜ϕ,  (8)  where the action L ≡ L(ϕ, ˜ϕ) is given by  eL(ϕ, ˜ϕ) ∝  �  eA(ϕ, ˜ϕ,S) D2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (9)  The expressions of p(U) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (7) and (8) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Either can be used and the large  U behavior of p(U) can equally be obtained from an instanton analysis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (8), instead of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As we want to determine the most probable realizations of S(x, z) when U is large,  it is natural to keep S explicit in the problem and to use the functional representation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  When U is large, the fluctuations of the fields around the instanton are small and can be  integrated out in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (7) as standard Gaussian fluctuations, yielding the leading asymptotic  behavior  ln p(U) ∼ ln  �  δ(U − |ϕinst(0, L)|2)  �  Sinst  (U → +∞),  (10)  where the subscript ‘inst’ stands for leading instanton and ⟨·⟩Sinst denotes the average over  the realizations of Sinst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In the diffraction-free case, it will be checked in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' II that Sinst  coincides with the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (3) and that the tail of p(U) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (10) is the same  as the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  6  The first attempt to obtain PDF tails from the highest saddle-point of the action in a  functional integral representation was made by Giles for Navier-Stokes turbulence in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Unfortunately, the perturbative approach followed in this work was doomed to fail as in-  stantons are nonperturbative objects by nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The proper, nonperturbative, instanton  analysis of intermittency in fluid turbulence was performed by Falkovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' See  also the instanton analyses of PDF tails in forced Burgers turbulence in [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Here, we  give the first instanton analysis of the stochastic amplifier (1) in the large amplification limit  of interest in the overcritical regime g > gc(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Our strategy is two-step:  write the MSR action A(ϕ, ˜ϕ, S) for the stochastic amplifier (1) and find the corre-  sponding leading instanton Sinst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The realizations of Sinst define the driver onto which  the realizations of S concentrate in the large U limit, which generalizes the diffraction-  free result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (3) to the case with diffraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  use the instanton ϕinst and Sinst obtained at the first step on the right-hand side of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (10) to get the tail of p(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Before entering the details of the calculations, it is useful to give a brief summary of the  main new results obtained in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  ⋄ We show that in the large U limit, the realizations of S concentrate onto large-scale  filamentary instantons running along specific non-random paths in B(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' These  paths are the ones maximizing the largest eigenvalue µ1[x(·)] of the covariance operator  Tx(·) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In the case of a ‘single-filament instanton’ (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' IV), we  prove that  S(x, z) ∼  1  µmax  � L  0  C(x − xinst(z′), z, z′)  � d1  �  ν=1  cνφν(z′)  �  dz′  (U → +∞),  (11)  where C(x − x′, z, z′) = ⟨S(x, z)S(x′, z′)∗⟩, xinst(·) maximizes µ1[x(·)], and µmax ≡  µ1[xinst(·)] with degeneracy d1 and orthonormal eigenfunctions φν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The cνs are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  complex Gaussian random variables with ⟨cν⟩ = ⟨c2  ν⟩ = 0 and ⟨|cν|2⟩ = µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The  instanton on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (11) — which lives within a long thin tube,  or filament, running along xinst(·) (see the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' IV A) — is less random than  the generic realizations of S for which U is in the bulk of p(U): it only takes the d1  7  (complex) random variables cν to characterize the instanton entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' If µmax is not  degenerate (d1 = 1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (11) reduces to  S(x, z)  ∥S∥2  ∼ Aei arg(c1)  � L  0  C(x − xinst(z′), z, z′) φ1(z′) dz′  (U → +∞),  where A > 0 is a constant, and the profile of S(x, z) defined by |S(x, z)|/∥S∥2 is  asymptotically non-random as U → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  ⋄ We determine the tail of p(U) for large U from the statistics of the instanton on the  right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We find that p(U) has a leading algebraic tail ∝ U −ζ with  exponent ζ = (1 + 1/2µmaxg), modulated by a slow varying amplitude (slower than  algebraic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Injecting this result into ⟨U⟩ =  � +∞  1  Up(U) dU, we find that ⟨U⟩ diverges  for all g > 1/2µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The critical coupling is thus given by gc(L) = 1/2µmax (where µmax  depends on L), in agreement with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We can then explain the intriguing presence  of the non-local quantity µmax in the expression of gc(L) as a direct consequence of  the fact that the realizations of S causing the divergence of ⟨U⟩ are realizations of the  non-local instanton (11), (rather than of localized hot spots, as is widely assumed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  ⋄ Finally, the emergence of the instanton in the realizations of S as U increases is  observed in numerical simulations (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For large but finite U in the sampled  range, the emerging large-scale instanton coexists with fluctuation induced small-scale  hot spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The presence of the instanton causes the hot spots to cluster in the instanton  region instead of being uniformly scattered in Λ × [0, L], and the level of |S(x, z)|2  between the hot spots remains significantly higher than it would be in the absence of  instanton (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6, 8(a), and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In Section II, we test the functional approach by  revisiting the diffraction-free problem where the results are already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In Section III,  we write the instanton equations for the full problem with diffraction in the case of one  transverse dimension (d = 1) and we specify the class of S we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Section IV is devoted  to the solution of the instanton equations in the case of ‘single-filament’ instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The  corresponding tail of p(U) is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In Section V, we verify our analytical predictions  via numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Finally, we discuss our results and their implications, especially  in laser-matter interaction physics, and we give potential perspectives in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Some  technical material is relegated to the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  8  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  AMPLIFICATION WITHOUT DIFFRACTION REVISITED  As a warm-up to the full problem (1), we test the functional approach on the simpler  problem without diffraction and see how the results in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (3) and (5) can also be obtained  from an instanton analysis of the appropriate MSR action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The MSR action A(ϕ, ˜ϕ, S)  In the diffraction-free limit, m−1 = 0, the equation (1) reduces to the 1D stochastic  amplifier (for fixed x, not written)  �  �  �  dzψ(z) − g|S(z)|2ψ(z) = 0,  0 ≤ z ≤ L and ψ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (12)  Let F[ψ(·)] be a functional of ψ(z) solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' From the general formalism devel-  oped in [15–20] it can be shown that F[ψ(·)] admits the functional integral representation  F[ψ(·)] =  �  ϕ(0)=b  F[ϕ(·)] e  i  2 (⟨ ˜ϕ|dz−g|S|2|ϕ⟩+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=') D2ϕ D2 ˜ϕ,  (13)  with Dirac’s bracket notation ⟨f|O|h⟩ =  � L  0 f(z)∗(Oh)(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note that since ψ(0) is real,  ψ(z) is also real for all z and a representation with real ϕ and ˜ϕ would have been sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (13) we have kept complex ϕ and ˜ϕ in anticipation of the generalization to the case  with diffraction where ψ is a complex field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Now, using (13) with F[ψ(·)] = δ(U − |ψ(L)|2)  in p(U) = ⟨δ(U − |ψ(L)|2)⟩S, where ⟨·⟩S denotes the average over the realizations of S, and  writing TC the covariance operator of S (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (2)), one gets  p(U) =  �  ϕ(0)=b  �  δ(U − |ϕ(L)|2) e  i  2 (⟨ ˜ϕ|dz−g|S|2|ϕ⟩+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=')�  S D2ϕ D2 ˜ϕ  ∝  �  ϕ(0)=b  δ(U − |ϕ(L)|2) e  i  2 (⟨ ˜ϕ|dz−g|S|2|ϕ⟩+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=')−⟨S|T −1  C |S⟩ D2ϕ D2 ˜ϕ D2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (14)  The functional integral representation of p(U) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (14) is of the same form as the one in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (7) with the MSR action  A(ϕ, ˜ϕ, S) = i  2  ��  ˜ϕ  ��dz − g|S|2�� ϕ  �  + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  �  −  �  S  ��T −1  C  �� S  �  = i  2  �� L  0  ˜ϕ∗(z)  �  dzϕ(z) − g|S(z)|2ϕ(z)  �  dz + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  �  −  � L  0  S∗(z)(T −1  C S)(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (15)  9  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Leading instanton and tail of p(U)  The leading instanton which determines the large U behavior of p(U) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (14) is a  stationary point of A(ϕ, ˜ϕ, S) under the restriction |ϕ(L)|2 = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' According to the usual  procedure of Lagrange multipliers [26], it can be found as a stationary point of the action  A′(ϕ, ˜ϕ, S) = A(ϕ, ˜ϕ, S) + λ|ϕ(L)|2 without restriction, where λ is a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Write δA′(ϕ, ˜ϕ, S) the variation of A′(ϕ, ˜ϕ, S) under variations of the fields and their com-  plex conjugates treated as independent variables, with endpoints ϕ(0) = ϕ∗(0) = 1, and  ˜ϕ(L+) = ˜ϕ∗(L+) = 0 (by causality principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The stationarity condition  δA′(ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' ˜ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' S) = 0 leads to the equations  dzϕ(z) − g|S(z)|2ϕ(z) = 0 with ϕ(0) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  dz ˜ϕ(z) + g|S(z)|2 ˜ϕ(z) = −2iλϕ(L)δ(z − L) with ˜ϕ(L+) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (16)  dz ˜ϕ∗(z) + g|S(z)|2 ˜ϕ∗(z) = −2iλϕ∗(L)δ(z − L) with ˜ϕ∗(L+) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  and  (T −1  C S)(z) = −ig  2 [ ˜ϕ∗(z)ϕ(z) + ˜ϕ(z)ϕ∗(z)] S(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (17)  or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' equivalently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  dzϕ(z) − g|S(z)|2ϕ(z) = 0 with ϕ(0) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  dz ˜ϕ(z) + g|S(z)|2 ˜ϕ(z) = 0 with ˜ϕ(L) = 2iλϕ(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (18)  dz ˜ϕ∗(z) + g|S(z)|2 ˜ϕ∗(z) = 0 with ˜ϕ∗(L) = 2iλϕ∗(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  and  [TC ( ˜ϕ∗ϕ + ˜ϕϕ∗) S] (z) = 2i  g S(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (19)  Note that if Re(λ) ̸= 0, ˜ϕ∗(z) as given by the third equation (18) is different from the complex  conjugate of the solution to the second equation (18) for ˜ϕ(z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' ˜ϕ∗(z) ̸= ˜ϕ(z)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' This  seemingly strange result is a consequence of treating the fields and their complex conjugates  as independent when varying the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  For the auxiliary (unphysical) field ˜ϕ, having  ˜ϕ∗(z) ̸= ˜ϕ(z)∗ in the instanton solution is not forbidden a priori, unlike the physical fields ϕ  and S for which instantons are observable realizations with ϕ∗(z) = ϕ(z)∗ and S∗(z) = S(z)∗  for all 0 ≤ z ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  10  The equations (18) are readily solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' One gets,  ϕ(z) = eg  � z  0 |S(z′)|2dz′,  ˜ϕ(z) = 2iλϕ(L)eg  � L  z |S(z′)|2dz′,  (20)  ˜ϕ∗(z) = 2iλϕ∗(L)eg  � L  z |S(z′)|2dz′,  and ˜ϕ∗(z)ϕ(z) = ˜ϕ(z)ϕ∗(z) = 4iλ|ϕ(L)|2 = 4iλU, independent of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Injecting this solution  onto the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (19), one obtains the eigenvalue equation  (TCS)(z) =  1  2λgU S(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (21)  It follows immediately from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (21) that an instanton solution for S is an eigenfunction of its  covariance operator TC, which fixes the value of λ for each instanton, namely λ = 1/(2µngU),  where µ1 > µ2 > · · · > 0 are the eigenvalues of TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The leading instanton Sinst corresponds to  the largest eigenvalue µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Writing d1 the degeneracy of µ1 and {φ1, · · · , φd1} an orthonormal  basis of the fundamental eigenspace of TC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', the eigenspace associated with µ1, one gets  Sinst(z) =  d1  �  ν=1  cνφν(z),  (22)  where cν =  � L  0 Sinst(z)φν(z)∗dz is a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The other components of the leading  instanton, ϕinst, ˜ϕinst, and ˜ϕ∗  inst, are the instanton solution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (20) with S = Sinst and  λ = 1/(2µ1gU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In the following, we will only need the expression of ϕinst(L),  ϕinst(L) = exp  �  g∥Sinst∥2  2  �  ,  (23)  where ∥ · ∥2 denotes the L2-norm over [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Integrating out the fluctuations of the fields around the leading instanton in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (14) and  using the expressions of Sinst and ϕinst in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (22) and (23), one obtains the diffraction-free  version of the general asymptotic expression (10),  ln p(U) ∼ ln  �  · ·  �  δ  �  U − exp  �  2g  d1  �  ν=1  |cν|2  ��  d1  �  ν=1  exp  �  −|cν|2  µ1  � d2cν  πµ1  = ln  1  Γ(d1)µ1  � +∞  0  δ  �  U − e2gη� � η  µ1  �d1−1  e−η/µ1dη  (U → +∞),  (24)  where we have made the change of variable �d1  ν=1 |cν|2 = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It can be seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (24) that the  cνs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (22) are independent (complex) Gaussian random variables with ⟨cν⟩ = ⟨c2  ν⟩ = 0  11  and ⟨|cν|2⟩ = µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Thus, writing cν = aν√η in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (22), one gets  Sinst(z) = √η  d1  �  ν=1  aνφν(z),  (25)  where η is a gamma-distributed random variable with p(η) = [Γ(d1)µ1]−1(η/µ1)d1−1e−η/µ1,  and the aνs define a random 2d1-dimensional (real) unit vector a with coordinates Re(aν)  and Im(aν) the direction of which is uniformly distributed over the unit (2d1−1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For  large U (hence large η), the realizations of S which contribute to the tail of p(U) concentrate  onto the leading instanton, S(z) ∼ Sinst(z) (η → +∞), and one recovers the result of [10, 11]  recalled in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It remains to perform the integral over η in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (24), which can be done  without difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' One obtains the asymptotic behavior given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (5),  ln p(U) = −  �  1 +  1  2µ1g  �  ln U + (d1 − 1) ln ln U + O(1)  (U → +∞),  as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  AMPLIFICATION WITH DIFFRACTION: GENERAL SETTING  The approach followed in the previous section to deal with the diffraction-free case is  completely different from the one in [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Having checked that both give the same  results, we can now move on to the next step and use the instanton analysis to deal with  the full problem with diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  MSR action and instanton equations  We consider the transversally one-dimensional (d = 1) version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1),  �  �  �  ∂zψ(x, z) −  i  2m∂2  x2ψ(x, z) = g|S(x, z)|2ψ(x, z),  0 ≤ z ≤ L, x ∈ Λ ⊂ R, and ψ(x, 0) = 1,  (26)  where we take for Λ the circle of length ℓ (and radius ℓ/2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The random field S is homo-  geneous along x with normalization L−1 � L  0 ⟨|S(x, z)|2⟩ dz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (The generalization to more  than one transverse dimension is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=') Our goal is to determine the realizations  of S and the tail of p(U) in the large U limit for U = |ψ(0, L)|2, with ψ(x, z) solution to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Write  Dz, x2 ≡ ∂z −  i  2m∂2  x2,  (27)  12  and TC the covariance operator of S defined by  (TCf)(x, z) =  �  Λ  � L  0  C(x − x′, z, z′) f(x′, z′) dz′ dx′,  f(x, z) ∈ L2(Λ × [0, L]),  (28)  with C(x − x′, z, z′) = ⟨S(x, z)S(x′, z′)∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The counterpart of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (14) in the problem with  diffraction reads  p(U) ∝  �  ϕ(x,0)=b  δ  �  U − |ϕ(0, L)|2�  e  i  2 (⟨ ˜ϕ|Dz, x2−g|S|2|ϕ⟩+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=')−⟨S|T −1  C |S⟩ D2ϕ D2 ˜ϕ D2S,  (29)  which is of the same form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (7) with MSR action  A(ϕ, ˜ϕ, S) = i  2  ��  ˜ϕ  ��Dz, x2 − g|S|2�� ϕ  �  + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  �  −  �  S  ��T −1  C  �� S  �  = i  2  ��  Λ  � L  0  ˜ϕ∗(x, z)  �  Dz, x2ϕ(x, z) − g|S(x, z)|2ϕ(x, z)  �  dz dx + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  �  (30)  −  �  Λ  � L  0  S∗(x, z)(T −1  C S)(x, z) dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The derivation of the instanton equations from the action in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (30) follows exactly the  same line as in the diffraction-free case in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Varying A(ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' ˜ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' S) with the Lagrange  multiplier term λ|ϕ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)|2 and setting the variation to zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' one obtains the equations  �  Dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x2 − g|S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)|2�  ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) = 0 with ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 0) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  �  Dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x2 + g|S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)|2�  ˜ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) = −2iλϕ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)δ(x)δ(z − L) with ˜ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L+) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (31)  �  D∗  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x2 + g|S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)|2�  ˜ϕ∗(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) = −2iλϕ∗(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)δ(x)δ(z − L) with ˜ϕ∗(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L+) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  and  (T −1  C S)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) = −ig  2 [ ˜ϕ∗(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) + ˜ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)ϕ∗(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)] S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (32)  or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' equivalently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  �  Dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x2 − g|S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)|2�  ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) = 0 with ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 0) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  �  Dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x2 + g|S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)|2�  ˜ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) = 0 with ˜ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L) = 2iλϕ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)δ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (33)  �  D∗  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x2 + g|S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z)|2�  ˜ϕ∗(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) = 0 with ˜ϕ∗(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L) = 2iλϕ∗(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)δ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  and  [TC ( ˜ϕ∗ϕ + ˜ϕϕ∗) S] (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z) = 2i  g S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (34)  The equations (33) are readily solved in terms of Feynman-Kac propagator,  K(x2, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x1, z1) =  � x(z2)=x2  x(z1)=x1  e  � z2  z1 [ im  2 ˙x(τ)2+g|S(x(τ),τ)|2] dτDx,  (35)  13  with z2 > z1, where the path-integral is over the set of all the continuous paths in Λ satisfying  x(z1) = x1 and x(z2) = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' One gets  ϕ(x, z) =  �  Λ  K(x, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' y, 0) dy,  ˜ϕ(x, z) = 2iλϕ(0, L) K(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x, z)∗,  (36)  ˜ϕ∗(x, z) = 2iλϕ∗(0, L) K(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Like in the diffraction-free case, one has ˜ϕ∗(x, z) ̸= ˜ϕ(x, z)∗ if Re(λ) ̸= 0 (see the discussion  below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Using the expressions (36) on the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (34), one obtains  ϕ∗(0, L) G1(x, z) + ϕ(0, L) G2(x, z) = 1  λg S(x, z),  (37)  with  G1(x, z) =  � L  0  �  Λ  �  Λ  K(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x′, z′)K(x′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' ξ, 0)  ×C(x − x′, z, z′) S(x′, z′) dx′ dξ dz′,  (38)  and  G2(x, z) =  � L  0  �  Λ  �  Λ  K(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' x′, z′)∗K(x′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' ξ, 0)∗  ×C(x − x′, z, z′) S(x′, z′) dx′ dξ dz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (39)  In the large U limit, S concentrates onto the leading instanton, Sinst, solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (37),  and ϕ(x, z) concentrates onto ϕinst(x, z) given by the Feynman-Kac path-integral for ϕ(x, z)  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (36) with S = Sinst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The key to solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (37) follows from the fact that, in this  limit, the path-integrals in Eqs (38) and (39) are dominated by the contribution of the  paths with the largest amplification, the contribution of the other paths being subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  These dominant trajectories run in the vicinity of ‘ridge paths’ along which |Sinst(x, z)|2 is  at a global maximum (for every given z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Assuming that the ridge paths are all continuous  (to be checked a posteriori, once Sinst is known), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (37) simplifies and for a class of S that  we will now specify, it can be solved explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Specification of S(x, z)  We assume that S(x, z) can be expressed as a finite random Fourier sum,  S(x, z) =  �  (n,j)∈I  s(n,j)  �σ(n,j)  ℓ  e2iπnx/ℓΦ(n,j)(z),  (40)  14  where I is a finite subset of Z × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The s(n,j)s are complex Gaussian random variables  with ⟨s(n,j)⟩ = ⟨s(n,j)s(m,k)⟩ = 0 and ⟨s(n,j)s∗  (m,k)⟩ = δnmδjk, the σ(n,j)s are positive constants  normalized to �  (n,j)∈I σ(n,j) = Lℓ, and, for fixed n, the Φ(n,j)s are orthonormal continuous  functions of 0 ≤ z ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (40) in C(x − x′, z, z′) = ⟨S(x, z)S(x′, z′)∗⟩ one gets  C(x − x′, z, z′) =  �  (n,j)∈I  σ(n,j)  ℓ  e2iπn(x−x′)/ℓΦ(n,j)(z)Φ(n,j)(z′)∗,  (41)  from which it follows that σ(n,j) and e2iπnx/ℓΦ(n,j)(z)/  √  ℓ are the eigenvalues and orthonormal  eigenfunctions of the covariance operator TC defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Equation (40) generalizes models of spatially smoothed laser beams in which laser light  is represented by a superposition of monochromatic beamlets the amplitudes of which are  independent random variables [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For a large number of beamlets these random variables  can be taken as Gaussian and the laser electric field takes on the form (40) in which the  sum over j reduces to j = 1 with Φ(n,1)(z) = (1/  √  L) exp[iα(2πn/ℓ)2z)] where α is a (real)  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Moreover, every centered Gaussian field with a continuous correlation function  has an expansion of the form (40), possibly with an infinite sum [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Combining this result  with the practically unavoidable existence of some natural cut-off making the sum finite  (like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', in numerical simulations), one can safely expects the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (40) to be  quite generic, at least from a practical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Let B(0, L) denote the set of all the continuous paths in Λ satisfying x(L) = 0 and define  M[x(·)] the |I| × |I| positive definite matrix with components  M(n,j)(m,k)[x(·)] =  √σ(n,j)σ(m,k)  ℓ  � L  0  e2iπ(m−n)x(z)/ℓΦ(n,j)(z)∗Φ(m,k)(z) dz,  (42)  in which x(·) ∈ B(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Write µ1[x(·)] > 0 the largest eigenvalue of M[x(·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We consider  cases fulfilling the following two assumptions:  (i) all the paths maximizing µ1[x(·)] are in B(0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (ii) there is a finite number of paths in B(0, L) maximizing µ1[x(·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  It is proved in [5] that the eigenvalues of M[x(·)] are equal to the ones of Tx(·) defined by  (Tx(·)f)(z) =  � L  0  C(x(z) − x(z′), z, z′) f(z′) dz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  f(z) ∈ L2([0, L]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (43)  15  It follows in particular that µ1[x(·)] is invariant under the path transformations leaving  C(x(z) − x(z′), z, z′) unchanged and that the image of a path maximizing µ1[x(·)] by such a  transformation is also a path maximizing µ1[x(·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Assumption (i) is a central feature of the class of S we consider in this paper, together  with the random Fourier representation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We don’t know whether Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (37)  could be solved analytically in the large U limit without this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The technical  restriction (ii) will be used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' IV B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Assumptions (i) and (ii) are fulfilled in most cases  of practical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Lifting (ii) raises tricky technical problems yet to be solved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' this will  be the subject of a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Finally, for notational convenience we define  µmax =  sup  x(·)∈B(0,L)  µ1[x(·)],  (44)  the supremum being reached in B(0, L), by Assumption (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  SINGLE-FILAMENT INSTANTON AND TAIL OF p(U)  In the following, we consider the simplest case where for each realization of Sinst there is  only one ridge path of |Sinst(x, z)|2 in B(0, L), denoted by xinst(·), dominating the large U  limit of the Feynman-Kac integrals in Eqs (38) and (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note that xinst(·) may be different  from one realization of Sinst to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The set of all the realizations of Sinst having the  same xinst(·) defines a random field, denoted by Sxinst(·)  inst  , referred to in the following as a  ‘single-filament instanton’ (the reason for this name will appear more clearly at the end of  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We will write ϕxinst(·)  inst  (x, z) the Feynman-Kac path-integral for ϕ(x, z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (36)  with S = Sxinst(·)  inst  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Single-filament instantons are not the only possible solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Multi-filament  instantons are also possible if realizations of |Sinst(x, z)|2 have more than one ridge path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The  conditions for single- or multi-filament instantons are specified below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) as well as at  the end of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The study of multi-filament instantons being excessively intricate,  we restrict ourselves to single-filament instantons for the sake of clarity and readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Leading instanton  Assume that xinst(·) is continuous (to be checked a posteriori).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As finite sums of continu-  ous functions, both S(x, z) and C(x − x′, z, z′) are continuous functions of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  16  It follows in particular that for fixed x, z, and z′, the product C(x − x′, z, z′) S(x′, z′) on the  right-hand side of Eqs (38) and (39) is a continuous function of x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Integrating over ξ and  x′ at fixed z′ and using the fact that, in the large U limit, only the vicinity of x′ = xinst(z′)  contributes, one gets the large U behavior of G1,2(x, z),  G1(x, z) ∼  �  Λ  K(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' ξ, 0) dξ  � L  0  C(x − xinst(z′), z, z′) Sxinst(·)  inst  (xinst(z′), z′) dz′  = ϕxinst(·)  inst  (0, L)  � L  0  C(x − xinst(z′), z, z′) Sxinst(·)  inst  (xinst(z′), z′) dz′  (U → +∞), (45)  and  G2(x, z) ∼  �  Λ  K(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' ξ, 0)∗ dξ  � L  0  C(x − xinst(z′), z, z′) Sxinst(·)  inst  (xinst(z′), z′) dz′  = ϕ(0, L)xinst(·) ∗  inst  � L  0  C(x − xinst(z′), z, z′) Sxinst(·)  inst  (xinst(z′), z′) dz′  (U → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (46)  Injecting these expressions onto the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (37), one obtains the instanton  equation  � L  0  C(x − xinst(z′), z, z′) Sxinst(·)  inst  (xinst(z′), z′) dz′ =  1  2λgU Sxinst(·)  inst  (x, z),  (47)  where we have used the equality |ϕxinst(·)  inst  (0, L)|2 = U imposed by the delta function on the  right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Equation (47) can be solved in two different ways, depending on wether or not the Fourier  decompositions (40) and (41) for Sxinst(·)  inst  and C(x−xinst(z′), z, z′) are used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Using  these decompositions, one gets the eigenvalue equation  �  (m,k)∈I  M(n,j)(m,k)[xinst(·)] s(m,k) =  1  2λgU s(n,j),  (48)  which fixes λ at λ = 1/(2µn[xinst(·)]gU), where µ1[xinst(·)] > µ2[xinst(·)] > · · · > 0 are the  eigenvalues of M[xinst(·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The leading instanton Sxinst(·)  inst  corresponds to the largest eigenvalue  µ1[xinst(·)] with xinst(·) maximizing µ1[x(·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The fact that xinst(·) exists and is continuous  is ensured by the assumption (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It is worth noticing that xinst(·) is a non-random path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Thus, for every path xinst(·) ∈ B(0, L) maximizing µ1[x(·)], there is a leading instanton  Sxinst(·)  inst  (x, z) =  �  (n,j)∈I  s(n,j)  �σ(n,j)  ℓ  e2iπnx/ℓΦ(n,j)(z),  (49)  17  where s (with components s(n,j)) is an eigenvector of M[xinst(·)] associated with the largest  eigenvalue µ1[xinst(·)] = µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It is checked in Appendix A that xinst(·) is indeed a ridge  path of |Sxinst(·)  inst  (x, z)|2, as it should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The calculation in Appendix A also specifies under what condition on S the leading  instanton in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) is a single-filament instanton: the fundamental eigenspace of M[xinst(·)]  and the one of M[x(·)] for every other path maximizing µ1[x(·)], if any, must be essentially  disjoint1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In particular, if the fundamental eigenspaces of M[x(·)] for all the different paths  maximizing µ1[x(·)] are essentially disjoint, all the instantons are single-filament instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  This is the case considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Conversely, if the fundamental eigenspaces of  M[x(·)] for different paths maximizing µ1[x(·)] have a non trivial intersection, then multi-  filament instantons come into play as possible solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We now solve the equation (47) without using the Fourier decompositions (40) and (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Taking x = xinst(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (47), one finds that Sxinst(·)  inst  (xinst(z), z) is an eigenfunction of  Txinst(·) with eigenvalue 1/2λgU, which fixes λ at λ = 1/(2µn[xinst(·)]gU), where µ1[xinst(·)] >  µ2[xinst(·)] > · · · > 0 are the eigenvalues of Txinst(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (Recall that Tx(·) and M[x(·)] have the  same eigenvalues with the same multiplicities [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Again, Sxinst(·)  inst  corresponds to the largest  eigenvalue µ1[xinst(·)] with xinst(·) maximizing µ1[x(·)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', µ1[xinst(·)] = µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Writing d1  the degeneracy of µ1[xinst(·)] and {φ1, · · · , φd1} an orthonormal basis of the fundamental  eigenspace of Txinst(·), one has  Sxinst(·)  inst  (xinst(z), z) =  d1  �  ν=1  cνφν(z),  (50)  where cν =  � L  0 Sxinst(·)  inst  (xinst(z), z)φν(z)∗dz is a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Injecting (50) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (47),  one obtains  Sxinst(·)  inst  (x, z) =  1  µmax  � L  0  C(x − xinst(z′), z, z′)  � d1  �  ν=1  cνφν(z′)  �  dz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (51)  Equivalence of the Fourier and convolution representations of Sxinst(·)  inst  The fact that the expressions of Sxinst(·)  inst  (x, z) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) and (51) are equivalent is proved  1 ‘essentially disjoint’ and ‘trivial intersection’ mean that the intersection reduces to the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  18  in Appendix B, with s(n,j) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) and cν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (51) related to each other by  s(n,j) =  1  √µmax  d1  �  ν=1  cνe(ν)  (n,j) and cν = √µmax  �  (n,j)∈I  s(n,j)e(ν) ∗  (n,j),  (52)  where {e(1), · · · , e(d1)} is an orthonormal basis of the fundamental eigenspace of M[xinst(·)]  the vectors of which are defined by their components  e(ν)  (n,j) =  � σ(n,j)  ℓ µmax  � L  0  e−2iπnxinst(z′)/ℓΦ(n,j)(z′)∗φν(z′) dz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (53)  Note that by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) for Sxinst(·)  inst  (xinst(z), z) and the definition of e(ν)  (n,j) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (53), the  expression of cν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (52) coincides with cν =  � L  0 Sxinst(·)  inst  (xinst(z), z)φν(z)∗dz, as it should  be (see below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (50)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Statistical properties of S in the large U limit  The statistical properties of the cνs and s(n,j)s are readily obtained from the ones of the  s(n,j)s in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Since the s(n,j)s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' standard complex Gaussian random variables  with ⟨s(n,j)⟩ = ⟨s2  (n,j)⟩ = 0 and ⟨|s(n,j)|2⟩ = 1, the orthogonal projection of the |I|-dimensional  (complex) vector s with coordinates s(n,j) onto any given direction is also a standard complex  Gaussian random variable statistically independent of the projections onto the orthogonal  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Thus, projecting s onto the direction of the vector e(ν) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (53)  (with given 1 ≤ ν ≤ d1) and writing cν = √µmax s · e(ν) ∗, one finds that the cνs are  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (complex) Gaussian random variables with ⟨cν⟩ = ⟨c2  ν⟩ = 0 and ⟨|cν|2⟩ = µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The statistical properties of the s(n,j)s are different from the ones of the s(n,j)s because s  is restricted to the d1-dimensional fundamental eigenspace of M[xinst(·)] (with d1 ≪ |I|,  typically), which induces correlations between the s(n,j)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' From the first Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (52) and the  statistical properties of the cνs, one finds that the s(n,j)s are correlated complex Gaussian  random variables with ⟨s(n,j)⟩ = ⟨s(n,j)s(m,k)⟩ = 0 and ⟨s(n,j)s∗  (m,k)⟩ = �d1  ν=1 e(ν)  (n,j)e(ν) ∗  (m,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (51) and S ∼ Sxinst(·)  inst  (U → +∞), it is clear that the realizations of S most likely  to produce a large value of U are less random than the unconditioned field S itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Besides  the (non-random) ridge path xinst(·), it only takes the 2d1 real Gaussian random variables  Re(cν) and Im(cν) to characterize these realizations entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For instance, if µ1[xinst(·)] is  not degenerate (d1 = 1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (51) and S ∼ Sxinst(·)  inst  yield  S(x, z)  ∥S∥2  ∼ Aeiθ  � L  0  C(x − xinst(z′), z, z′) φ1(z′) dz′  (U → +∞),  (54)  19  where ∥ · ∥2 denotes the L2-norm over Λ × [0, L], A is a positive constant, and θ is a  random phase uniformly distributed over [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (54) it follows immediately that  |S(x, z)|/∥S∥2 is non-random, which means that the profile of S(x, z) is purely deterministic  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' This result generalizes the diffraction-free deterministic profile of S in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (4)  when diffraction is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Typical shape of Sxinst(·)  inst  Although the Fourier representation of Sxinst(·)  inst  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) is very useful to deal with  technical points like in Appendix A, it is far from clear as to the structure of Sxinst(·)  inst  in real  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' By contrast, it is easier to figure out the shape of Sxinst(·)  inst  (x, z) from the convolution  representation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (51), knowing the correlation function C(x − x′, z, z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As a simple  illustration, take, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', C(x−x′, z, z′) = f[(x−x′)/xc, (z−z′)/zc], where xc and zc respectively  denote transverse and axial correlation lengths, f(x, z) being negligibly small outside the  domain defined by both |x| ≤ 1 and |z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Assuming xc ≪ ℓ, zc ≪ L and a ‘gentle’  ridge path with | ˙xinst(z)| ≲ ℓ/L for all 0 ≤ z ≤ L, it is not difficult to show from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (51)  that Sxinst(·)  inst  lives within a thin tube, or filament, of radius ρ ≲ xc + ℓzc/L ≪ ℓ along the  path xinst(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' This is the reason for the name ‘single-filament instanton’ given to Sxinst(·)  inst  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Numerical simulations confirm the elongated profile of Sxinst(·)  inst  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 1 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Tail of p(U)  Write Ninst the number of single-filament instantons (Ninst is the number of paths maxi-  mizing µ1[x(·)], which is finite by assumption (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As mentioned below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49), we consider  cases where the fundamental eigenspaces of M[x(·)] for all the different paths maximizing  µ1[x(·)] are essentially disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It means that the instantons — that are all single-filament  instantons — are mutually exclusive realizations of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  As a result, the total instanton  contribution to the tail of p(U) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (29) is the sum of the contributions of the Ninst single-  filament instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Let π(i)(U) denotes the contribution of the ith single-filament instanton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  It will be seen below that the leading term of ln π(i)(U) in the large U limit does not depend  on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Writing ln f(U) this term and π(i)(U) = f(U)A(i)(U) with ln A(i)(U) = o[ln f(U)] as  20  U → +∞, one has  ln p(U) ∼ ln  Ninst  �  i=1  π(i)(U) = ln  Ninst  �  i=1  f(U)A(i)(U)  = ln f(U) + ln  Ninst  �  i=1  A(i)(U)  (55)  = ln f(U) + o[ln f(U)]  (U → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Thus, at leading order, ln p(U) ∼ ln f(U) where ln f(U) is the leading term of ln π(i)(U) for  all 1 ≤ i ≤ Ninst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Let [LT](U→+∞)(·) denote the leading term of the asymptotic expansion  of (·) as U → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Picking a i and integrating out the fluctuations of the fields around the  corresponding ith single-filament instanton (with ridge-path xinst(·)) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (29), one obtains  ln f(U) = [LT](U→+∞) ln  �  δ(U − |ϕxinst(·)  inst  (0, L)|2)  �  Sxinst(·)  inst  (56)  = [LT](U→+∞) ln  ��  · ·  �  δ  �  U − |ϕxinst(·)  inst  (0, L)|2�  d1  �  ν=1  exp  �  − |cν|2  µmax  �  d2cν  πµmax  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  To go further we need the behavior of |ϕxinst(·)  inst  (0, L)|2 as a function of the cνs in the large  U limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' First, we make the change of variables η = �d1  ν=1 |cν|2 and cν = aν√η, where η is  a gamma-distributed random variable with p(η) = [Γ(d1)µmax]−1(η/µmax)d1−1e−η/µmax, and  the aνs define a random 2d1-dimensional (real) unit vector a with coordinates Re(aν) and  Im(aν) the direction of which is uniformly distributed over the unit (2d1 − 1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For  finite L and ℓ, a large U implies a large η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Writing  |ϕxinst(·)  inst  (0, L)|2 = A(η, a) e2gη,  (57)  without loss of generality, on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (56) (with variables η and a), one  obtains  ln f(U) = [LT](U→+∞) ln  �� +∞  0  δ  �  U − A(η, a) e2gη�  p(η) dη  �  a  = [LT](U→+∞) ln  �� +∞  0  δ (η − η(U, a))  |2g + ∂η ln A(η, a)| U p(η) dη  �  a  (58)  = [LT](U→+∞) ln  K(d1)  U 1+1/2µmaxg  �η(U, a) A(η(U, a), a)1/2µmaxg  |2g + ∂η ln A(η(U, a), a)|  �  a  where ⟨·⟩a denotes the average over the direction of a, K(d1) = [Γ(d1)µd1  max]−1, and η(U, a)  is the solution to A(η, a) e2gη = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It is shown in Appendix C that  limη→+∞  1  η ln A(η, a) = 0,  limη→+∞ ∂η ln A(η, a) = 0,  (59)  21  for every direction of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' If d1 = 1, A(η, a) ≡ A(η) does not depend on a and the limits  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (59) are trivially uniform with respect to the direction of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' If d1 > 1, it is not  unreasonable to expect that uniform convergence also applies to non-pathological instantons,  like in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (51) (with variables η and a), where no particular direction of a stands out  significantly from the others, which could jeopardize uniform convergence in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We  therefore assume that the limits in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (59) are uniform in a also for d1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Proving this  conjecture is another problem that we are unable to solve at the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Under this  assumption, it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (59) that at leading order in the large U limit, η(U, a) ∼  (2g)−1 ln U and |2g + ∂η ln A(η(U, a), a)| ∼ 2g uniformly in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Uniform asymptotics makes  it possible to interchange asymptotics and average over the direction of a on the right-hand  side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (58), which yields  ln f(U) = [LT](U→+∞) ln  �K(d1)  4g2  ln U  U 1+1/2µmaxg ⟨A((2g)−1 ln U, a)1/2µmaxg⟩a  �  = −  �  1 +  1  2µmaxg  �  ln U,  (60)  where we have used ln⟨A((2g)−1 ln U, a)1/2µmaxg⟩a = o(ln U) (see the end of appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Note that the expression of ln f(U) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (60) is independent of i, as announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (60) on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (55), one finally obtains the tail of p(U) as  ln p(U) = −  �  1 +  1  2µmaxg  �  ln U + o(ln U)  (U → +∞),  (61)  from which it follows that p(U) has a leading algebraic tail ∝ U −ζ modulated by a slow  varying amplitude (slower than algebraic) with exponent ζ = (1 + 1/2µmaxg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Injecting this  result into ⟨U⟩ =  � +∞  1  Up(U) dU, one finds that the critical coupling in the case of single-  filament instantons is given by gc(L) = 1/2µmax (where µmax depends on L), in agreement  with the general result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [5] recalled in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  NUMERICAL RESULTS  In this section, we report on numerical simulations we have performed to test our ana-  lytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' First of all, we emphasize on the fact that all our predictions are asymptotic  results the convergence of which is notoriously slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The very large values of U correspond-  ing to the instanton dominated regime are too rarely sampled for our analytical results to  be observed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' What can be observed, however, as a good indication of their  22  validity is the emergence of a statistical bias of S towards the instanton as the amplification  increases within the sampled range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  For definiteness, we have taken  S(x, z) =  50  �  n=−50  sn  √ςn exp i  �  2πn  ℓ x +  �2πn  ℓ  �2 z  2  �  ,  (62)  where the sns are complex Gaussian random variables with ⟨sn⟩ = ⟨snsm⟩ = 0 and ⟨sns∗  m⟩ =  δnm, and the spectral density ςn is normalized to �50  n=−50 ςn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Equation (62) — which is  of the form (40) in which the sum over j reduces to j = 1 and ςn = σ(n,1)/Lℓ — is reminiscent  of models of spatially smoothed laser beams [6], where S is a solution to the paraxial wave  equation  ∂zS(x, z) + i  2∂2  x2S(x, z) = 0,  (63)  here with boundary condition S(x, 0) = �50  n=−50 sn√ςn exp(2iπnx/ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' To ensure that the  space average ℓ−1 �  Λ S(x, z) dx is zero for all z and every realization of S, as expected for  the electric field of a smoothed laser beam, the mode at n = 0 is excluded by taking ς0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Here we show the results for the Gaussian spectrum  ςn̸=0 ∝ exp  �  −  �πn  ℓ  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (64)  (Other widely used spectra, like top-hat and Cauchy spectra, give similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=')  For each realization of S on a cylinder of length L = 10 and circumference ℓ = 20, we have  solved Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1) by using a symmetrized z-split method [28] which propagates the diffraction  term, (i/2m)∂2  x2ψ(x, z), in Fourier space and the amplification term, g|S(x, z)|2ψ(x, z), in  real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We have taken m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7, and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  To get a better statistics of large  amplification values, we have considered Umax = |ψ(xmax, L)|2 instead of U = |ψ(0, L)|2,  where xmax is the value of x maximizing |ψ(x, L)|2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', the location of the highest peak  of |ψ(x, L)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As Umax increases, S concentrates onto the leading instanton(s) arriving at  x(L) ≃ xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  By statistical invariance under x-translation, the leading instanton(s) arriving at x(L) = y  is simply given by Sy  inst(x, z) = Sinst(x − y, z), where Sinst(x, z) is the leading instanton(s)  arriving at x(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We have determined Sinst from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49), (62), and (64), with numer-  ically computed eigenvector(s) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We have found a unique single-filament instanton with  xinst(·) ≡ 0, µmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='34984 and d1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The critical coupling is gc(L) = 1/(2µmax) ≃  23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='11495 and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='5 is in the above critical regime with g/gc(L) ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='35 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Figure 1 shows  the contour plots of |Sinst|2 and ‘hot spot profile’ |C|2 [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  0  2  4  6  8  10  10  5  0  5  10  z  x  (a)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='5  1  4  2  0  2  4  10  5  0  5  10  z - L/2  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 1: (a) Contour plot of |Sinst(x, z)|2 normalized to maxΛ×[0,L] |Sinst(x, z)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (b) Contour  plot of the ‘hot spot profile’ |C(x, z)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Define ˆSy  inst = Sy  inst/∥Sy  inst∥2,Λ×[0,L] and ˆS = S/∥S∥2,Λ×[0,L] where ∥ · ∥2,Λ×[0,L] is the L2-  norm on Λ × [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Write Sy  ∥ =  �  ˆSy  inst, ˆS  �  ˆSy  inst the component of ˆS along Sy  inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We have  measured the difference between S and the instanton through the minimized L2-distance  D ≡ d2  �  ˆS, Symin  ∥  �  = min  y∈Λ ∥ ˆS − Sy  ∥∥2,Λ×[0,L]  =  �  1 − max  y∈Λ |( ˆS, ˆSy  inst)|2,  (65)  where ymin is the value of y minimizing ∥ ˆS − Sy  ∥∥2,Λ×[0,L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The smaller D, the closer S to the  instanton arriving at x(L) = ymin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The fact that ymin can be different from xmax is due to  the fluctuations of S away from the instanton, the relative amplitude of which is measured  by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For D smaller than average, ymin ≃ xmax with a relatively small dispersion of the  data points about ymin = xmax, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Using the Fourier representations  (62) for both S and Sy  Inst on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (65), one gets  D =  �  1 − maxy∈Λ | �  n ςnˆsnˆs∗  ne2iπny/ℓ|2  (�  n ςn|sn|2) (�  n ςn|sn|2) ,  (66)  which is the expression we have used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  24  We drew 105 independent realizations of S denoted in the following by {S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Figure 2  shows the probability distribution of D estimated from {S} and the realizations in {S} with  Umax above the 90th and 99th percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The last two are conditional probabilities knowing  that Umax ≥ 5 1012 and Umax ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='6 1017, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' One can see a clear tendency of D  to decrease with increasing Umax: the subsamples of {S} conditioned on a large Umax are  statistically biased toward the instanton compared with the unconditioned sample {S} itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  This numerical result for large but finite Umax is consistent with the predicted concentration  of S onto the instanton for U → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  p (\uf78d)  (a)  (b)  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='0  0  2  4  6  8  \uf78d  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 2: Probability distribution of D estimated from (a) {S}, (b) the realizations in {S} with Umax  above the 90th percentile, and (c) the realizations in {S} with Umax above the 99th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  p (log10Umax)  (a)  (b)  (c)  8  18  28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='14  log10Umax  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 3: Probability distribution of log10 Umax estimated from (a) {S}, (b) {S}10%, and (c) {S}1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  25  To study this bias phenomenon in more detail, we have used the two samples {S}10%  and {S}1% respectively defined as the realizations in {S} with D below the 10th and 1st  percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  These samples correspond to D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='8 for {S}10% and D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7 for {S}1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Figure 3 shows the probability distribution of log10 Umax estimated from (a) {S}, (b) {S}10%,  and (c) {S}1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The last two are conditional probabilities knowing that D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='8 and D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In this figure, the statistical bias of S, already observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 2, appears as  the clear tendency of Umax to increase with decreasing D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The concentration of S onto the instanton implies that for all ε > 0 and 0 < a < b,  one has lima→+∞ Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (D ≤ ε| a ≤ Umax < b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Thus, for a large enough it is not  unreasonable to expect Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (D ≤ ε| a ≤ Umax < b) to increase with increasing a, which  should be possible to check numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (D ≤ ε| a ≤ Umax < b) can be estimated by  the percentage of realizations of S with D ≤ ε among the realizations with a ≤ Umax < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In  Figure 4 we show the results for ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='8 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', S in {S}10% and {S}1%, respectively),  a = 10n, and b = 10n+2, with n an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It can be seen that both curves increase with  increasing Umax, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', that 30% of the realizations with Umax ≃ 1028 are  in {S}1% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', have D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7), when {S}1% represents only 1% of all the realizations in {S}:  the emergence of a statistical bias of S with increasing amplification is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  relative histogram of Umax  108  1018  1028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='0  {S}10 %  {S}1 %  Umax  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 4: Percentage of realizations of S in {S}10% (orange, upper curve) and {S}1% (green, lower  curve) for Umax in [10n, 10n+2), with n varying from 0 to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As guides to the eyes, the solid lines  are nonlinear fits of the corresponding data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Dashed lines are continuations of these fits to  higher Umax (disregarding the data points in this domain)  26  Figure 5 shows a scatter plot of xmax and ymin for the 103 realizations in {S} with the  largest Umax (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', Umax ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='6 1017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Red circles and gray squares correspond to realizations  with D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7 and D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7, respectively (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', realizations in {S}1% and {S} \\ {S}1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It can  be checked that the dispersion of the data points about ymin = xmax is indeed smaller for  smaller D, as announced below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note that due to the periodic boundary condition  in Λ, the distance between xmax and ymin is min(|xmax −ymin|, ℓ−|xmax −ymin|) and the data  points in the left-upper and right-lower corners are actually close to the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='ymin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='■  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
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+page_content='■ ●● 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
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+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='xmax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 5: Scatter plot of xmax and ymin for the realizations in {S} with Umax ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='6 1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Red circles  and gray squares are realizations in {S}1% (D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7) and {S} \\ {S}1% (D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We have compared the realizations in {S}1% and {S}\\{S}1% near the edge of the sampled  domain of Umax, where {S}1% becomes statistically significant according to the results in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We have considered realizations with Umax ≥ 1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' There are 15 such realizations in  {S} among which 5 in {S}1% (D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7) and 10 in {S} \\ {S}1% (D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In Figures 6 and  7 we show two pairs of typical realizations picked in {S}1% and {S} \\ {S}1%, respectively  (technical details are given in the captions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For each realization, the theoretical instanton  arriving at x(L) = ymin is indicated by a dashed contour, solution to |Sinst(x−ymin, z)|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='75  with |Sinst|2 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Intense localized hot spots similar to the theoretical one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 1(b)  are clearly visible in both figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6 (D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7), hot spots occur inside the dashed  line, in the instanton region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note also that the level of |S(x, z)|2 is significantly higher than  average throughout the instanton region (∼ 6, while ⟨|S(x, z)|2⟩ = 1), which seems difficult  to explain by generic fluctuations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' independent, small-scale hot spots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' On the other  hand, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 7 (D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7), hot spots occur anywhere and the levels of |S(x, z)|2 inside and  27  outside the instanton region are quite comparable (hot spots excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  0  2  4  6  8  10  10  5  0  5  10  z  x - xmax  (a)  fraction of max  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='5  1  0  2  4  6  8  10  10  5  0  5  10  z  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6: Contour plots of |S(x, z)|2 for two realizations in {S}1% (D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7) with Umax ≥ 1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The dashed contours indicate the theoretical instanton arriving at x(L) = ymin for the considered  realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (a): Umax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='3 1029, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='68, and maxΛ×[0,L] |S(x, z)|2 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (b): Umax =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='15 1027, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='537, and maxΛ×[0,L] |S(x, z)|2 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  0  2  4  6  8  10  10  5  0  5  10  z  x - xmax  (a)  fraction of max  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='5  1  0  2  4  6  8  10  10  5  0  5  10  z  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 7: Plots similar to the ones shown in Figure 6 for two realizations in {S} \\ {S}1% (D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (a): Umax = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='92 1029, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='8, and maxΛ×[0,L] |S(x, z)|2 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (b): Umax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='95 1027, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='82,  and maxΛ×[0,L] |S(x, z)|2 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  28  The robustness of these observations from one realization to the other can be tested  through the sample mean of |S(x + xmax, z)|2 in which the realizations are translated to  align the maxima of |ψ(x, L)|2 with each other at the same position (here, x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  0  2  4  6  8  10  10  5  0  5  10  z  x  (a)  fraction of max  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='5  1  0  2  4  6  8  10  10  5  0  5  10  z  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 8: Contour plots of the sample mean of |S(x + xmax, z)|2 for the realizations in (a) {S}1% and  (b) {S} \\ {S}1%, with Umax ≥ 1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The dashed contours indicate the region where the sample  mean of |Sinst(x + xmax − ymin, z)|2 is greater than 75% of its maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  In Figure 8, we show the results for the same 15 realizations with Umax ≥ 1027 as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The region of the sample mean of |Sinst(x+xmax−ymin, z)|2 is indicated by a dashed contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The red spot seen in both figures (a) and (b) at x = 0 and z ≃ L is a cumulative effect  reflecting the tendency of |S(x, z)|2 to have a bump — not necessarily high — right behind  the maximum of |ψ(x, L)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Such recurring bumps can be observed in the four realizations  shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6 and 7: plots are never blue at x = xmax and z ≃ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Figure 8(a) shows the  result for the realizations in {S}1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In substance, it confirms the observations already made  about the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' namely, the observed level of |S(x, z)|2 inside the dashed contour is the  superposition of an average elevation of the level (the emerging instanton) and fluctuations of  comparable amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The presence of such an average elevation inside the dashed contour  results in a larger probability that high maxima of |S(x, z)|2 occur inside the instanton  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It is a pure statistical effect similar to the well known enhancement of correlations  of peaks in Gaussian fields [29, 30], the large-scale instanton playing the same role as the  ‘signal’ and ‘background field’ in [29] and [30], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As a consequence, one observes  29  (i) a tendency for the hot spots to cluster in the instanton region and (ii) a level of |S(x, z)|2  between the hot spots significantly higher than the average level outside the instanton region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  These two points (i) and (ii) are strong indications of the presence of the instanton in the  realizations of {S}1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Hot spot clustering is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 9 for the same realizations as  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' By contrast, no particular structure is observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 8(b) for the realizations  in {S} \\ {S}1% (except the red spot at x = 0 and z ≃ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It means that neither instanton  nor clustering of hot spots are significant in those realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  0  2  4  6  8  10  10  5  0  5  10  z  x - xmax  (a)  0  2  4  6  8  10-10  5  0  5  10  z  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 9: Positions of the local maxima of |S(x, z)|2 higher than 75% of the global maximum (blue  circles) for the same realizations as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' High maxima cluster in the instanton region indicated  by the dashed contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (See caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=')  Combining numerical results with analytical predictions, one can now infer what realiza-  tions of S(x, z) contribute to p(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As long as the value of U is in the bulk of p(U), an  overwhelming majority of the realizations of S(x, z) contributing to p(U) are generic real-  izations with hot spots uniformly scattered in Λ × [0, L] and D close to its typical value at  D ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The situation changes gradually as U increases into the tail of p(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Namely, the  larger U the larger the percentage of atypical realizations with D smaller than, say, its first  percentile — like the ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6 — to the detriment of generic realizations — like the ones  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In those atypical realizations, the hot spots cluster in the instanton region instead  of being uniformly scattered in Λ × [0, L] and the level of |S(x, z)|2 between the hot spots  remains abnormally high (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 6, 8(a), and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Letting U → +∞, the percentage of  atypical realizations goes up to 100% while D and the relative fluctuations-to-instanton am-  30  plitude decrease to zero with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In this limit, the tail of p(U) is asymptotically  dominated by the instanton which determines the the critical coupling gc(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  DISCUSSION AND PERSPECTIVES  In this paper, we have studied the large amplification limit of a linear amplifier driven by  the square of a Gaussian random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We have considered the same model as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [1]  and [5] in which the propagation is that of a free Schrödinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' By performing the  first instanton analysis of the corresponding MSR action, we have identified the realizations  of the Gaussian field most likely to produce a large amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We have found that  when U = |ψ(0, L)|2 gets large, for ψ solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1) with S defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' III B,  the realizations of S concentrate onto large-scale filamentary instantons running along the  path(s) maximizing the largest eigenvalue of the covariance operator defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  This result explains the otherwise mysterious presence of this maximized eigenvalue in the  expression of gc(L) found in [5] (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' We have then derived the tail of p(U) for large U  from the instanton contribution and checked that the resulting critical coupling does coincide  with the one in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' From this analysis, it follows in particular that the realizations of  S causing the divergence of ⟨|ψ(0, L)|2⟩ for g > gc(L) are long filamentary structures (the  instantons) rather than localized hot-spots, as assumed in hot-spot models [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' This result  extends the conclusions of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [10] to the full problem (1) with diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Our analytical predictions are supported by numerical simulations that clearly show a  statistical bias of S towards the instanton, as U increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The larger U in the sampled  range, the larger the fraction of atypical realizations of S in which a large-scale instanton  coexists with fluctuation induced localized hot spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (See [10] for a quantitative comparison  of hot spot and instanton contributions to the amplification in the diffraction-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=') In  those atypical realizations, hot spots are not uniformly distributed in Λ × [0, L] but tend  to cluster in the instanton region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For the parameters used in our simulations we failed to  sample values of U large enough that the fluctuations of S away from the instanton could be  neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Hot spot clustering and nonlinear evolution of the coupled hot spots/instanton  system are interesting subjects that would deserve to be dealt with in more depth, especially  in laser-plasma interaction physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The work presented here is only a first step toward a comprehensive study of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1) in  31  the large amplification limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' There are various directions along which investigations could  be pushed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Obviously, trying to lift all or part of the assumptions made in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' III B  appears as a natural next step, especially the technical restriction (ii), as well as proving the  conjecture that the limits in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (59) are uniform also for degenerate µmax (d1 > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The  second natural next step is to investigate the possibility of multi-filament instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Another challenging line of research is the study of a possible intermittency of |ψ(x, L)|2  and its connection with our results, as we will now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Typically, a unique realization  of S is available in a given experimental environment, like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', in a laser-plasma interaction  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' In this case, ⟨|ψ(0, L)|2⟩ is replaced with the space average |Λ|−1 �  Λ |ψ(x, L)|2 dx  for a generic realization of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' As rare events, instantons are very unlikely to contribute to  the latter quantity unless |Λ| is large and the space average is dominated by the contribution  of scarce, intense peaks of |ψ(x, L)|2 the high amplitude of which outbalances their scarcity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  The question is then whether such a peak-dominated behavior — called ‘intermittency’ in the  literature on random media [31] — can be observed in the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (1) for g > gc(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  If so, our results imply that S(x, z) in the region of Λ×[0, L] upstream from a dominant peak  of |ψ(x, L)|2 is a filament instanton arriving at the peak location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Intermittency of |ψ(x, L)|2  is thus important as connecting our instanton analysis approach with experimental results in  the above critical regime g > gc(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' The interested reader will find a detailed introduction  to intermittency in random media in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Finally, it would also be interesting to investigate the small m behavior of the same  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' For m → 0, it can be shown that ψ(x, z) reduces to  ψ(x, z) = exp  � g  |Λ| ∥S∥2  2,Λ×[0,L]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (67)  Thus, in this limit, the realizations of S giving rise to a large U are the ones with a large  L2-norm, which are known to concentrate onto the fundamental eigenspace of the covariance  operator TC defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (28), as U → +∞ [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' If S is given by the random Fourier  sum (40) with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=', σ(n,j) < σ(0,0) for all (n, j) ̸= (0, 0), the fundamental eigenspace of TC  reduces to the functions ∝ Φ(0,0)(z) independent of x, and the realizations of S in the large  U limit are completely delocalized in Λ, in striking contrast to the filamentary instantons we  have found for a fixed m ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' This simple example indicates that the two limits U → +∞  and m → 0 do not commute, which raises the natural question of how precisely the crossover  occurs between the limits ‘U → +∞ then m → 0’ and ‘m → 0 then U → +∞’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Answering  32  this question will elucidate the intriguing transition suggested by the above example, from  filamentary to delocalized instantons, as m goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  In conclusion, it may be noted that the number of highly non-trivial questions raised  by the seemingly simple linear problem (1) is quite remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Following on from the  work presented here, we hope that those questions will motivate interesting research in both  statistical physics and laser-matter interaction physics where the linear amplifier model (1)  first appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Acknowledgments  The author warmly thanks Satya N Majumdar, Denis Pesme, and Grégory Schehr for  their interest and valuable advice about the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' He also thanks Harvey A Rose and  Joel L Lebowitz for the inspiring discussions he had with them on related subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Appendix A: Paths maximizing µ1[x(·)] and ridge paths of |Sinst(x, z)|2  In this appendix we show that xinst(·) is a ridge path of |Sxinst(·)  inst  (x, z)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (42)  and (49), one gets  � L  0  |Sxinst(·)  inst  (xinst(z), z)|2 dz = s†M[xinst(·)]s = µmax∥s∥2,  (A1)  where s is in the fundamental eigenspace of M[xinst(·)], and  � L  0  |Sxinst(·)  inst  (x(z), z)|2 dz = s†M[x(·)]s ≤ µ1[x(·)]∥s∥2  = µ1[x(·)]  µmax  � L  0  |Sxinst(·)  inst  (xinst(z), z)|2 dz ≤  � L  0  |Sxinst(·)  inst  (xinst(z), z)|2 dz,  (A2)  yielding  sup  x(·)∈B(0,L)  � L  0  |Sxinst(·)  inst  (x(z), z)|2 dz =  � L  0  |Sxinst(·)  inst  (xinst(z), z)|2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (A3)  Equation (A3) means that in the path-integral for ϕxinst(·)  inst  (0, L), xinst(·) is a path along which  the amplification is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Now, assume that there is A ⊂ [0, L] with |A| ≡  � L  0 1z∈Adz >  0 such that for all z ∈ A, there is x ∈ Λ with |Sxinst(·)  inst  (x, z)|2 > |Sxinst(·)  inst  (xinst(z), z)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It follows  immediately that  � L  0  sup  x∈Λ  |Sxinst(·)  inst  (x, z)|2dz >  � L  0  |Sxinst(·)  inst  (xinst(z), z)|2 dz,  (A4)  33  and from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (A3) and (A4) one should have  � L  0  sup  x∈Λ  |Sxinst(·)  inst  (x, z)|2dz >  sup  x(·)∈B(0,L)  � L  0  |Sxinst(·)  inst  (x(z), z)|2 dz,  (A5)  in contradiction with the Lemma A1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' [5] according to which one must have an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Thus, there is no such A and since every given realization of Sxinst(·)  inst  (x, z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) is  a continuous function of x and z, one has |Sxinst(·)  inst  (x, z)|2 ≤ |Sxinst(·)  inst  (xinst(z), z)|2 for all  0 ≤ z ≤ L and x ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' This proves that for all the realizations of Sxinst(·)  inst  (x, z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49)  with s in the fundamental eigenspace of M[xinst(·)], xinst(·) is a ridge path of |Sxinst(·)  inst  (x, z)|2  along which the amplification is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Assume that there is a s in the fundamental eigenspace of M[xinst(·)] and yinst(·) ∈  B(0, L) with yinst(·) ̸= xinst(·) such that yinst(·) is also a ridge path of |Sxinst(·)  inst  (x, z)|2 along  which the amplification is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Then, yinst(·) maximizes µ1[x(·)] and s belongs to the  fundamental eigenspace of M[yinst(·)] (otherwise, the amplification along yinst(·) would be  less than along xinst(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Since s belongs to the fundamental eigenspaces of both M[xinst(·)]  and M[yinst(·)], their intersection is necessarily non trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' It shows that the number of  ridge paths depends on the relative structure of the fundamental eigenspaces of M[x(·)] for  the different paths maximizing µ1[x(·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' If the fundamental eigenspaces of M[x(·)] for all  the paths maximizing µ1[x(·)] are essentially disjoint, s cannot belong to more than one  fundamental eigenspace and each realization of the instanton has only one ridge path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' This  is the case considered in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' On the other hand, if the fundamental eigenspaces of  M[x(·)] for different paths maximizing µ1[x(·)] have a non trivial intersection, then for all  the realizations with s in the intersection, the instanton has more than one ridge path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' This  case corresponds to multi-filament instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Appendix B: Equivalence of the Fourier and convolution representations of Sinst  In this appendix, we prove the equivalence of the expressions of Sxinst(·)  inst  (x, z) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49)  and (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Permuting the sum and the integral on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (51) and using  the Fourier decomposition (41) for C(x − xinst(z′), z, z′), one readily finds that the equation  (51) can be rewritten as  Sxinst(·)  inst  (x, z) =  d1  �  ν=1  cνΩν(x, z),  (B1)  34  with  Ων(x, z) =  �  (n,j)∈I  e(ν)  (n,j)  � σ(n,j)  ℓ µmax  e2iπnx/ℓΦ(n,j)(z),  (B2)  where e(ν) is a vector defined by its components  e(ν)  (n,j) =  � σ(n,j)  ℓ µmax  � L  0  e−2iπnxinst(z′)/ℓΦ(n,j)(z′)∗φν(z′) dz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (B3)  Showing that the equation (49) can also be written in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (B1) requires a little  more work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (41), (42), and (B3) it can be checked that  �  M[xinst(·)]e(ν)�  (n,j) =  �  (m,k)∈I  M(n,j)(m,k)[xinst(·)] e(ν)  (m,k)  =  � σ(n,j)  ℓ µmax  � L  0  e−2iπnxinst(z)/ℓΦ(n,j)(z)∗ ⟨z|Txinst(·)|φν⟩ dz  = µmax e(ν)  (n,j),  (B4)  and  e(µ) ∗ · e(ν) =  �  (n,j)∈I  e(µ) ∗  (n,j)e(ν)  (n,j) =  1  µmax  ⟨φµ|Txinst(·)|φν⟩  = ⟨φµ|φν⟩ = δµν,  (B5)  which means that {e(1), · · · , e(d1)} is an orthonormal basis of the fundamental eigenspace of  M[xinst(·)] on which s can be decomposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Writing  s(n,j) =  1  √µmax  d1  �  ν=1  cνe(ν)  (n,j) with cν = √µmax  �  (n,j)∈I  s(n,j)e(ν) ∗  (n,j),  (B6)  on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) and permuting the sums over (n, j) and ν, one obtains the  same equation (B1), as expected, which proves the equivalence of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (49) and (51) with  s(n,j) and cν related to each other by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  Appendix C: Limit of η−1 ln A(η, a) and ∂η ln A(η, a) as η → +∞  In this appendix we derive the two limits in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We will use the convolution  representation (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' To make the dependence of Sxinst(·)  inst  on c explicit we write Sxinst(·)  inst  (x, z) ≡  Sxinst(·)  inst  (x, z, c) = √η Sxinst(·)  inst  (x, z, a), with η = �d1  ν=1 |cν|2 and cν = aν√η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Deriving the  Feynman-Kac path-integral representation of ϕxinst(·)  inst  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  ϕxinst(·)  inst  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L) =  �  x(·)∈B(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='L)  e  � L  0  �  im  2 ˙x(τ)2+gη|Sxinst(·)  inst  (x(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='a)|2�  dτDx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (C1)  35  with respect to η for fixed a and using the fact that  � L  0 |Sxinst(·)  inst  (xinst(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' a)|2dτ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' one  gets  ∂ηϕxinst(·)  inst  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L) = g  �  x(·)∈B(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='L)  �� L  0  |Sxinst(·)  inst  (x(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' a)|2dτ  �  × e  � L  0  �  im  2 ˙x(τ)2+gη|Sxinst(·)  inst  (x(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='a)|2�  dτDx  ∼ g  �  x(·)∈B(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='L)  �� L  0  |Sxinst(·)  inst  (xinst(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' a)|2dτ  �  × e  � L  0  �  im  2 ˙x(τ)2+gη|Sxinst(·)  inst  (x(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='a)|2�  dτDx  = g  �  x(·)∈B(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='L)  e  � L  0  �  im  2 ˙x(τ)2+gη|Sxinst(·)  inst  (x(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='a)|2�  dτDx  = gϕxinst(·)  inst  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)  (η → +∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (C2)  from which it follows that  ∂η ln |ϕxinst(·)  inst  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)|2 = 2Re  �  ∂ηϕxinst(·)  inst  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)  ϕxinst(·)  inst  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' L)  �  ∼ 2g  (η → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (C3)  Thus, for all ε > 0 there is η0(ε, a) > 0 such that for every η ≥ η0(ε, a),  2g(1 − ε) ≤ ∂η ln |ϕxinst(·)  inst  (0, L)|2 ≤ 2g(1 + ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  (C4)  Writing |ϕxinst(·)  inst  (0, L)|2 = A(η, a) e2gη in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (C4), one obtains  −2gε ≤ ∂η ln A(η, a) ≤ 2gε,  (C5)  for every η ≥ η0(ε, a), and since ε can be taken arbitrarily small, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (C5) reduces to  lim  η→+∞ ∂η ln A(η, a) = 0,  (C6)  which is the second limit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' To get the first limit, we integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (C5) from  η0(ε, a) to any η > η0(ε, a), which yields  −2gεη + K+(ε, a) ≤ ln A(η, a) ≤ 2gεη + K−(ε, a),  (C7)  with K±(ε, a) = ln A(η0(ε, a), a) ± 2gεη0(ε, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Note that ln A(η0(ε, a), a) exists, otherwise  ln A(η, a) would have a vertical asymptote at η = η0(ε, a), in contradiction with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (C5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  It remains to divide Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (C7) by η:  −2gε − |K+(ε, a)|  η  ≤ 1  η ln A(η, a) ≤ 2gε + |K−(ε, a)|  η  ,  (C8)  36  where we have used K+(ε, a) ≥ −|K+(ε, a)| and K−(ε, a) ≤ |K−(ε, a)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Now, for η large  enough, namely η ≥ max[η0(ε, a), ε−1|K+(ε, a)|, ε−1|K−(ε, a)|], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (C8) gives  −3gε ≤ 1  η ln A(η, a) ≤ 3gε,  (C9)  and since ε can be taken arbitrarily small, one finally obtains  lim  η→+∞  1  η ln A(η, a) = 0,  (C10)  which is the first limit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content='  We now prove the relation ln⟨A((2g)−1 ln U, a)1/2µmaxg⟩a = o(ln U) used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Mak-  ing the conjecture that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (C10) holds uniformly with respect to the direction of a (see the  discussion below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (59)), one finds that for all ε > 0 there is η0(ε) > 0 independent of a  such that for every η ≥ η0(ε),  exp(−εη) ≤ A(η, a) ≤ exp(εη),  (C11)  whence,  exp  �  −  εη  2µmaxg  �  ≤ A(η, a)1/2µmaxg ≤ exp  �  εη  2µmaxg  �  ,  (C12)  and, averaging over a then taking the logarithm,  −  εη  2µmaxg ≤ ln⟨A(η, a)1/2µmaxg⟩a ≤  εη  2µmaxg,  (C13)  for every η ≥ η0(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Since ε can be taken arbitrarily small, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' (C13) reduces to  lim  η→+∞  1  η ln⟨A(η, a)1/2µmaxg⟩a = 0,  (C14)  which means that ln⟨A(η, a)1/2µmaxg⟩a = o(η), as announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
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+page_content=' (Letters) 284 L9  [30] Bardeen J M, Bond J R, Kaiser N and Szalay A S 1986 Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 304 15  [31] Molchanov S A 1991 Acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
+page_content=' 22 139 and references therein  38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf'}
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+Large voltage-induced magnetic anisotropy
+change in a few atomic layers of iron
+T. Maruyama1, Y. Shiota1, T. Nozaki1, K. Ohta1, N. Toda1, M. Mizuguchi1†, A. A. Tulapurkar1, T. Shinjo1,
+M. Shiraishi1, S. Mizukami2, Y. Ando3 and Y. Suzuki1*
+In the field of spintronics, researchers have manipulated mag-
+netization using spin-polarized currents1–3. Another option is
+to use a voltage-induced symmetry change in a ferromagnetic
+material to cause changes in magnetization or in magnetic
+anisotropy4–14. However, a significant improvement in efficiency
+is needed before this approach can be used in memory devices
+with ultralow power consumption. Here, we show that a
+relatively small electric field (less than 100 mV nm21) can
+cause a large change (�40%) in the magnetic anisotropy of
+a bcc Fe(001)/MgO(001) junction. The effect is tentatively
+attributed to the change in the relative occupation of 3d
+orbitals of Fe atoms adjacent to the MgO barrier. Simulations
+confirm that voltage-controlled magnetization switching in
+magnetic tunnel junctions is possible using the anisotropy
+change demonstrated here, which could be of use in the
+development of low-power logic devices and non-volatile
+memory cells.
+To develop voltage-driven spintronic devices, several areas of
+investigation have been suggested, including voltage control of mag-
+netic anisotropy4,7, ferromagnetism in ferromagnetic semiconduc-
+tors8,9,
+magnetoelectric
+switching
+of
+exchange
+bias10,11
+and
+anisotropy12, magnetoelectric interface effects5,6, multiferroic prop-
+erties13 and magnetostriction in a hybrid system with piezoelectric
+materials14. Most work to date has been carried out only at low
+temperatures, using GaMnAs or perovskite systems, or needed
+piezoelectric distortions, which may limit the endurance of any
+devices. Weisheit and colleagues7 observed up to 4.5% coercivity
+change in FePt(Pd) films with the application of voltage.
+However, they required the use of a liquid electrolyte to apply a
+high electric field at the surface. In this study, we have overcome
+these difficulties by using ultrathin Fe/MgO junctions. The system
+is built from all-solid-state and distortion-free materials that have
+controllable perpendicular surface anisotropy15–20.
+As shown in Fig. 1, the sample structure stack layers comprised a
+MgO substrate/MgO(10 nm)/Cr(10 nm)/Au(50 nm)/Fe(2–4 ML)/
+MgO(10 nm)/polyimide(1,500 nm)/ITO(100 nm) (ML, monatomic
+layer; see Methods). Because the influence of the electric field on
+the perpendicular anisotropy is effective only at the interface, the fer-
+romagnetic layer had to be composed of only a few monatomic layers.
+In addition, to control the perpendicular anisotropy, the film had to
+have a moderate crystalline and surface anisotropy in the absence of
+the bias voltage. In this regard, we used an ultrathin Fe layer for the
+anisotropy change. Because the Fe grew almost in a layer-by-layer
+mode onto the Au(001) surface at room temperature, we were able
+to precisely control the layer thickness21. The ultrathin epitaxial Fe
+layer deposited on the Au(001) buffer layer exhibited a transition in
+magnetic anisotropy from in-plane to perpendicular, depending on
+the film thickness20. Such features are ideal for the observation of
+the anisotropy change in response to the electric field, because we
+could systematically control the strength of the perpendicular aniso-
+tropy. The insulating layers comprised MgO and polyimide. MgO
+was used because it can be epitaxially grown onto an Fe(001)
+surface and exhibits a high breakdown voltage as a barrier material
+in magnetic tunnel junctions22. The polyimide layer was used to
+ensure a pinhole-free barrier over an extended area.
+Figure 2 shows representative magnetic hysteresis loops in a
+0.48-nm-thick Fe layer under the application of a bias voltage,
+obtained from Kerr ellipticity, hK, measurements. Under two differ-
+ent bias voltages, U ¼ þ200 and –200 V, a significant change in
+perpendicular anisotropy was observed. The perpendicular mag-
+netic anisotropy energy per unit volume of the film, Eperp, was cal-
+culated from the hK–H curve, assuming a linear relation between
+hK and magnetization:
+Eperp ¼ �m0
+Ms
+hs
+ðhs
+0
+HdhK;
+ð1Þ
+where m0, MS, H, hK and hs are permeability of free space, satur-
+ation magnetization, external magnetic field, Kerr ellipticity and sat-
+uration Kerr ellipticity, respectively. If the film possesses uniaxial
+crystalline anisotropy Ku, and surface anisotropy Ks, the Eperp is
+expressed as
+Eperpd ¼
+� 1
+2 m0M2
+s þ Ku
+�
+�
+d þ KS;MgO=Fe þ KS;Fe=Au þ DKSðVÞ
+ð2Þ
+where d is the film thickness. DKS(V) is a surface anisotropy,
+induced by application of a voltage. When a positive voltage
+U ¼ þ200 V
+was
+applied,
+then
+decreased
+to
+U ¼ –200 V,
+perpendicular anisotropy was induced and the magnetic anisotropy
+energy was changed from 231.3 to 213.7 kJ m23. The ratio of
+the magnetic anisotropy energy change, defined as DEperp/
+(2Eperp,ave) ¼ (Eperp,200V–Eperp,–200V)/(Eperp,200V þ Eperp,–200V), was
+39%. If we count this change as a change in the surface
+anisotropy energy, that is, DKs(V) in equation (2), it corresponds to
+8.4 mJ m22.
+To precisely measure the thickness dependence of the effect, we
+used a modulation technique (see Methods). The inset in Fig. 2
+shows the external field dependence of the dhK/dV signal that
+was obtained for the same sample. The dhK/dV signal displays a
+maximum value, (dhK/dV)max, where the hK hysteresis curves,
+obtained for positive (B) and negative (A) bias voltages, show a
+1Graduate School of Engineering Science, Osaka University, Toyonaka, Japan,
+2WPI Advanced Institute for Materials Research, Tohoku University, Sendai,
+Japan,
+3Graduate School of Engineering, Tohoku University, Sendai, Japan; †Present address: Institute for Materials Research, Tohoku University, Sendai,
+Japan; *e-mail: suzuki-y@mp.es.osaka-u.ac.jp
+LETTERS
+PUBLISHED ONLINE: 18 JANUARY 2009 | DOI: 10.1038/NNANO.2008.406
+NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.nature.com/naturenanotechnology
+158
+© 2009 Macmillan Publishers Limited.  All rights reserved. 
+ 
+
+maximum difference (C point in Fig. 2, inset). Figure 3a shows
+(dhK/dV)max as a function of film thickness. The effect was
+largest for an Fe film with a thickness of 0.48 nm, and was smaller
+for both thinner and thicker films. Because the influence of the elec-
+tric field is effective only at the metal/insulator interface, it is natural
+to observe a smaller effect for the thicker Fe films. Figure 3b shows
+the dependence of the saturation Kerr ellipticity, hs, on Fe thickness.
+The linear dependence of hs down to 0.25 nm (1.8 ML on average)
+proves that continuous films had been grown, even if they were only
+a few atomic layers in thickness. In addition, this behaviour also
+proves that the film was uniform across the thickness. In Fig. 3c,
+Eperpd is plotted as a function of Fe layer thickness d at zero bias
+voltage, together with a linear fit using equation (2). The fit indicates
+that Ks,MgO/Fe þ Ks,Fe/Au ¼ 580 mJ m22 and Ms ¼ 1.5 MA m21,
+neglecting a contribution from Ku. Ms is about 83% that of the
+bulk value. The reduction in the apparent Ms could be caused by
+a contribution from a positive Ku, produced by a lattice mismatch
+between Fe and Au (1.6%). The value of Ks,MgO/Fe þ Ks,Fe/Au
+observed here was a little higher than previous observations made
+on a Au/Fe(001) interface23, Ks,Fe/Au ¼ 470, 400, 540 mJ m22.
+This suggests that the MgO/Fe interface also has a positive contri-
+bution to Ks. The experimental data deviate from the linear fit line
+below 0.48 nm. This is well-documented behaviour for ultrathin
+films and may have many origins24. The thickness at which the
+linear fit, using equation (2), starts to deviate corresponds to the
+thickness where the maximum of (dhK/dV)max is obtained. One
+of the possible reasons for this is a deterioration of the film
+quality in this ultrathin thickness region. Clarification of the mech-
+anism of the deviation requires further investigation.
+One possible origin of the effect is in the influence of an electric
+field on electron filling of the Fe layer, which should affect the mag-
+netic anisotropy (see Supplementary Information). Kyuno and col-
+leagues pointed out that surface magnetic anisotropies in 3d
+ferromagnetic metal/noble metal interfaces were very sensitive to
+the electron filling of 3d orbitals25. In our case, from the capacitance
+of the junction, we estimate that we could change electron filling by
+2 � 1023 electrons per Fe surface atom by the application of 200 V.
+From the density of states, this corresponds to about 1 meV change
+in chemical potential (see Supplementary Information). This small
+change, however, may produce a non-negligible change in the
+surface anisotropy energy. From our experiment, we could change
+anisotropy energy by 4 meV per surface Fe atom. This magnitude
+of change can be reproduced from Kyuno’s calculation using
+1 meV change in the chemical potential. Kyuno also noted that
+the effect originates mainly from the large density of states (DOS)
+of a dxy and dx2�y2 character (jmzj ¼ 2) in the Fermi energy in the
+Fe/Au (001) system, in which Au has a large spin–orbit coup-
+ling26–29. In our case, because the Fe has two interfaces, with
+Au(001) and MgO(001), the situation is not completely the same,
+but a similar mechanism may occur. As schematically shown in
+Fig. 3d, the application of a negative voltage, for example, may
+cause an increase in the energy of the d3z2�r2 (mz ¼ 0) states,
+because of higher electron density at the barrier/Fe interface,
+leading to a reduction in the electron occupancy in those
+states. Therefore, the electron occupancy in the dxy and dx2�y2
+states could be changed relative to one another, leading to a modu-
+lation of the magnetic anisotropy. Further discussion requires first
+principles calculations.
+In ourexperiment, we needed to applya large voltage because ofthe
+thickness of the polyimide layer. The estimated voltage drop across the
+MgO layer, however, was �45 mV nm21 if we can neglect charge
+accumulation in the barrier (see Supplementary Information). As we
+know that more than 2 V can be applied to a 2-nm-thick MgO
+barrier, a much larger effect can be expected for conventional tunnel
+magnetoresistance junctions with a MgO barrier.
+In the latter part of this letter, we suggest a novel magnetization
+switching technique, using the voltage-induced magnetic anisotropy
+change explored in this work (see Supplementary Information).
+Figure 4 shows a result of macro-spin model simulation of voltage-
+controlled magnetization switching for a 0.48-nm Fe film. Here, we
+used parameters obtained from the above described experiments
+and an additional ferromagnetic resonance (FMR) experiment
+(see Fig. 4 caption). An external magnetic field of 8 kA m21 was
+applied normal to the film plane to tilt the magnetization towards
+the perpendicular direction. Initially, the bias voltage was held off
+(point A in Fig. 4). If we then apply a bias voltage with a slow
+rise time, the perpendicular anisotropy field changes and the mag-
+netization changes its direction to point B. However, if the rise time
+of the pulse is short enough (less than 1 ns), a dynamic precession
+and switching to another energetically stable point is achieved
+(point C in Fig. 4). When the voltage pulse is switched off with a
+slow fall time, the magnetization stabilizes at point D (Fig. 4) after
+the relaxation process. This simulation clearly shows that if we
+MgO (001) substrate
+MgO (10 nm)
+Cr (10 nm)
+Au (50 nm)
++
+−
+Bottom electrode (Au)
+Top view
+20 mm
+ITO (1 mm) × 24 
+Fe (2−4 ML)
+MgO (10 nm)
+Polyimide (1,500 nm)
+ITO (100 nm)
+Magnetic field
+Figure 1 | Schematic of the sample used for a voltage-induced magnetic
+anisotropy change. a, A positive voltage is defined as a positive voltage on
+the top electrode with respect to the bottom electrode. A perpendicular
+magnetic anisotropy was induced by a negative voltage. The magnetic field
+was applied perpendicular to the film plane for Kerr ellipticity
+measurements. b, We fabricated the wedge-shaped Fe layer, incorporating
+24 samples on the substrate, to investigate the dependence of the effect on
+Fe thickness.
+−1,000
+−500
+0
+500
+1,000
+−200 V
+200 V
+−1,000
+0
+1,000
+B
+A
+Fe
+A
+B
+C
+C
+Kerr ellipticity, η
+K (a.u.)
+Magnetic field (Oe)
+Magnetic field (Oe)
+dη
+K/dV (a.u.)
+Figure 2 | Magneto-optical Kerr ellipticity hk for different applied voltages
+as a function of applied field. The thickness of the Fe film was 0.48 nm.
+A significant change in the hysteresis curve indicated a large change in
+perpendicular anisotropy following application of the bias voltage. The right
+inset shows the voltage modulation response of the Kerr ellipticity, dhK/dV.
+The left inset illustrates the magnetization direction at points A and B in the
+hysteresis curves.
+NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2008.406
+LETTERS
+NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.nature.com/naturenanotechnology
+159
+© 2009 Macmillan Publishers Limited.  All rights reserved. 
+ 
+
+fabricate a Fe/MgO/Fe/Au junction with radio-frequency signal
+access, successful magnetization switching by the application of
+high-speed, low-voltage pulse should be possible. We believe that
+this novel magnetization switching will be demonstrated in the
+near future and, by combining it with magnetoresistive structures,
+could prove to be a highly successful technique.
+Our results show that it is possible to control magnetic aniso-
+tropy by the application of an electric field in Fe/MgO junctions,
+which can be combined with high-quality Fe/MgO/Fe magnetic
+tunnel junctions. This approach provides a technique for voltage-
+controlled magnetization switching and could lead to innovations
+in ultra-low-power spintronic devices.
+Methods
+In this study, we focused on a structure comprising a solid insulator and an ultrathin
+epitaxial ferromagnetic layer with moderate perpendicular surface anisotropy15–20.
+The sample structure consisted of MgO substrate/MgO(10 nm)/Cr(10 nm)/
+Au(50 nm)/Fe(2–4 ML)/MgO(10 nm)/polyimide(1,500 nm)/ITO(100 nm)
+layers. All layers, except for the top thick polyimide and ITO layers, were grown
+epitaxially by a molecular beam epitaxy method, using electron beam evaporators in
+an ultra-high vacuum. The Au buffer layer was annealed at 250 8C, after deposition
+at room temperature, to obtain an atomically flat surface. The ultrathin Fe layer
+and insulating MgO layer were grown on the Au(001) surface at room temperature.
+The sample was coated with polyimide using a spin coater, and annealed at 200 8C.
+ITO was used for the top electrodes, 1 mm in diameter, and was deposited using a
+metal mask. A bias voltage was applied between the top ITO and the bottom
+Au electrode.
+Magnetic hysteresis loops of ultrathin Fe films were measured using the magneto
+optical Kerr effect (MOKE) in a polar configuration30. The voltage dependence of
+the hysteresis was detected either by a direct observation of Kerr ellipticity, hK,
+signals for different bias voltages, or by a lock-in detection of the small change in
+Kerr ellipticity with respect to an applied bias voltage modulation, dhK/dV.
+The modulation amplitude and frequency were 160 V (peak-to-peak) and
+37 Hz, respectively.
+Received 24 June 2008; accepted 9 December 2008;
+published online 18 January 2009
+B
+A
+D
+C
+Figure 4 | A macro spin model simulation of voltage-controlled
+magnetization switching. The green line indicates the trajectory of the
+spin. The parameters used for the calculation include a damping constant
+a ¼ 0.025, perpendicular anisotropy fields of 12 kA m21 for the on state and
+22 kA m21 for the off state. An external perpendicular field of 8 kA m21 is
+also applied to tilt the magnetization. The magnetic cell is assumed
+to have a rectangular shape, with a 1.6 kA m21 in-plane hard axis
+demagnetization field.
+0.35
+−
+−
+−
+−
+−
+−
+−
+−
+−
+−
+Fe
+Electrode
+d 3z2−r2 
+d x2−y2
+0
+0.5
+1
+1.5
+0.4
+0.45
+0.5
+0.55
+0.6
+0.65
+(dηK/dV)max (a.u.)
+Fe thickness (nm)
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+Fe thickness (nm)
+−800
+−600
+−400
+−200
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+Eperpd (μJ m−2)
+Fe thickness (nm)
+ηs (a.u.)
+e−
+e−
+Figure 3 | Fe layer thickness dependencies of the voltage modulation response of hK, saturation Kerr ellipticity hs, and Eperpd. a, Maximum dhK /dV signal
+as a function of Fe layer thickness. The lock-in modulation technique was used for the precise measurement of the voltage response. The line through the
+data is a visual aid. b, Fe layer thickness dependence of the saturation Kerr ellipticity hs. c, Plot to calculate surface anisotropy energy and bulk anisotropy
+energy. The Y-cut corresponds to the surface anisotropy energy. The slope corresponds to the bulk anisotropy (see text). d, Schematic of the effect of the
+electric field on electron filling of the 3d orbitals in the ultrathin Fe layer. Application of a negative voltage, for example, may suppress the number of
+electrons in the mz ¼ 0 states, because of the quadrupole effect.
+LETTERS
+NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2008.406
+NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.nature.com/naturenanotechnology
+160
+© 2009 Macmillan Publishers Limited.  All rights reserved. 
+ 
+
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+Acknowledgements
+The authors would like to thank D. Yamaguchi, Y. Sobajima, T. Toyama and H. Okamoto
+for their assistance in ITO deposition. The authors also acknowledge H. Kubota, W. Van
+Roy, S. Blu¨gel and T. Miyazaki for their valuable comments. A part of the research was
+conducted under the financial support of Grant-in-Aid for Scientific Research
+(A19206002) and G-COE program of Ministry of Education, Culture, Sports, Science and
+Technology-Japan (MEXT).
+Author contributions
+Y.S. conceived and designed the experiments and performed micro magnetic calculation.
+T.M. and Y.S. performed the experiments and analysis. T.N. and A.A.T. led experiments
+and physical discussions. K.O., N.T. and M.M. established experimental techniques. S.M.
+and Y.A. performed FMR measurements. M.S. and T.S. contributed to general discussions.
+T.M. wrote the paper with review and input from Y.S., T.N. and A.A.T.
+Additional information
+Supplementary Information accompanies this paper at www.nature.com/
+naturenanotechnology. Reprints and permission information is available online at
+http://npg.nature.com/reprintsandpermissions/. Correspondence and requests for materials
+should be addressed to Y.S.
+NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2008.406
+LETTERS
+NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.nature.com/naturenanotechnology
+161
+© 2009 Macmillan Publishers Limited.  All rights reserved. 
+ 
+
diff --git a/kb_37/content/tmp_files/load_file.txt b/kb_37/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..7c8c87d946c76146fc54224bab80817424291fe9
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf,len=535
+page_content='Large voltage-induced magnetic anisotropy  change in a few atomic layers of iron  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Maruyama1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Shiota1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Nozaki1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Ohta1, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Toda1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Mizuguchi1†, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Tulapurkar1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Shinjo1,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Shiraishi1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Mizukami2, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Ando3 and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Suzuki1*  In the field of spintronics, researchers have manipulated mag-  netization using spin-polarized currents1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Another option is  to use a voltage-induced symmetry change in a ferromagnetic  material to cause changes in magnetization or in magnetic  anisotropy4–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' However, a significant improvement in efficiency  is needed before this approach can be used in memory devices  with ultralow power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Here, we show that a  relatively small electric field (less than 100 mV nm21) can  cause a large change (�40%) in the magnetic anisotropy of  a bcc Fe(001)/MgO(001) junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The effect is tentatively  attributed to the change in the relative occupation of 3d  orbitals of Fe atoms adjacent to the MgO barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Simulations  confirm that voltage-controlled magnetization switching in  magnetic tunnel junctions is possible using the anisotropy  change demonstrated here, which could be of use in the  development of low-power logic devices and non-volatile  memory cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  To develop voltage-driven spintronic devices, several areas of  investigation have been suggested, including voltage control of mag-  netic anisotropy4,7, ferromagnetism in ferromagnetic semiconduc-  tors8,9,  magnetoelectric  switching  of  exchange  bias10,11  and  anisotropy12, magnetoelectric interface effects5,6, multiferroic prop-  erties13 and magnetostriction in a hybrid system with piezoelectric  materials14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Most work to date has been carried out only at low  temperatures, using GaMnAs or perovskite systems, or needed  piezoelectric distortions, which may limit the endurance of any  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Weisheit and colleagues7 observed up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='5% coercivity  change in FePt(Pd) films with the application of voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  However, they required the use of a liquid electrolyte to apply a  high electric field at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' In this study, we have overcome  these difficulties by using ultrathin Fe/MgO junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The system  is built from all-solid-state and distortion-free materials that have  controllable perpendicular surface anisotropy15–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 1, the sample structure stack layers comprised a  MgO substrate/MgO(10 nm)/Cr(10 nm)/Au(50 nm)/Fe(2–4 ML)/  MgO(10 nm)/polyimide(1,500 nm)/ITO(100 nm) (ML, monatomic  layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Because the influence of the electric field on  the perpendicular anisotropy is effective only at the interface, the fer-  romagnetic layer had to be composed of only a few monatomic layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  In addition, to control the perpendicular anisotropy, the film had to  have a moderate crystalline and surface anisotropy in the absence of  the bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' In this regard, we used an ultrathin Fe layer for the  anisotropy change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Because the Fe grew almost in a layer-by-layer  mode onto the Au(001) surface at room temperature, we were able  to precisely control the layer thickness21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The ultrathin epitaxial Fe  layer deposited on the Au(001) buffer layer exhibited a transition in  magnetic anisotropy from in-plane to perpendicular, depending on  the film thickness20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Such features are ideal for the observation of  the anisotropy change in response to the electric field, because we  could systematically control the strength of the perpendicular aniso-  tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The insulating layers comprised MgO and polyimide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' MgO  was used because it can be epitaxially grown onto an Fe(001)  surface and exhibits a high breakdown voltage as a barrier material  in magnetic tunnel junctions22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The polyimide layer was used to  ensure a pinhole-free barrier over an extended area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  Figure 2 shows representative magnetic hysteresis loops in a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='48-nm-thick Fe layer under the application of a bias voltage,  obtained from Kerr ellipticity, hK, measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Under two differ-  ent bias voltages, U ¼ þ200 and –200 V, a significant change in  perpendicular anisotropy was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The perpendicular mag-  netic anisotropy energy per unit volume of the film, Eperp, was cal-  culated from the hK–H curve, assuming a linear relation between  hK and magnetization:  Eperp ¼ �m0  Ms  hs  ðhs  0  HdhK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  ð1Þ  where m0, MS, H, hK and hs are permeability of free space, satur-  ation magnetization, external magnetic field, Kerr ellipticity and sat-  uration Kerr ellipticity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' If the film possesses uniaxial  crystalline anisotropy Ku, and surface anisotropy Ks, the Eperp is  expressed as  Eperpd ¼  � 1  2 m0M2  s þ Ku  �  �  d þ KS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='MgO=Fe þ KS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='Fe=Au þ DKSðVÞ  ð2Þ  where d is the film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' DKS(V) is a surface anisotropy,  induced by application of a voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' When a positive voltage  U ¼ þ200 V  was  applied,  then  decreased  to  U ¼ –200 V,  perpendicular anisotropy was induced and the magnetic anisotropy  energy was changed from 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='3 to 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='7 kJ m23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The ratio of  the magnetic anisotropy energy change, defined as DEperp/  (2Eperp,ave) ¼ (Eperp,200V–Eperp,–200V)/(Eperp,200V þ Eperp,–200V), was  39%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' If we count this change as a change in the surface  anisotropy energy, that is, DKs(V) in equation (2), it corresponds to  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='4 mJ m22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  To precisely measure the thickness dependence of the effect, we  used a modulation technique (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 2  shows the external field dependence of the dhK/dV signal that  was obtained for the same sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The dhK/dV signal displays a  maximum value, (dhK/dV)max, where the hK hysteresis curves,  obtained for positive (B) and negative (A) bias voltages, show a  1Graduate School of Engineering Science, Osaka University, Toyonaka, Japan,  2WPI Advanced Institute for Materials Research, Tohoku University, Sendai,  Japan,  3Graduate School of Engineering, Tohoku University, Sendai, Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' †Present address: Institute for Materials Research, Tohoku University, Sendai,  Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' *e-mail: suzuki-y@mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='jp  LETTERS  PUBLISHED ONLINE: 18 JANUARY 2009 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='1038/NNANO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='406  NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='com/naturenanotechnology  158  © 2009 Macmillan Publishers Limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  maximum difference (C point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 2, inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Figure 3a shows  (dhK/dV)max as a function of film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The effect was  largest for an Fe film with a thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='48 nm, and was smaller  for both thinner and thicker films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Because the influence of the elec-  tric field is effective only at the metal/insulator interface, it is natural  to observe a smaller effect for the thicker Fe films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Figure 3b shows  the dependence of the saturation Kerr ellipticity, hs, on Fe thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  The linear dependence of hs down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='25 nm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='8 ML on average)  proves that continuous films had been grown, even if they were only  a few atomic layers in thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' In addition, this behaviour also  proves that the film was uniform across the thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 3c,  Eperpd is plotted as a function of Fe layer thickness d at zero bias  voltage, together with a linear fit using equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The fit indicates  that Ks,MgO/Fe þ Ks,Fe/Au ¼ 580 mJ m22 and Ms ¼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='5 MA m21,  neglecting a contribution from Ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Ms is about 83% that of the  bulk value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The reduction in the apparent Ms could be caused by  a contribution from a positive Ku, produced by a lattice mismatch  between Fe and Au (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The value of Ks,MgO/Fe þ Ks,Fe/Au  observed here was a little higher than previous observations made  on a Au/Fe(001) interface23, Ks,Fe/Au ¼ 470, 400, 540 mJ m22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  This suggests that the MgO/Fe interface also has a positive contri-  bution to Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The experimental data deviate from the linear fit line  below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='48 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' This is well-documented behaviour for ultrathin  films and may have many origins24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The thickness at which the  linear fit, using equation (2), starts to deviate corresponds to the  thickness where the maximum of (dhK/dV)max is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' One  of the possible reasons for this is a deterioration of the film  quality in this ultrathin thickness region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Clarification of the mech-  anism of the deviation requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  One possible origin of the effect is in the influence of an electric  field on electron filling of the Fe layer, which should affect the mag-  netic anisotropy (see Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Kyuno and col-  leagues pointed out that surface magnetic anisotropies in 3d  ferromagnetic metal/noble metal interfaces were very sensitive to  the electron filling of 3d orbitals25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' In our case, from the capacitance  of the junction, we estimate that we could change electron filling by  2 � 1023 electrons per Fe surface atom by the application of 200 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  From the density of states, this corresponds to about 1 meV change  in chemical potential (see Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' This small  change, however, may produce a non-negligible change in the  surface anisotropy energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' From our experiment, we could change  anisotropy energy by 4 meV per surface Fe atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' This magnitude  of change can be reproduced from Kyuno’s calculation using  1 meV change in the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Kyuno also noted that  the effect originates mainly from the large density of states (DOS)  of a dxy and dx2�y2 character (jmzj ¼ 2) in the Fermi energy in the  Fe/Au (001) system, in which Au has a large spin–orbit coup-  ling26–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' In our case, because the Fe has two interfaces, with  Au(001) and MgO(001), the situation is not completely the same,  but a similar mechanism may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' As schematically shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 3d, the application of a negative voltage, for example, may  cause an increase in the energy of the d3z2�r2 (mz ¼ 0) states,  because of higher electron density at the barrier/Fe interface,  leading to a reduction in the electron occupancy in those  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Therefore, the electron occupancy in the dxy and dx2�y2  states could be changed relative to one another, leading to a modu-  lation of the magnetic anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Further discussion requires first  principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  In ourexperiment, we needed to applya large voltage because ofthe  thickness of the polyimide layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The estimated voltage drop across the  MgO layer, however, was �45 mV nm21 if we can neglect charge  accumulation in the barrier (see Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' As we  know that more than 2 V can be applied to a 2-nm-thick MgO  barrier, a much larger effect can be expected for conventional tunnel  magnetoresistance junctions with a MgO barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  In the latter part of this letter, we suggest a novel magnetization  switching technique, using the voltage-induced magnetic anisotropy  change explored in this work (see Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  Figure 4 shows a result of macro-spin model simulation of voltage-  controlled magnetization switching for a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='48-nm Fe film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Here, we  used parameters obtained from the above described experiments  and an additional ferromagnetic resonance (FMR) experiment  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 4 caption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' An external magnetic field of 8 kA m21 was  applied normal to the film plane to tilt the magnetization towards  the perpendicular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Initially, the bias voltage was held off  (point A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' If we then apply a bias voltage with a slow  rise time, the perpendicular anisotropy field changes and the mag-  netization changes its direction to point B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' However, if the rise time  of the pulse is short enough (less than 1 ns), a dynamic precession  and switching to another energetically stable point is achieved  (point C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' When the voltage pulse is switched off with a  slow fall time, the magnetization stabilizes at point D (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' 4) after  the relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' This simulation clearly shows that if we  MgO (001) substrate  MgO (10 nm)  Cr (10 nm)  Au (50 nm)  +  −  Bottom electrode (Au)  Top view  20 mm  ITO (1 mm) × 24  Fe (2−4 ML)  MgO (10 nm)  Polyimide (1,500 nm)  ITO (100 nm)  Magnetic field  Figure 1 | Schematic of the sample used for a voltage-induced magnetic  anisotropy change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' a, A positive voltage is defined as a positive voltage on  the top electrode with respect to the bottom electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' A perpendicular  magnetic anisotropy was induced by a negative voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The magnetic field  was applied perpendicular to the film plane for Kerr ellipticity  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' b, We fabricated the wedge-shaped Fe layer, incorporating  24 samples on the substrate, to investigate the dependence of the effect on  Fe thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  −1,000  −500  0  500  1,000  −200 V  200 V  −1,000  0  1,000  B  A  Fe  A  B  C  C  Kerr ellipticity, η  K (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=')  Magnetic field (Oe)  Magnetic field (Oe)  dη  K/dV (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=')  Figure 2 | Magneto-optical Kerr ellipticity hk for different applied voltages  as a function of applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The thickness of the Fe film was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='48 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  A significant change in the hysteresis curve indicated a large change in  perpendicular anisotropy following application of the bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The right  inset shows the voltage modulation response of the Kerr ellipticity, dhK/dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  The left inset illustrates the magnetization direction at points A and B in the  hysteresis curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  NATURE NANOTECHNOLOGY DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='1038/NNANO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='406  LETTERS  NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='com/naturenanotechnology  159  © 2009 Macmillan Publishers Limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  fabricate a Fe/MgO/Fe/Au junction with radio-frequency signal  access, successful magnetization switching by the application of  high-speed, low-voltage pulse should be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' We believe that  this novel magnetization switching will be demonstrated in the  near future and, by combining it with magnetoresistive structures,  could prove to be a highly successful technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  Our results show that it is possible to control magnetic aniso-  tropy by the application of an electric field in Fe/MgO junctions,  which can be combined with high-quality Fe/MgO/Fe magnetic  tunnel junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' This approach provides a technique for voltage-  controlled magnetization switching and could lead to innovations  in ultra-low-power spintronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  Methods  In this study, we focused on a structure comprising a solid insulator and an ultrathin  epitaxial ferromagnetic layer with moderate perpendicular surface anisotropy15–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  The sample structure consisted of MgO substrate/MgO(10 nm)/Cr(10 nm)/  Au(50 nm)/Fe(2–4 ML)/MgO(10 nm)/polyimide(1,500 nm)/ITO(100 nm)  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' All layers, except for the top thick polyimide and ITO layers, were grown  epitaxially by a molecular beam epitaxy method, using electron beam evaporators in  an ultra-high vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The Au buffer layer was annealed at 250 8C, after deposition  at room temperature, to obtain an atomically flat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The ultrathin Fe layer  and insulating MgO layer were grown on the Au(001) surface at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  The sample was coated with polyimide using a spin coater, and annealed at 200 8C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  ITO was used for the top electrodes, 1 mm in diameter, and was deposited using a  metal mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' A bias voltage was applied between the top ITO and the bottom  Au electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  Magnetic hysteresis loops of ultrathin Fe films were measured using the magneto  optical Kerr effect (MOKE) in a polar configuration30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The voltage dependence of  the hysteresis was detected either by a direct observation of Kerr ellipticity, hK,  signals for different bias voltages, or by a lock-in detection of the small change in  Kerr ellipticity with respect to an applied bias voltage modulation, dhK/dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  The modulation amplitude and frequency were 160 V (peak-to-peak) and  37 Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  Received 24 June 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' accepted 9 December 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  published online 18 January 2009  B  A  D  C  Figure 4 | A macro spin model simulation of voltage-controlled  magnetization switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The green line indicates the trajectory of the  spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The parameters used for the calculation include a damping constant  a ¼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='025, perpendicular anisotropy fields of 12 kA m21 for the on state and  22 kA m21 for the off state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' An external perpendicular field of 8 kA m21 is  also applied to tilt the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The magnetic cell is assumed  to have a rectangular shape, with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='6 kA m21 in-plane hard axis  demagnetization field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='35  −  −  −  −  −  −  −  −  −  −  Fe  Electrode  d 3z2−r2  d x2−y2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='65  (dηK/dV)max (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=')  Fe thickness (nm)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='5  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='8  1  Fe thickness (nm)  −800  −600  −400  −200  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='8  1  Eperpd (μJ m−2)  Fe thickness (nm)  ηs (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=')  e−  e−  Figure 3 | Fe layer thickness dependencies of the voltage modulation response of hK, saturation Kerr ellipticity hs, and Eperpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' a, Maximum dhK /dV signal  as a function of Fe layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The lock-in modulation technique was used for the precise measurement of the voltage response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The line through the  data is a visual aid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' b, Fe layer thickness dependence of the saturation Kerr ellipticity hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' c, Plot to calculate surface anisotropy energy and bulk anisotropy  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The Y-cut corresponds to the surface anisotropy energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The slope corresponds to the bulk anisotropy (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' d, Schematic of the effect of the  electric field on electron filling of the 3d orbitals in the ultrathin Fe layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Application of a negative voltage, for example, may suppress the number of  electrons in the mz ¼ 0 states, because of the quadrupole effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  LETTERS  NATURE NANOTECHNOLOGY DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='1038/NNANO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='406  NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='com/naturenanotechnology  160  © 2009 Macmillan Publishers Limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
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+page_content='  Acknowledgements  The authors would like to thank D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Yamaguchi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Sobajima, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Toyama and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Okamoto  for their assistance in ITO deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' The authors also acknowledge H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Kubota, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Van  Roy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Blu¨gel and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Miyazaki for their valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' A part of the research was  conducted under the financial support of Grant-in-Aid for Scientific Research  (A19206002) and G-COE program of Ministry of Education, Culture, Sports, Science and  Technology-Japan (MEXT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  Author contributions  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' conceived and designed the experiments and performed micro magnetic calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' performed the experiments and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' led experiments  and physical discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' established experimental techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' performed FMR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' contributed to general discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' wrote the paper with review and input from Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  Additional information  Supplementary Information accompanies this paper at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='com/  naturenanotechnology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Reprints and permission information is available online at  http://npg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='com/reprintsandpermissions/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' Correspondence and requests for materials  should be addressed to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='  NATURE NANOTECHNOLOGY DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='1038/NNANO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='406  LETTERS  NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content='com/naturenanotechnology  161  © 2009 Macmillan Publishers Limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_37/content/kb_37.pdf'}
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+Dissipation Dynamics Driven Transitions of the Density Matrix Topology
+Liang Mao, Fan Yang, and Hui Zhai∗
+Institute for Advanced Study, Tsinghua University, Beijing,100084, China
+(Dated: January 12, 2023)
+The dynamical evolution of an open quantum system can be governed by the Lindblad equation
+of the density matrix. In this letter, we propose that the density matrix topology can undergo a
+transition during the Lindbladian dynamical evolution. Here we characterize the density matrix
+topology by the topological invariant of its modular Hamiltonian. We focus on the fermionic Gaus-
+sian state, where the modular Hamiltonian is a quadratic operator of a set of fermionic operators.
+The topological classification of such Hamiltonians depends on their symmetry classes. Hence, a pri-
+mary issue we deal with in this work is to determine the requirement for the Lindbladian operators,
+under which the modular Hamiltonian can maintain its symmetry class during the dynamical evo-
+lution. When these conditions are satisfied, along with a nontrivial topological classification of the
+symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We
+present two examples of dissipation driven topological transitions where the modular Hamiltonian
+lies in the AIII class with U(1) symmetry and in the DIII class without U(1) symmetry, respectively.
+As a manifestation of the topological transition, we present the signature of the eigenvalues of the
+density matrix at the transition point.
+In the past decades, topology has been extensively used
+to characterize the ground state wavefunction of a quan-
+tum Hamiltonian.
+In such situations, the topology of
+the wavefunction and the topology of the Hamiltonian
+are directly related.
+One of the most well-established
+situations is the insulators of free fermions [1–3].
+An-
+other important lesson we have learned is the close rela-
+tionship between the symmetry of a Hamiltonian and its
+topological classification [4–6]. In many cases, topologi-
+cally nontrivial states exist only when certain symmetries
+are enforced. For free fermion insulators and supercon-
+ductors, this leads to the celebrated Altland-Zirnbauer
+ten-fold way classification [7] of topological states using
+time-reversal, particle-hole, and chiral symmetries [4–6].
+In recent years, there has been an increasing interest in
+studying topological properties for non-equilibrium dy-
+namics. In some non-equilibrium situations, the quan-
+tum state remains a pure state, but is no longer an eigen-
+state of the system. A typical situation is quench dynam-
+ics [8–15], where an initial pure state undergoes unitary
+dynamical evolution governed by the system’s Hamilto-
+nian. It has been shown that a properly defined topology
+of the wavefunction dynamics can still reveal the topol-
+ogy of the system’s Hamiltonian [8–15].
+Nevertheless,
+in more generic non-equilibrium situations, the quantum
+state is not even a pure state but a mixed state described
+by a density matrix. These situations include finite tem-
+perature and open systems with dissipation.
+An open quantum system is described by a density
+matrix ˆρ whose evolution is governed by the Lindblad
+equation [16, 17]. Since the Lindbladian evolution can be
+viewed as evolution under non-hermitian Hamiltonian in
+doubled Hilbert space, there are works on open system
+topology by considering the topology of Lindbladian itself
+[18–20]. At sufficiently long time, an open system can
+reach a non-equilibrium steady state that does not evolve.
+There are also extensive works focusing on the topological
+properties of the non-equilibrium steady states [21–29].
+On the other hand, the topology of density matrix itself
+attracts attention [30–37]. The basic idea is to consider
+the modular Hamiltonian ˆK by writing ˆρ = e− ˆ
+K. The
+modular Hamiltonian is always a Hermitian operator,
+and the topology of the modular Hamiltonian can char-
+acterize the topology of the density matrix [34, 36, 37].
+The modular Hamiltonian changes in time when the den-
+sity matrix evolves under the Lindbladian dynamics. In
+this letter, we address the issue of whether the modular
+Hamiltonian can undergo a topological transition as time
+evolves in the Lindbladian dynamics of an open system.
+This question concerns the entire dissipation-driven dy-
+namical process instead of the long-time steady state.
+The answer to this question should depend on both the
+Lindbladian operator and the choice of the initial state.
+This distinguishes our work from the previous studies of
+open-system topology [18–29].
+Here we consider the Gaussian state of a set of fermion
+operators.
+A gaussian state can be viewed as a non-
+equilibrium generalization of free-fermion state, whose
+modular Hamiltonian is a quadratic operator of a set of
+fermion operators [38, 39]. Hence, we can utilize the ex-
+isting knowledge of classifying such quadratic fermionic
+Hamiltonian. It is known that the topological classifica-
+tion of such quadratic Hamiltonians requires understand-
+ing its symmetry class [4–6]. In this work, we consider
+a generic non-equilibrium situation where the modular
+Hamiltonian of the initial state have no relation with the
+system’s Hamiltonian and can lie in a different symmetry
+class. Hence, primarily to address the dissipation-driven
+topological transition, we first need to consider another
+issue: under what conditions the Lindbladian evolution
+can preserve the symmetry of a modular Hamiltonian?
+Symmetry Preserving Lindbladian. The time evolution
+of the density matrix of an open system is governed by
+arXiv:2301.04345v1  [cond-mat.mes-hall]  11 Jan 2023
+
+2
+the Lindblad equation[16, 17]
+dˆρ
+dt = −i[ ˆH, ˆρ] +
+�
+µ
+(2ˆLµˆρˆL†
+µ − {ˆL†
+µ ˆLµ, ˆρ}).
+(1)
+We consider Gaussian initial state ˆρ(t = 0) of a set
+of fermion operators {ˆc†
+i}.
+The Hamiltonian
+ˆH is a
+quadratic operator of these fermion operators, and all
+ˆLµ are linear in these fermion operators, under which
+a Gaussian state can remain a Gaussian one during the
+Lindblad time evolution [39–41]. In this work, we deal
+with the most generic situations that ˆK, ˆH and ˆLµ do
+not commute with each other.
+We first study the cases that both the modular Hamil-
+tonian and the Lindbladian possess a global U(1) sym-
+metry. This U(1) symmetry can be either a spin U(1) or
+a charge U(1) symmetry. When interpreted as a charge
+(spin) U(1) symmetry, the corresponding class describes
+insulators (superconductors). For superconductors with
+the spin U(1) symmetry, one can always apply a particle-
+hole transformation to map the spin U(1) symmetry into
+a charge U(1) symmetry. Therefore, we present the fol-
+lowing discussions in the context of the charge U(1) sym-
+metry.
+With the presence of the charge U(1) symmetry, we
+can write ˆK = �
+ij Kijˆc†
+i ˆcj, and ˆH = �
+ij hijˆc†
+i ˆcj. Each
+dissipation operator should be either pure loss term as
+a superposition of annihilation operators, denoted by
+ˆLl
+µ = �
+µ,i Dl
+µiˆci, or pure gain term as a superposi-
+tion of creation operators ˆLg
+µ = �
+µ,i Dg
+µiˆc†
+i.
+In other
+words, each single dissipation operator cannot contain
+both creation and annihilation operators. Kij, hij, Dl
+µi
+and Dg
+µi are matrices or vectors written in the single par-
+ticle bases.
+Below we should study how the modular Hamiltonian
+matrix K evolves in time. Nevertheless, the matrix K
+obeys a non-linear equation which complicates the ana-
+lyzing process. On the other hand, another important
+property of a Gaussian state is that the two-point corre-
+lation contains all the information of this state, and all
+the higher-order correlations can be expressed in terms of
+two-point corrections. Thus, instead of considering the
+dynamics of the matrix Kij, we can consider the corre-
+lation matrix C, defined as Cij = Tr(ˆc†
+i ˆcj ˆρ). It is im-
+portant to note that the correlation matrix C obeys a
+linear equation. For cases with the U(1) symmetry, the
+correlation matrix and the modular Hamiltonian matrix
+are related by [40, 41]
+C =
+1
+eKT + 1,
+(2)
+where “T” stands for transpose. Following the Lindblad
+equation, it can be shown that the correlation matrix
+obeys the following linear equation [40–42]
+dC
+dt = XC + CX† + 2M g
+(3)
+where the matrix X is defined as X = iHT − (M l)T −
+M g and H is the physical Hamiltonian matrix.
+Here
+the matrix M l is from the loss terms, defined as M l
+ij =
+�
+µ Dl∗
+µiDl
+µj, and the matrix M g is from the gain terms
+M g
+ij = �
+µ Dg∗
+µiDg
+µj.
+Following the ten-fold way classification, we consider
+the time-reversal symmetry T , the particle-hole sym-
+metry C and the chiral symmetry S of the modular
+Hamiltonian matrix K. Here we should clarify that the
+time-reversal symmetry should not be understood as a
+physical time reversal of the system. Instead, it merely
+means an antiunitary transformation that changes K by
+U †
+T KUT = K∗, where UT is the unitary part of T sym-
+metry.
+First we derive the conditions to preserve T symme-
+try. Using Eq. 2, T symmetry leads to CTUT = UT C.
+That is to say, initially, CTUT − UT C = 0. If the Lind-
+blad evolution can keep the T symmetry of the modular
+Hamiltonian, it requires d(CTUT − UT C)/dt = 0. It is
+easy to show that
+d
+dt(CTUT − UT C) = (X∗UT − UT X)C
++ CT(XTUT − UT X†) + 2(M gUT − UT M g).
+(4)
+Since we consider generic initial states with T symme-
+try, the requirement d(CTUT − UT CT)/dt = 0 leads to
+X∗UT − UT X = 0, XTUT − UT X† = 0 and M gUT −
+UT M g = 0. These conditions can be further simplified
+as
+U †
+T HUT = −H∗, U †
+T M lUT = (M l)∗, U †
+T (M g)∗UT = M g.
+(5)
+In other words, if the modular Hamiltonian of the initial
+state has T symmetry, and the Hamiltonian and the dis-
+sipation operators satisfy Eq. 5, the T symmetry will be
+preserved during the entire Lindbladian dynamics. Espe-
+cially, we note that the required symmetry property for
+the physical Hamiltonian H is different compared with
+the symmetry property of the modular Hamiltonian K.
+Next, we consider the particle-hole symmetry C. With
+this symmetry the modular Hamiltonian matrix trans-
+fers as U †
+CKUC = −K∗ under a unitary matrix UC. This
+is equivalent to CTUC + UCC − UC = 0. Hence, in or-
+der to preserve the particle-hole symmetry, we require
+d(CTUC + UCC − UC)/dt = 0. Similar analysis as above
+leads to following conditions
+U †
+CHUC = −H∗, U †
+CM lUC = M g, U †
+C(M g)∗UC = (M l)∗.
+(6)
+We note that preserving the particle-hole symmetry re-
+quires simultaneously presence both the loss and the gain
+terms.
+Finally we consider the chiral symmetry S under which
+the modular Hamiltonian matrix transfers as U †
+SKUS =
+
+3
+−K, equivalent to CTUS +USCT−U = 0. Hence, to pre-
+serve the chiral symmetry, we require d(CTUS +USCT −
+US)/dt = 0. Following the same spirit, we arrive at
+U †
+SHUS = H, U †
+SM lUS = (M g)∗, U †
+S(M g)∗US = M l.
+(7)
+Time-reversal and the particle-hole symmetries automat-
+ically guarantees the chiral symmetry by taking US =
+UCU ∗
+T . As a self-consistent check, it is easy to prove that
+Eq. 5 and Eq. 6 automatically ensure Eq. 7.
+Eq.
+5, Eq.
+6 and Eq.
+7 respectively give the con-
+ditions for preserving T , C and S symmetries of the
+modular Hamiltonian when the system and initial states
+are U(1) symmetric.
+Next, we move to the situation
+without charge or spin U(1) symmetry, such as the
+cases with fermion pairing between same spins. In this
+case, we use the Nambu spinor by introducing ˆΨ =
+(ˆc1, . . . , ˆcN, ˆc†
+1, . . . , ˆc†
+N)T, and we write the physical and
+the modular Hamiltonians into the Bogoliubov form as
+ˆH = ˆΨ†H ˆΨ and ˆK = ˆΨ†K ˆΨ. Unlike the cases with U(1)
+symmetry, the dissipation operators can be a superposi-
+tion of both loss and gain as ˆLµ = Dl
+µiˆci +Dg
+µiˆc†
+i. We in-
+troduce the correlation matrix as C as Cij = Tr(ˆΨ†
+i ˆΨj ˆρ)
+that includes anomalous correlations. Now the correla-
+tion matrix and the modular Hamiltonian matrix are re-
+lated by
+C =
+1
+e2KT + 1.
+(8)
+Similarly, we can define the matrix M l and M g as intro-
+duced above and two extra matrices M lg
+ij = �
+µ Dl∗
+µiDg
+µj
+and M gl
+ij = �
+µ Dg∗
+µiDl
+µj. Following the Lindblad equa-
+tion, the correlation matrix C now obeys the linear equa-
+tion
+dC
+dt = XC + CX† + 2W.
+(9)
+Here the matrix X = 2iHT −M T −W, and the matrices
+M and W are respectively written
+M =
+� M l
+M lg
+M gl M g
+�
+,
+W =
+� M g
+M gl
+M lg
+M l
+�
+.
+(10)
+Since the Bogoliubov Hamiltonian automatically pos-
+sesses the particle-hole symmetry, we only need to inves-
+tigate the time-reversal symmetry. The derivation of the
+symmetry preserving condition is very similar to the case
+with U(1) symmetry above. The results are
+U †
+T HUT = −H∗, U †
+T MUT = M ∗, U †
+T W ∗UT = W. (11)
+Hence we have succeeded in deriving the conditions for
+Lindbladian to preserve T , C and S symmetries with and
+without U(1) symmetry.
+Examples of Topological Transition. Now we give con-
+crete examples of topological transition. It is easy to see
+FIG. 1: Topological transition of the density matrix during
+the Lindbladian evolution. The vertical axis is the topological
+invariant of the modular Hamiltonian and the horizontal axis
+is time t in unit of hopping J. (a-b) Example I. The modular
+Hamiltonian is a one-dimension SSH model in the AIII class.
+J = (J1 + J2)/2. (a) J1 = 0.8J and J2 = 1.2J, γ1 = 0.8J
+and γ2 = 0.4J; (b) J1 = 1.2J and J2 = 0.8J, γ1 = 0.4J and
+γ2 = 0.8J. δ = 0.1J for both (a) and (b). (c) Example II.
+The modular Hamiltonian belongs to two-dimensional square
+lattice pairing model in DIII class. We set µ = 1.0J, ∆0 =
+1.0J, w = 0.5J, γ1 = 0.4J and γ2 = 0.6J. The insets show
+the ratio λi/λ0 between a few of the largest eigenvalues of the
+density matrix, with λ0 the largest eigenvalue.
+that when the following three conditions are satisfied, the
+
+1.0
+(a)
+1.0
+2:/2o
+0.0
+1.0
+1.0
+2:/20
+2:/2o
+0.0
+0.0
+7
+0.0
+0.00
+0.05
+0.10
+0.15
+0.20
+1.0
+(b)
+1.0
+1.0
+2:/20
+2:/20
+0.0
+0.0
+1.0
+2;/ 20
+0.0
+0.0
+0.1
+0.2
+0.3
+1.0
+(c)
+1.0
+1.0
+2;/20
+2/20
+0.0
+0.0
+1.0
+2;/20
+0.0
+-1.0
+0.00
+0.05
+0.10
+0.15
+0.20
+tJ4
+modular Hamiltonian must undergo a topological tran-
+sition during the dissipation dynamics. First, the initial
+modular Hamiltonian must lie in a symmetry class and
+dimension which hosts nontrivial topological classifica-
+tion. Secondly, the Lindbladian must satisfy the above-
+mentioned conditions to preserve this symmetry class.
+Thirdly, the modular Hamiltonians of the initial state
+and the long-time steady state are both gapped and have
+different topological numbers. Below we will discuss two
+examples.
+Example I: Our first example is a one-dimensional
+model in the AIII class, which is the celebrated Su-
+Schrieffer-Heeger (SSH) model [43] with dissipations.
+This one-dimensional lattice contains two sites in each
+unit cell, denoted by site-A and -B, and the modular
+Hamiltonian takes the form of the SSH model as
+ˆK(t = 0) =
+�
+i
+J1ˆc†
+i,Aˆci,B + J2ˆc†
+i,Bˆci+1,A + h.c..
+(12)
+This class possesses the chiral symmetry and the SSH
+model has the charge U(1) symmetry. Hence, the phys-
+ical Hamiltonian and the dissipation operators have to
+satisfy Eq. 7. Here we choose the set of fermion opera-
+tor basis as {. . . , ˆci,A, ˆciB, ˆci+1,A, ˆci+1,B, . . . }. Under this
+basis, the K matrix and the corresponding US matrix are
+written as
+K =
+�
+�
+�
+�
+�
+�
+�
+J1
+J1
+J2
+J2
+J1
+J1
+...
+�
+�
+�
+�
+�
+�
+�
+; US =
+�
+�
+�
+�
+�
+�
+�
+1
+−1
+1
+−1
+...
+�
+�
+�
+�
+�
+�
+�
+It is easy to see that this modular Hamiltonian obeys the
+chiral symmetry condition U †
+SKUS = −K.
+We consider a simple physical Hamiltonian
+ˆH
+=
+δ �
+i(ˆc†
+i,Aˆci,A − ˆc†
+i,Bˆci,B).
+The H-matrix is a diagonal
+matrix denoted by H = diag(δ, −δ, δ, −δ, . . . ) and it sat-
+isfies U †
+SHUS = H. We choose two types of loss opera-
+tors ˆLl
+i = √γ1(ˆci,A + ˆci,B) and ˆLl
+i = √γ2(ˆci,B + ˆci+1,A),
+and two types of gain operators ˆLg
+i = √γ1(ˆc†
+i,A − ˆc†
+i,B),
+ˆLg
+i = √γ2(ˆc†
+i,B − ˆc†
+i+1,A). Hence, the two dissipation ma-
+trices M l and M g are respectively given by
+M l =
+�
+�
+�
+�
+�
+γ1 + γ2
+γ1
+γ1
+γ1 + γ2
+γ2
+γ2
+γ1 + γ2 γ1
+...
+�
+�
+�
+�
+� ;
+M g =
+�
+�
+�
+�
+�
+γ1 + γ2
+−γ1
+−γ1
+γ1 + γ2
+−γ2
+−γ2
+γ1 + γ2 −γ1
+...
+�
+�
+�
+�
+� .
+It is easy to verify that these two matrices satisfy
+U †
+SM lUS = (M g)∗ and U †
+S(M g)∗US = M l. Hence, we
+show that this physical Hamiltonian and the dissipation
+operators can keep the chiral symmetry of the modular
+Hamiltonian during the evolution.
+Moreover, for the Lindbladian we considered, we find
+that, when γ1 > γ2, the modular Hamiltonian of the
+steady state is topologically trivial, and when γ2 > γ1,
+the modular Hamiltonian of the steady state is topolog-
+ically nontrivial. Hence, if we choose the initial modu-
+lar Hamiltonian as topologically nontrivial for the former
+case and topological trivial for the latter case, a dissipa-
+tion dynamics driven topological transition should occur,
+as shown in Fig. 1(a) and (b). In these figures, we plot
+the topological winding number of the modular Hamilto-
+nian as time evolves, where a jump of the winding num-
+ber can be found in the intermediate time [44]. We note
+that in the case, the topological invariant of the modular
+Hamiltonian is directly measurable through the ensemble
+geometric phase [34].
+Example
+II:
+This
+example
+considers
+a
+two-
+dimensional model in the DIII class. Here we consider
+spin-1/2 fermions in two-dimensional square lattice, with
+i = (ix, iy) labeling each site. The initial modular Hamil-
+tonian is chosen as follows:
+ˆK = − J
+�
+⟨ij⟩σ
+ˆc†
+iσˆcjσ − µ
+�
+i
+ˆc†
+iσˆciσ
++
+�
+⟨ij⟩σ
+∆⟨ij⟩σˆciσˆcjσ + ∆∗
+⟨ij⟩σˆc†
+jσˆc†
+iσ.
+(13)
+Here ⟨ij⟩ denotes pairs of two nearest neighboring sites.
+For σ =↑, we have ∆⟨ij⟩↑ = ±∆0 if jx = ix±1 and jy = iy
+and ∆⟨ij⟩↑ = ±i∆0 if jy = iy ± 1 and jx = ix. This gives
+a px + ipy pairing for spin-↑ between two neighboring
+sites. Similarly, we introduce a px − ipy pairing for spin-
+↓ between neighboring sites. The topological invariant of
+this Hamiltonian is given by the Fu-Kane invariant [45–
+49], protected by the time-reversal symmetry T . Here
+the topological invariant is a Z2 index, where +1 and
+−1 stand for topologically trivial and nontrivial cases,
+respectively. Moreover, it is easy to see that this model
+has neither the charge U(1) nor the spin U(1) symme-
+tries.
+Here we consider the physical Hamiltonian and the dis-
+sipation operator as follows
+ˆH = w
+�
+⟨ij⟩
+(−1)σˆc†
+iσˆcjσ,
+(14)
+ˆLi = √γ1ˆciσ + (−1)σ√γ2(ˆc†
+iσ + αijˆc†
+jσ),
+(15)
+where (−1)σ = 1 for σ =↑ and (−1)σ = −1 for σ =↓. In
+the definition of ˆLi, αij = 1 if jx = ix+1 and jy = iy, and
+αij = (−1)σi if jy = iy + 1 and jx = ix, and αij = 0 oth-
+erwise. It can be shown that this choice of the Lindblad
+operator satisfies the condition Eq.
+11 for preserving
+the time-reversal symmetry in this model. The steady
+state of this Lindblad operator is topologically trivial.
+
+5
+Hence, when the modular Hamiltonian of the initial state
+is topologically nontrivial, a transition must occur in the
+intermediate time, as shown in Fig. 1(c).
+In both examples, the modular Hamiltonian has the
+chiral symmetry and its spectrum is symmetric between
+positive and negative energies. In the inset of Fig. 1,
+we plot the eigenvalues of the density matrix in de-
+scending order. The largest eigenvalue is denoted by λ0.
+These plots show that when the modular Hamiltonian is
+gapped, all other eigenvalues are separated from λ0 by
+a finite purity gap. Nevertheless, at the transition time,
+the modular Hamiltonian becomes gapless, and therefore,
+the purity gap vanishes in the thermodynamic limit.
+Outlook. In summary, we have revealed a novel phe-
+nomenon of dissipation dynamics driven transition of the
+density matrix topology, characterized by the topolog-
+ical invariant of the modular Hamiltonian.
+So far, no
+general framework has been established to measure the
+density matrix topology. However, physical observables
+of density matrix topology have been proposed for spe-
+cific cases, such as the ensemble geometric phase for AIII
+class in one dimension [34]. How to experimentally ob-
+serve such transitions in more general situations is still
+an open question. To this end, we show signatures in
+density matrix eigenvalues at the transition, which can
+inspire experimental protocol design. Moreover, our dis-
+cussion so far is limited to Gaussian states. It will be
+interesting to study more general situations where the
+modular Hamiltonian hosts generic symmetry protected
+topological phases. We leave these exciting issues for fu-
+ture studies.
+Acknowledgement.
+We thank Ying-Fei Gu, Zhong
+Wang and Tian-Shu Deng for helpful discussion. This
+work is supported by Innovation Program for Quan-
+tum Science and Technology 2021ZD0302005, the Beijing
+Outstanding Young Scholar Program and the XPLORER
+Prize.
+F.Y. is supported by Chinese International
+Postdoctoral Exchange Fellowship Program (Talent-
+introduction Program) and Shuimu Tsinghua Scholar
+Program at Tsinghua University.
+∗ Electronic address: hzhai@tsinghua.edu.cn
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf,len=648
+page_content='Dissipation Dynamics Driven Transitions of the Density Matrix Topology  Liang Mao, Fan Yang, and Hui Zhai∗  Institute for Advanced Study, Tsinghua University, Beijing,100084, China  (Dated: January 12, 2023)  The dynamical evolution of an open quantum system can be governed by the Lindblad equation  of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In this letter, we propose that the density matrix topology can undergo a  transition during the Lindbladian dynamical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Here we characterize the density matrix  topology by the topological invariant of its modular Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' We focus on the fermionic Gaus-  sian state, where the modular Hamiltonian is a quadratic operator of a set of fermionic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  The topological classification of such Hamiltonians depends on their symmetry classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, a pri-  mary issue we deal with in this work is to determine the requirement for the Lindbladian operators,  under which the modular Hamiltonian can maintain its symmetry class during the dynamical evo-  lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' When these conditions are satisfied, along with a nontrivial topological classification of the  symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' We  present two examples of dissipation driven topological transitions where the modular Hamiltonian  lies in the AIII class with U(1) symmetry and in the DIII class without U(1) symmetry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  As a manifestation of the topological transition, we present the signature of the eigenvalues of the  density matrix at the transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  In the past decades, topology has been extensively used  to characterize the ground state wavefunction of a quan-  tum Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  In such situations, the topology of  the wavefunction and the topology of the Hamiltonian  are directly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  One of the most well-established  situations is the insulators of free fermions [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  An-  other important lesson we have learned is the close rela-  tionship between the symmetry of a Hamiltonian and its  topological classification [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In many cases, topologi-  cally nontrivial states exist only when certain symmetries  are enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' For free fermion insulators and supercon-  ductors, this leads to the celebrated Altland-Zirnbauer  ten-fold way classification [7] of topological states using  time-reversal, particle-hole, and chiral symmetries [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  In recent years, there has been an increasing interest in  studying topological properties for non-equilibrium dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In some non-equilibrium situations, the quan-  tum state remains a pure state, but is no longer an eigen-  state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' A typical situation is quench dynam-  ics [8–15], where an initial pure state undergoes unitary  dynamical evolution governed by the system’s Hamilto-  nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' It has been shown that a properly defined topology  of the wavefunction dynamics can still reveal the topol-  ogy of the system’s Hamiltonian [8–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Nevertheless,  in more generic non-equilibrium situations, the quantum  state is not even a pure state but a mixed state described  by a density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' These situations include finite tem-  perature and open systems with dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  An open quantum system is described by a density  matrix ˆρ whose evolution is governed by the Lindblad  equation [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Since the Lindbladian evolution can be  viewed as evolution under non-hermitian Hamiltonian in  doubled Hilbert space, there are works on open system  topology by considering the topology of Lindbladian itself  [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' At sufficiently long time, an open system can  reach a non-equilibrium steady state that does not evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  There are also extensive works focusing on the topological  properties of the non-equilibrium steady states [21–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  On the other hand, the topology of density matrix itself  attracts attention [30–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The basic idea is to consider  the modular Hamiltonian ˆK by writing ˆρ = e− ˆ  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The  modular Hamiltonian is always a Hermitian operator,  and the topology of the modular Hamiltonian can char-  acterize the topology of the density matrix [34, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  The modular Hamiltonian changes in time when the den-  sity matrix evolves under the Lindbladian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In  this letter, we address the issue of whether the modular  Hamiltonian can undergo a topological transition as time  evolves in the Lindbladian dynamics of an open system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  This question concerns the entire dissipation-driven dy-  namical process instead of the long-time steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  The answer to this question should depend on both the  Lindbladian operator and the choice of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  This distinguishes our work from the previous studies of  open-system topology [18–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Here we consider the Gaussian state of a set of fermion  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  A gaussian state can be viewed as a non-  equilibrium generalization of free-fermion state, whose  modular Hamiltonian is a quadratic operator of a set of  fermion operators [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, we can utilize the ex-  isting knowledge of classifying such quadratic fermionic  Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' It is known that the topological classifica-  tion of such quadratic Hamiltonians requires understand-  ing its symmetry class [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In this work, we consider  a generic non-equilibrium situation where the modular  Hamiltonian of the initial state have no relation with the  system’s Hamiltonian and can lie in a different symmetry  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, primarily to address the dissipation-driven  topological transition, we first need to consider another  issue: under what conditions the Lindbladian evolution  can preserve the symmetry of a modular Hamiltonian?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Symmetry Preserving Lindbladian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The time evolution  of the density matrix of an open system is governed by  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='04345v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='mes-hall]  11 Jan 2023  2  the Lindblad equation[16, 17]  dˆρ  dt = −i[ ˆH, ˆρ] +  �  µ  (2ˆLµˆρˆL†  µ − {ˆL†  µ ˆLµ, ˆρ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (1)  We consider Gaussian initial state ˆρ(t = 0) of a set  of fermion operators {ˆc†  i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  The Hamiltonian  ˆH is a  quadratic operator of these fermion operators, and all  ˆLµ are linear in these fermion operators, under which  a Gaussian state can remain a Gaussian one during the  Lindblad time evolution [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In this work, we deal  with the most generic situations that ˆK, ˆH and ˆLµ do  not commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  We first study the cases that both the modular Hamil-  tonian and the Lindbladian possess a global U(1) sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' This U(1) symmetry can be either a spin U(1) or  a charge U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' When interpreted as a charge  (spin) U(1) symmetry, the corresponding class describes  insulators (superconductors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' For superconductors with  the spin U(1) symmetry, one can always apply a particle-  hole transformation to map the spin U(1) symmetry into  a charge U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Therefore, we present the fol-  lowing discussions in the context of the charge U(1) sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  With the presence of the charge U(1) symmetry, we  can write ˆK = �  ij Kijˆc†  i ˆcj, and ˆH = �  ij hijˆc†  i ˆcj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Each  dissipation operator should be either pure loss term as  a superposition of annihilation operators, denoted by  ˆLl  µ = �  µ,i Dl  µiˆci, or pure gain term as a superposi-  tion of creation operators ˆLg  µ = �  µ,i Dg  µiˆc†  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  In other  words, each single dissipation operator cannot contain  both creation and annihilation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Kij, hij, Dl  µi  and Dg  µi are matrices or vectors written in the single par-  ticle bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Below we should study how the modular Hamiltonian  matrix K evolves in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Nevertheless, the matrix K  obeys a non-linear equation which complicates the ana-  lyzing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' On the other hand, another important  property of a Gaussian state is that the two-point corre-  lation contains all the information of this state, and all  the higher-order correlations can be expressed in terms of  two-point corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Thus, instead of considering the  dynamics of the matrix Kij, we can consider the corre-  lation matrix C, defined as Cij = Tr(ˆc†  i ˆcj ˆρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' It is im-  portant to note that the correlation matrix C obeys a  linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' For cases with the U(1) symmetry, the  correlation matrix and the modular Hamiltonian matrix  are related by [40, 41]  C =  1  eKT + 1,  (2)  where “T” stands for transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Following the Lindblad  equation, it can be shown that the correlation matrix  obeys the following linear equation [40–42]  dC  dt = XC + CX† + 2M g  (3)  where the matrix X is defined as X = iHT − (M l)T −  M g and H is the physical Hamiltonian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Here  the matrix M l is from the loss terms, defined as M l  ij =  �  µ Dl∗  µiDl  µj, and the matrix M g is from the gain terms  M g  ij = �  µ Dg∗  µiDg  µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Following the ten-fold way classification, we consider  the time-reversal symmetry T , the particle-hole sym-  metry C and the chiral symmetry S of the modular  Hamiltonian matrix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Here we should clarify that the  time-reversal symmetry should not be understood as a  physical time reversal of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Instead, it merely  means an antiunitary transformation that changes K by  U †  T KUT = K∗, where UT is the unitary part of T sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  First we derive the conditions to preserve T symme-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 2, T symmetry leads to CTUT = UT C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  That is to say, initially, CTUT − UT C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' If the Lind-  blad evolution can keep the T symmetry of the modular  Hamiltonian, it requires d(CTUT − UT C)/dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' It is  easy to show that  d  dt(CTUT − UT C) = (X∗UT − UT X)C  + CT(XTUT − UT X†) + 2(M gUT − UT M g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (4)  Since we consider generic initial states with T symme-  try, the requirement d(CTUT − UT CT)/dt = 0 leads to  X∗UT − UT X = 0, XTUT − UT X† = 0 and M gUT −  UT M g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' These conditions can be further simplified  as  U †  T HUT = −H∗, U †  T M lUT = (M l)∗, U †  T (M g)∗UT = M g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (5)  In other words, if the modular Hamiltonian of the initial  state has T symmetry, and the Hamiltonian and the dis-  sipation operators satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 5, the T symmetry will be  preserved during the entire Lindbladian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Espe-  cially, we note that the required symmetry property for  the physical Hamiltonian H is different compared with  the symmetry property of the modular Hamiltonian K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Next, we consider the particle-hole symmetry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' With  this symmetry the modular Hamiltonian matrix trans-  fers as U †  CKUC = −K∗ under a unitary matrix UC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' This  is equivalent to CTUC + UCC − UC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, in or-  der to preserve the particle-hole symmetry, we require  d(CTUC + UCC − UC)/dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Similar analysis as above  leads to following conditions  U †  CHUC = −H∗, U †  CM lUC = M g, U †  C(M g)∗UC = (M l)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (6)  We note that preserving the particle-hole symmetry re-  quires simultaneously presence both the loss and the gain  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Finally we consider the chiral symmetry S under which  the modular Hamiltonian matrix transfers as U †  SKUS =  3  −K, equivalent to CTUS +USCT−U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, to pre-  serve the chiral symmetry, we require d(CTUS +USCT −  US)/dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Following the same spirit, we arrive at  U †  SHUS = H, U †  SM lUS = (M g)∗, U †  S(M g)∗US = M l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (7)  Time-reversal and the particle-hole symmetries automat-  ically guarantees the chiral symmetry by taking US =  UCU ∗  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' As a self-consistent check, it is easy to prove that  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 5 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 6 automatically ensure Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  5, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  6 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  7 respectively give the con-  ditions for preserving T , C and S symmetries of the  modular Hamiltonian when the system and initial states  are U(1) symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Next, we move to the situation  without charge or spin U(1) symmetry, such as the  cases with fermion pairing between same spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In this  case, we use the Nambu spinor by introducing ˆΨ =  (ˆc1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' , ˆcN, ˆc†  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' , ˆc†  N)T, and we write the physical and  the modular Hamiltonians into the Bogoliubov form as  ˆH = ˆΨ†H ˆΨ and ˆK = ˆΨ†K ˆΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Unlike the cases with U(1)  symmetry, the dissipation operators can be a superposi-  tion of both loss and gain as ˆLµ = Dl  µiˆci +Dg  µiˆc†  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' We in-  troduce the correlation matrix as C as Cij = Tr(ˆΨ†  i ˆΨj ˆρ)  that includes anomalous correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Now the correla-  tion matrix and the modular Hamiltonian matrix are re-  lated by  C =  1  e2KT + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (8)  Similarly, we can define the matrix M l and M g as intro-  duced above and two extra matrices M lg  ij = �  µ Dl∗  µiDg  µj  and M gl  ij = �  µ Dg∗  µiDl  µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Following the Lindblad equa-  tion, the correlation matrix C now obeys the linear equa-  tion  dC  dt = XC + CX† + 2W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (9)  Here the matrix X = 2iHT −M T −W, and the matrices  M and W are respectively written  M =  � M l  M lg  M gl M g  �  ,  W =  � M g  M gl  M lg  M l  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (10)  Since the Bogoliubov Hamiltonian automatically pos-  sesses the particle-hole symmetry, we only need to inves-  tigate the time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The derivation of the  symmetry preserving condition is very similar to the case  with U(1) symmetry above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The results are  U †  T HUT = −H∗, U †  T MUT = M ∗, U †  T W ∗UT = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' (11)  Hence we have succeeded in deriving the conditions for  Lindbladian to preserve T , C and S symmetries with and  without U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Examples of Topological Transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Now we give con-  crete examples of topological transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' It is easy to see  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 1: Topological transition of the density matrix during  the Lindbladian evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The vertical axis is the topological  invariant of the modular Hamiltonian and the horizontal axis  is time t in unit of hopping J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' (a-b) Example I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The modular  Hamiltonian is a one-dimension SSH model in the AIII class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  J = (J1 + J2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' (a) J1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='8J and J2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='2J, γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='8J  and γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='4J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' (b) J1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='2J and J2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='8J, γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='4J and  γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='8J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='1J for both (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' (c) Example II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  The modular Hamiltonian belongs to two-dimensional square  lattice pairing model in DIII class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' We set µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='0J, ∆0 =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='0J, w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='5J, γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='4J and γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='6J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The insets show  the ratio λi/λ0 between a few of the largest eigenvalues of the  density matrix, with λ0 the largest eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  that when the following three conditions are satisfied, the  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='0  (a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='20  tJ4  modular Hamiltonian must undergo a topological tran-  sition during the dissipation dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' First, the initial  modular Hamiltonian must lie in a symmetry class and  dimension which hosts nontrivial topological classifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Secondly, the Lindbladian must satisfy the above-  mentioned conditions to preserve this symmetry class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Thirdly, the modular Hamiltonians of the initial state  and the long-time steady state are both gapped and have  different topological numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Below we will discuss two  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Example I: Our first example is a one-dimensional  model in the AIII class, which is the celebrated Su-  Schrieffer-Heeger (SSH) model [43] with dissipations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  This one-dimensional lattice contains two sites in each  unit cell, denoted by site-A and -B, and the modular  Hamiltonian takes the form of the SSH model as  ˆK(t = 0) =  �  i  J1ˆc†  i,Aˆci,B + J2ˆc†  i,Bˆci+1,A + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='.  (12)  This class possesses the chiral symmetry and the SSH  model has the charge U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, the phys-  ical Hamiltonian and the dissipation operators have to  satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Here we choose the set of fermion opera-  tor basis as {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' , ˆci,A, ˆciB, ˆci+1,A, ˆci+1,B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Under this  basis, the K matrix and the corresponding US matrix are  written as  K =  �  �  �  �  �  �  �  J1  J1  J2  J2  J1  J1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  �  �  �  �  �  �  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' US =  �  �  �  �  �  �  �  1  −1  1  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  �  �  �  �  �  �  �  It is easy to see that this modular Hamiltonian obeys the  chiral symmetry condition U †  SKUS = −K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  We consider a simple physical Hamiltonian  ˆH  =  δ �  i(ˆc†  i,Aˆci,A − ˆc†  i,Bˆci,B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  The H-matrix is a diagonal  matrix denoted by H = diag(δ, −δ, δ, −δ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' ) and it sat-  isfies U †  SHUS = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' We choose two types of loss opera-  tors ˆLl  i = √γ1(ˆci,A + ˆci,B) and ˆLl  i = √γ2(ˆci,B + ˆci+1,A),  and two types of gain operators ˆLg  i = √γ1(ˆc†  i,A − ˆc†  i,B),  ˆLg  i = √γ2(ˆc†  i,B − ˆc†  i+1,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, the two dissipation ma-  trices M l and M g are respectively given by  M l =  �  �  �  �  �  γ1 + γ2  γ1  γ1  γ1 + γ2  γ2  γ2  γ1 + γ2 γ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  �  �  �  �  � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  M g =  �  �  �  �  �  γ1 + γ2  −γ1  −γ1  γ1 + γ2  −γ2  −γ2  γ1 + γ2 −γ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  It is easy to verify that these two matrices satisfy  U †  SM lUS = (M g)∗ and U †  S(M g)∗US = M l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, we  show that this physical Hamiltonian and the dissipation  operators can keep the chiral symmetry of the modular  Hamiltonian during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Moreover, for the Lindbladian we considered, we find  that, when γ1 > γ2, the modular Hamiltonian of the  steady state is topologically trivial, and when γ2 > γ1,  the modular Hamiltonian of the steady state is topolog-  ically nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hence, if we choose the initial modu-  lar Hamiltonian as topologically nontrivial for the former  case and topological trivial for the latter case, a dissipa-  tion dynamics driven topological transition should occur,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 1(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In these figures, we plot  the topological winding number of the modular Hamilto-  nian as time evolves, where a jump of the winding num-  ber can be found in the intermediate time [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' We note  that in the case, the topological invariant of the modular  Hamiltonian is directly measurable through the ensemble  geometric phase [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Example  II:  This  example  considers  a  two-  dimensional model in the DIII class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Here we consider  spin-1/2 fermions in two-dimensional square lattice, with  i = (ix, iy) labeling each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The initial modular Hamil-  tonian is chosen as follows:  ˆK = − J  �  ⟨ij⟩σ  ˆc†  iσˆcjσ − µ  �  i  ˆc†  iσˆciσ  +  �  ⟨ij⟩σ  ∆⟨ij⟩σˆciσˆcjσ + ∆∗  ⟨ij⟩σˆc†  jσˆc†  iσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  (13)  Here ⟨ij⟩ denotes pairs of two nearest neighboring sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  For σ =↑, we have ∆⟨ij⟩↑ = ±∆0 if jx = ix±1 and jy = iy  and ∆⟨ij⟩↑ = ±i∆0 if jy = iy ± 1 and jx = ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' This gives  a px + ipy pairing for spin-↑ between two neighboring  sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Similarly, we introduce a px − ipy pairing for spin-  ↓ between neighboring sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The topological invariant of  this Hamiltonian is given by the Fu-Kane invariant [45–  49], protected by the time-reversal symmetry T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Here  the topological invariant is a Z2 index, where +1 and  −1 stand for topologically trivial and nontrivial cases,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Moreover, it is easy to see that this model  has neither the charge U(1) nor the spin U(1) symme-  tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Here we consider the physical Hamiltonian and the dis-  sipation operator as follows  ˆH = w  �  ⟨ij⟩  (−1)σˆc†  iσˆcjσ,  (14)  ˆLi = √γ1ˆciσ + (−1)σ√γ2(ˆc†  iσ + αijˆc†  jσ),  (15)  where (−1)σ = 1 for σ =↑ and (−1)σ = −1 for σ =↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In  the definition of ˆLi, αij = 1 if jx = ix+1 and jy = iy, and  αij = (−1)σi if jy = iy + 1 and jx = ix, and αij = 0 oth-  erwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' It can be shown that this choice of the Lindblad  operator satisfies the condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  11 for preserving  the time-reversal symmetry in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The steady  state of this Lindblad operator is topologically trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  5  Hence, when the modular Hamiltonian of the initial state  is topologically nontrivial, a transition must occur in the  intermediate time, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  In both examples, the modular Hamiltonian has the  chiral symmetry and its spectrum is symmetric between  positive and negative energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 1,  we plot the eigenvalues of the density matrix in de-  scending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' The largest eigenvalue is denoted by λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  These plots show that when the modular Hamiltonian is  gapped, all other eigenvalues are separated from λ0 by  a finite purity gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Nevertheless, at the transition time,  the modular Hamiltonian becomes gapless, and therefore,  the purity gap vanishes in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' In summary, we have revealed a novel phe-  nomenon of dissipation dynamics driven transition of the  density matrix topology, characterized by the topolog-  ical invariant of the modular Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  So far, no  general framework has been established to measure the  density matrix topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' However, physical observables  of density matrix topology have been proposed for spe-  cific cases, such as the ensemble geometric phase for AIII  class in one dimension [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' How to experimentally ob-  serve such transitions in more general situations is still  an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' To this end, we show signatures in  density matrix eigenvalues at the transition, which can  inspire experimental protocol design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Moreover, our dis-  cussion so far is limited to Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' It will be  interesting to study more general situations where the  modular Hamiltonian hosts generic symmetry protected  topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' We leave these exciting issues for fu-  ture studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  We thank Ying-Fei Gu, Zhong  Wang and Tian-Shu Deng for helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' This  work is supported by Innovation Program for Quan-  tum Science and Technology 2021ZD0302005, the Beijing  Outstanding Young Scholar Program and the XPLORER  Prize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' is supported by Chinese International  Postdoctoral Exchange Fellowship Program (Talent-  introduction Program) and Shuimu Tsinghua Scholar  Program at Tsinghua University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  ∗ Electronic address: hzhai@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='cn  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hasan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Kane, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 82, 3045  (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  [2] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Qi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Zhang, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 83, 1057  (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Bernevig and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Hughes, Topological insulators and  topological superconductors (Princeton University Press  2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Schnyder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Ryu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Furusaki, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Lud-  wig, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Chiu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Ryu,  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Zirnbauer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' B 55, 1142  (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Zhai, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Breuer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Petruccione, The Theory of Open  Quantum Systems (Oxford University Press, Oxford,  2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Huelga, Open Quantum Systems: An  Introduction (Springer, Heidelberg, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' McGinley, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Cooper, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Kulkarni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Numasawa, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Baranov, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Zoller, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Baranov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Kraus, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Diehl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Martin-Delgado, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Budich and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Altland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Diehl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Gong, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Fleischhauer,  6  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
+page_content=' 12  065007 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf'}
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+ON RETROSPECTIVE k-SPACE SUBSAMPLING SCHEMES FOR
+DEEP MRI RECONSTRUCTION
+George Yiasemis
+Netherlands Cancer Institute,
+Amsterdam, 1066 CX,
+the Netherlands
+g.yiasemis@nki.nl
+Clara I. Sánchez
+qurAI group
+University of Amsterdam,
+Amsterdam, 1012 WX,
+the Netherlands
+c.i.sanchezgutierrez@uva.nl
+Jan-Jakob Sonke
+Netherlands Cancer Institute,
+Amsterdam, 1066 CX,
+the Netherlands
+j.sonke@nki.nl
+Jonas Teuwen
+Netherlands Cancer Institute,
+Amsterdam, 1066 CX,
+the Netherlands
+j.teuwen@nki.nl
+January 23, 2023
+ABSTRACT
+Purpose: The MRI k-space acquisition is time consuming. Traditional techniques aim to acquire
+accelerated data, which in conjunction with recent DL methods, aid in producing high-fidelity images
+in truncated times. Conventionally, subsampling the k-space is performed by utilizing Cartesian-
+rectilinear trajectories, which even with the use of DL, provide imprecise reconstructions, though, a
+plethora of non-rectilinear or non-Cartesian trajectories can be implemented in modern MRI scanners.
+This work investigates the effect of the k-space subsampling scheme on the quality of reconstructed
+accelerated MRI measurements produced by trained DL models.
+Methods: The RecurrentVarNet was used as the DL-based MRI-reconstruction architecture. Carte-
+sian fully-sampled multi-coil k-space measurements from three datasets with different accelerations
+were retrospectively subsampled using eight distinct subsampling schemes (four Cartesian-rectilinear,
+two Cartesian non-rectilinear, two non-Cartesian). Experiments were conducted in two frameworks:
+Scheme-specific, where a distinct model was trained and evaluated for each dataset-subsampling
+scheme pair, and multi-scheme, where for each dataset a single model was trained on data randomly
+subsampled by any of the eight schemes and evaluated on data subsampled by all schemes.
+Results: In the scheme-specific setting RecurrentVarNets trained and evaluated on non-rectilinearly
+subsampled data demonstrated superior performance especially for high accelerations, whilst in the
+multi-scheme setting, reconstruction performance on rectilinearly subsampled data improved when
+compared to the scheme-specific experiments.
+Conclusion: Training DL-based MRI reconstruction algorithms on non-rectilinearly subsampled
+measurements can produce more faithful reconstructions. Our findings demonstrate the potential
+for using DL-based methods trained on prospective acquisitions with non-rectilinearly subsampled
+measurements to optimize scan time and image quality.
+Keywords Deep MRI Reconstruction, Retrospective k-space Subsampling, Non-rectilinear Subsampling, Non-
+Cartesian Subsampling, Recurrent Variational Network
+arXiv:2301.08365v1  [eess.IV]  20 Jan 2023
+
+Submitted to Magnetic Resonance in Medicine
+1
+Introduction
+Magnetic Resonance Imaging (MRI) is one of the most important imaging modalities in medicine. MRI’s non-invasive
+nature, non-use of ionizing radiation, and ability to produce high-resolution images make it a valuable technique for a
+wide range of clinical applications, including diagnosis, treatment planning, and dynamic tasks such as MR-guided
+surgery or radiotherapy. However, the application of MRI to dynamic tasks has been limited by the long acquisition
+times required. MRI measurements, known as the k-space, are acquired sequentially, resulting in prolonged scanning
+times.
+Over the past two decades, several methods have been put to use in clinical practice for accelerating the MRI acquisition.
+The two most conventionally applied methods to-date are Parallel Imaging (PI) and Compressed Sensing (CS), which
+are usually both incorporated in modern state-of-the-art MRI scanners.
+Compressed Sensing aims in reconstructing images from subsampled k-space measurements [1, 2, 3, 4]. Subsampling
+the k-space is, in general, a violation of the Nyquist-Shannon sampling criterion [5] and reconstructions of subsampled
+data are prone to producing aliasing artifacts. CS reconstruction algorithms attempt to solve minimization problems
+such as Total Variation (TV) optimization [6] that given a sparse low-dimensional input signals, aim to reconstruct
+high-dimensional images.
+Parallel Imaging on the other hand, employs an array of multiple - instead of one - radio-frequency receiver coils which
+measure reduced sets of spatially localised k-space frequencies while maintaining the same spatial resolution [7, 8, 9].
+Each independent receiver coil receives distinct measurements corresponding to their spatial location in relation to
+the scanned object. Hence, for each coil a unique sensitivity profile-map exists, that encodes its spatial sensitivity.
+Sensitivity maps are either known or estimated by performing a pre-scan. In Figure 1 we provide an example of a PI
+acquisition from a MRI scanner with nc = 16 coils.
+Figure 1: Parallel Imaging: Acquisition of k-space measurements from a 16-coil array machine. The reconstructed
+image can be obtained by combining the individual coil reconstructions using the root-sum-of-squares (RSS) method.
+With the recent advancements in Deep Learning (DL) and Computer Vision (CV), a plethora of algorithms have emerged
+targeting to solve imaging inverse problems, with Accelerated MRI Reconstruction being a par excellence example.
+Combined with CS and PI, numerous DL-based methods involving convolutional neural networks (CNNs) have been
+proposed in the literature [10, 11, 12, 13, 14, 15, 16, 17] applied to the task of Accelerated MRI Reconstruction. These
+methods are usually trained in a supervised manner using retrospectively subsampled (from available fully-sampled)
+k-space datasets and their target is to make a prediction of the fully-sampled k-space or its image reconstruction.
+Rectilinear Cartesian patterns constitute the most commonly employed (prospective) sampling and subsampling
+techniques applied in clinical settings. Subsequently, DL-based Accelerated MRI Reconstruction applications utilize
+rectilinear subsampling masks to retrospectively subsample fully-sampled data. However, a variety of prospective and
+retrospective sampling and subsampling patterns exist with the majority of them not being rectilinear Cartesian. For
+instance, non-Cartesian patterns such as radial or spiral have been shown to be applied in real-time MRI acquisitions due
+to the fact that they are less susceptible to motion compared to Cartesian ones [18]. The authors in [19] by employing a
+deep neural network architecture, namely the Recurrent Inference Machine (RIM) [13], explored the effects of training
+RIMs by applying either rectilinear or radial retrospective subsampling and concluded that the RIM trained using the
+latter can produce higher-fidelity reconstructions.
+In this work, we aim to investigate and compare the effects of employing miscellaneous retrospective subsampling
+schemes on the quality of DL-based learned image reconstructions. To that end, we trained and tested Recurrent
+Variational Networks [10] (RecurrentVarNets) on retrospectively subsampled k-space measurements. We performed
+experiments under either scheme-specific or multi-scheme setups, in which models were trained and evaluated on data
+subsampled with either individual and multiple, respectively, subsampling schemes.
+2
+
+(a) Individual coil k-space measurements (b) Individual coil reconstructions
+(c) Individual coil sensitivity maps
+(d) Reconstruction using all coil dataSubmitted to Magnetic Resonance in Medicine
+The contributions and findings of our work can be summarized as follows:
+• We provide a review of eight currently employed (retrospective) subsampling techniques.
+• We experimentally show that DL models trained and evaluated on non-rectilinearly, compared to rectilinearly,
+subsampled data output superior reconstructions, especially for high acceleration factors.
+• We demonstrate that models trained on data subsampled with multiple instead of individual patterns, can reconstruct
+rectilinearly subsampled data with higher fidelity.
+2
+Background - Theory
+2.1
+MRI Acquisition
+MRI reconstruction is an inverse problem on account of the fact that MR scanners acquire MRI measurements in the
+frequency domain, also known as the Fourier space, and an inversion procedure is required to produce the desired MR
+image.
+Let n = nx × ny denote the spatial size of the reconstructed data. In the case of single-coil acquisition, the relationship
+between the underlying (vectorized) image x ∈ Cn and the (vectorized) single-channel k-space y ∈ Cn is given by
+y = F(x) + e,
+(1)
+where F denotes the two-dimensional (Fast) Fourier Transform (FFT) and e ∈ Cn some measurement noise.
+2.1.1
+Parallel MRI Acquisition
+In PI multiple receiver coils are placed around the subject to speed up the acquisition. Assuming a number of nc coils,
+the acquired (multi-channel) k-space measurements are given by
+y =
+�
+y1, · · · , ync�
+∈ Cn×nc,
+yk = F(Skx) + ek,
+k = 1, 2, · · · , nc,
+(2)
+where ek denotes noise measured by the kth coil and Sk ∈ Cn×n the sensitivity map of the kth coil expressed as a
+diagonal complex matrix. Within each coil’s reception region, these maps encode their spatial sensitivity by measuring
+the relative weighting of signals acquired from various locations around the subject. The sensitivity maps are usually
+normalized as follows
+nc
+�
+k=1
+Sk∗Sk = In,
+(3)
+where Sk∗ indicates the complex conjugate of Sk and In ∈ Rn×n denotes the n-rank identity matrix.
+Obtaining an image from multi-channel measurements y can done by either using the root-sum-of-squares (RSS) or the
+SENSE methods which operate as follows:
+xrss = RSS
+�ˆx1, · · · , ˆxnc�
+= (
+nc
+�
+k=1
+| ˆxk |2)
+1
+2
+(4)
+and,
+xsense =
+��SENSE
+�ˆx1, · · · , ˆxnc��� =
+��
+nc
+�
+k=1
+Sk∗ˆxk�� ,
+ˆxk = F−1(yk),
+k = 1, 2, · · · , nc.
+(5)
+2.2
+Accelerated MRI Acquisition
+To accelerate the MRI acquisition, in CS settings the k-space is subsampled by collecting fewer than necessary
+measurements. The subsampling procedure can be described as the application of a subsampling operator U on the
+fully-sampled k-space measurements. The subsampled k-space is given by
+˜yk = Uyk = UF(Skx) + ˜ek,
+k = 1, 2, · · · , nc,
+(6)
+3
+
+Submitted to Magnetic Resonance in Medicine
+where U ∈ {0, 1}n is expressed as a binary diagonal mask, and indicates which measurements are sampled as follows:
+zU := (Uz)i =
+�zi,
+Uii = 1
+0,
+Uii = 0.
+(7)
+The magnitude of the acceleration is determined by an acceleration factor R. For a specific R, U can be chosen such
+that
+n ·
+�
+n
+�
+i=1
+Uii
+�−1 ≈ R.
+(8)
+2.3
+Accelerated MRI Reconstruction
+2.3.1
+Sensitivity Map Estimation
+The sensitivity maps S = (S1, · · · , Snc) can be estimated by various methods found in the literature [20, 8, 21, 22].
+A common method for estimating them is by fully-sampling a small region of the center of the k-space, also known as
+the autocalibration signal (ACS) which includes low frequencies [11, 10].
+Let UACS ∈ {0, 1}n denote the ACS-subsampling operator such that when applied on k-space data it outputs the
+fully-sampled ACS region, i.e.:
+zacs := (UACSz)i =
+�zi,
+i ∈ ACS region
+0,
+otherwise.
+(9)
+Subsequently, to obtain an initial approximation of the sensitivity maps we use the root-sum-of-squares (RSS) method:
+˜Sk ≈ diag
+�
+xk
+acs ⊘ xacs
+�
+,
+k = 1, 2, · · · , nc,
+(10)
+where ⊘ denotes the element-wise division, and
+xk
+acs = F−1�
+UACS˜yk�
+,
+xacs = RSS
+��
+xk
+acs
+�nc
+k=1
+�
+.
+(11)
+2.3.2
+Accelerated MRI Reconstruction as an Inverse Problem
+Obtaining a reconstruction from accelerated multicoil k-space measurements is an inverse problem, with a forward
+problem given by Eq. 6. We can rewrite Eq. 6 in a more compact notation:
+˜y = AU,S(x),
+AU,S := U ◦ F ◦ ES
+(12)
+where AU,S : Cn → Cn×nc denotes the forward operator and ES : Cn → Cn×nc is called the expand operator which
+maps an image w ∈ Cn to the individual coil images using S:
+ES(w) =
+�
+S1w, · · · , Sncw
+�
+.
+(13)
+The backward operator of AU,S is given by
+A∗
+U,S := RS ◦ F−1 ◦ U : Cn×nc → Cn,
+(14)
+where RS : Cn×nc → Cn is called the reduce operator that combines individual coil images z ∈ Cn×nc using S as
+follows:
+RS(z) =
+nc
+�
+k=1
+Sk∗zk.
+(15)
+Note that the operators U, F and F−1 in Eq. 12 and Eq. 14 are applied coil-wise.
+Subsampling the k-space causes obtaining a solution to Eq. 12 to be an ill-posed inverse problem [23, 24], and therefore,
+a solution through direct inversion is not feasible. Conventionally, in CS recovering an estimation of the ground truth
+image x from the subsampled MRI measurements ˜y can be formulated as a solution to a variational optimization
+problem as follows:
+ˆx = argmin
+w
+����AU,S(w) − ˜y
+����2
+2 + α G(w),
+(16)
+where G : Cn → R is a regularization function which can impose prior information about the solution and α > 0 is a
+regularization parameter. In the literature various choices of G and algorithms for solving Eq. 16 have been employed
+[25, 26, 27, 28].
+4
+
+Submitted to Magnetic Resonance in Medicine
+2.3.3
+Deep Learning-based Accelerated MRI Reconstruction
+With the advent of the involvement of DL in MRI reconstruction tasks, the need for handcrafting a specific regularization
+function has been replaced with CNN-based architectures. A plethora of approaches solve Eq. 16 by unrolling it into a
+gradient descent iterative optimization scheme over T time-steps:
+wt+1 = wt − αt+1 A∗
+U,S
+�
+AU,S(wt) − ˜y
+�
++ Hθt+1(wt),
+t = 0, · · · , T − 1,
+(17)
+where αt denotes a (trainable) learning rate and Hθt a CNN-based architecture with trainable parameters θt. The initial
+image w0 in Eq. 17 can be chosen as a zero-filled reconstruction using ˜y as in Eq. 4 or Eq. 5.
+Sensitivity maps can be estimated as in Eq. 10 and/or can be refined using another CNN-based model Sψ with trainable
+parameters ψ which takes as input the estimation ˜S = (˜S1, · · · , ˜Snc) as in Eq. 10:
+S = Sψ(˜S).
+(18)
+Optimization of Eq. 17 may alternatively be performed in the k-space domain as demonstrated by some authors [10, 11]:
+yt+1 = yt − αt+1 U
+�
+yt − ˜y
+�
++
+F ◦ ES ◦ Hθt+1 ◦ RS ◦ F−1�
+yt
+�
+,
+y0 = ˜y.
+(19)
+The architecture we opted for is based on Eq. 19.
+2.4
+k-space Sampling
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+Figure 2: Top: k-space sampling trajectories. Cartesian: (a) Rectilinear: k-space is filled in a line-by-line scheme, (b)
+EPI: k-space is filled in a rectilinear way but in one shot ("zig-zag"). Non-Cartesian (c) Radial: k-space is filled with
+radial spokes passing through the center, (d) Spiral: k-space is filled by one or multiple hellical curves. Each line in
+(a), (b) and (c) and each curve in (d) represents a separate filling. The dark blue arrows show the direction of each
+readout. Bottom: Subsampled k-space trajectories for different acceleration factors: (e) Rectilinear, R ≈ 1.5. (f) EPI,
+R ≈ 1.75. (g) Radial, R ≈ 2. (h) Spiral, R ≈ 3.
+A sampling scheme or k-space trajectory refers to the course of filling up a complex array with k-space frequencies
+acquired over a sequence of time-steps during the MRI acquisition. There exist a wide range of k-space trajectories
+implemented in clinical settings which can be split into two groups: Cartesian and non-Cartesian trajectories. Figure
+2(a)-(d) depicts two of each.
+Cartesian trajectories aim in collecting samples on a Cartesian or equispaced and rectangular grid. The most common
+Cartesian trajectory is the rectilinear one in which k-space samples are acquired in a line-by-line scheme as illustrated
+by Figure 2(a) with resulting samples being equidistant in both axes. Note that usually in the literature the Cartesian
+5
+
+ASubmitted to Magnetic Resonance in Medicine
+rectilinear trajectory is referred to as simply Cartesian. In this work we use the characterization Cartesian to refer to
+any trajectory acquired on a Cartesian grid. Other Cartesian trajectories include the Echo-planar imaging (EPI) in which
+k-space lines are acquired in a rectilinear fashion but in a "zig-zag" pattern as shown by Figure 2(b).
+Non-Cartesian trajectories include schemes such as the radial or the spiral. In the former the k-space signal samples are
+acquired along several spokes crossing its center (see Figure 2(c)) with a result the center being sampled multiple times,
+while the latter includes acquiring data on single or multiple helical curves starting from the center of the k-space (see
+Figure 2(d)). In non-Cartesian trajectories k-space measurements are acquired on a non-Cartesian grid and are therefore
+not equidistant with each other. For instance, in the radial filling samples closer to the center are more dense compared
+to samples further on the radial spokes.
+To accelerate the MRI acquisition, the k-space is subsampled by an acceleration factor R, resulting in fewer measure-
+ments being collected than those strictly required by the Nyquist-Shannon criterion for perfect reconstruction. This can
+lead to a degradation of the quality of the reconstruction, depending on the magnitude of the acceleration factor. For
+instance, for R = 2, half of the necessary k-space measurements are acquired. In Figure 2(e)-(h) we provide examples
+of subsampled k-space trajectories: Cartesian rectilinear and EPI (Figures 2(e)-(f)), radial and spiral (Figure 2(g)-(h)).
+3
+Methods
+3.1
+Retrospective k-space Subsampling
+Figure 3: Examples of subsampling masks for Cartesian data for acceleration factor R = 5. (a)-(d) Rectilinear:
+generated by first selecting a fraction of ACS columns and then the rest of the columns (a) are selected uniformly at
+random, (b) are equispaced with a fixed distance, (c) are equispaced but symmetric, (d) are selected from the Gaussian
+distribution (e) Variable-density Poisson Disk: generated by fully-sampling a centered disk for the ACS region and then
+applying the Bridson’s fast algorithm [29]. (f) Gaussian 2D: samples selected from a 2D Gaussian distribution. (g)-(h)
+Simulated non-Cartesian using the CIRCUS algorithm [30]: (g) Radial, (h) Spiral.
+The data used in this work consisted of volumes of fully-sampled raw k-space measurements acquired on a Cartesian
+grid. To simulate prospective subsampling we generated subsampling masks which we retrospectively applied onto
+the fully-sampled multi-coil k-space data to produce subsampled/masked measurements. The generated subsampling
+masks were binary signifying that a sample from the fully-sampled data was masked if and only if the corresponding
+mask entry was zero.
+As in this work we are interested in studying the role of the subsampling pattern on the quality of DL-based recon-
+structions of subsampled MRI data, we focused on the following retrospective subsampling patterns on the Cartesian
+grid:
+6
+
+(a) Random Rectilinear
+(b) Equispaced Rectilinear
+(c) Equispaced+ Rectilinear
+(d) Gaussian 1D
+(e) VDPD
+(f) Gaussian 2D
+(g) Radial
+(h) SpiralSubmitted to Magnetic Resonance in Medicine
+Cartesian Subsampling
+• Rectilinear: Achieved by including some and omitting other horizontal (phase encoding direction) lines on the
+Cartesian grid. For the autocalibration region we use a number of racs · ny lines, where 0 < racs < 1. We used four
+distinct rectilinear sampling patterns:
+– Random (Figure 3(a)): Lines were included uniformly at random with possible overlap with the ACS lines.
+– Equispaced (Figure 3(b)): Lines were included with a fixed distancing that satisfied the desired acceleration.
+– Equispaced+ (Figure 3(c)) : Improved rectilinear equispaced pattern by exploiting the k-space symmetry [31].
+– Gaussian 1D (Figure 3(d)): Lines were drawn from the Gaussian distribution with mean µ = ny
+2 and standard
+deviation σ = 4 · √µ.
+• Variable Density Poisson Disk (VDPD, Figure 3(e)): Combines both random sampling and denser center sampling.
+For our implementation we followed Bridson’s algorithm in [29], which is a fast algorithm of order O(n). Samples
+were drawn with a density
+1
+1 + s·|r|, i.e. inversely proportional to the k-space radius r and a slope s which were
+determined by the prescribed acceleration. For the autocalibration signal we fully-sampled a centered disk with a
+radius nx·ny·racs
+π
+.
+• Gaussian 2D (Figure 3(f)): Samples were drawn from the Gaussian distribution with mean µ = 1
+2(nx, ny) and
+covariance Σ = 4 · I2 · √µT .
+Code for VDPD and Gaussian 2D schemes was implemented in Cython for fast and efficient sampling.
+Simulated non-Cartesian Subsampling
+To simulate non-Cartesian subsampling, we applied the CIRcular Cartesian
+UnderSampling (CIRCUS) [30] technique to produce the following retrospective subsampling patterns for Cartesian
+data:
+• Radial (Figure 3(g)): Simulates radial subsampling on the Cartesian grid.
+• Spiral (Figure 3(h)): Simulates spiral subsampling on the Cartesian grid.
+For randomization CIRCUS’ offset parameter as defined in [30] can be set to produce random radial and spiral patterns.
+CIRCUS was modified to output masks by specifying the acceleration factor. Note that in contrast to the rest of the
+subsampling patterns above, for the non-Cartesian case we did not sample the ACS region exclusively, as these patterns
+already fully-sample a great portion of the k-space center. Therefore, for the ACS subsampling mask Uacs we calculated
+the largest sampled centered disk from U.
+3.2
+Deep MRI-reconstruction Model Architecture
+3.2.1
+The Recurrent Variational Network
+To compare and evaluate the aforementioned subsampling techniques we employed a DL-based reconstruction network,
+namely the Recurrent Variational Network [10] (RecurrentVarNet). The RecurrentVarNet is a DL-based inverse
+problem solver previously applied on the task of Accelerated MRI Reconstruction [10] with state-of-the-art performance
+(MC-MRI reconstruction challenge winning solution [32]). It iteratively solves the gradient descent scheme in
+the measurements domain as portrayed by Eq. 19 using convolutional recurrent neural networks (ConvRNNs) as
+a regularizer. The RecurrentVarNet takes subsampled multi-coil k-space as input and outputs a prediction of the
+fully-sampled multi-coil k-space measurements. It comprises of three main modules:
+• Recurrent Variational Block (RecurrentVarNet Block). The RecurrentVarNet consists of T RecurrentVarNet
+Blocks which are the main blocks of the RecurrentVarNet each responsible for performing an unrolled gradient
+descent optimization time-step as in Eq. 19 by replacing Hθt with a recurrent unit denoted as RNNθt:
+wt, ht+1 = RNNθt+1
+�
+RS ◦ F−1�
+yt
+�
+, ht
+�
+,
+yt+1 = yt − αt+1 U
+�
+yt − ˜y
+�
++ F ◦ ES
+�
+wt
+�
+,
+y0 = ˜y,
+t = 0, · · · , T − 1.
+(20)
+Each RNNθt is consisted of a convolutional layer (Conv) with a 5 × 5 kernel followed by nl cascades of alternating
+Convs with a 3 × 3 kernel and convolutional gated recurrent units (ConvGRUs). A rectified linear unit is applied after
+each Conv excluding the last. RNNθt takes as input intermediate quantities of the image projection of the refined
+k-space RS ◦ F−1�
+yt−1
+�
+and the hidden state ht−1 from the previous time-step.
+7
+
+Submitted to Magnetic Resonance in Medicine
+• Recurrent State Initializer (RSI). It produces an initialization for the first hidden state h0 to be used by RNNθ1
+provided as input the SENSE reconstruction of the image projection of y0:
+h0 = RSI
+�
+SENSE
+�
+F−1(y0)
+��
+.
+(21)
+• Sensitivity Estimation - Refinement (SER). The RecurrentVarNet for Accelerated MRI Reconstruction also es-
+timates at each iteration the coil sensitivity maps as in Eq. 10 and refines them using a U-Net [33] with trainable
+parameters ψ denoted as Sψ:
+S = SER
+�˜S
+�
+:
+Sk = Sψ
+� ˜
+Sk�
+,
+k = 1, · · · , nc.
+(22)
+3.3
+Experimental Setup
+Recurrent 
+Variational
+Network
+ 
+Subsampled k-space
+Prediction of Fully
+Sampled k-space     
+Fully Sampled k-space
+Ground Truth Image
+Image Prediction
+Training &
+Inference
+Training
+Select Subsampling Scheme
+Figure 4: Experiments Pipeline: For each subsampling scheme (e.g. here: radial) the fully-sampled multi-coil k-space
+is retrospectively subsampled and used as input to a RecurrentVarNet which outputs a prediction of the fully-sampled
+k-space. The predicted xT and ground truth ˆx images are produced by applying the RSS ◦ F−1 operator onto yT and
+y, respectively. During training, the loss L is calculated using xT and ˆx.
+To perform our experiments, we retrospectively subsampled the fully-sampled k-space data by generating subsampling
+masks as introduced in Section 3.1. We carried out two classes of experiments: Scheme-specific and Multi-scheme. An
+overview of our experimental setup is illustrated in Figure 4.
+3.3.1
+Scheme-specific Setup
+To compare the individual subsampling patterns and demonstrate their effect on the quality of DL-based reconstruction,
+we first performed experiments in a scheme-specific setting: for each dataset-pattern pair we ran individual experiments
+by training and evaluating (twenty-four) distinct RecurrentVarNets (with the same choice of hyper-parameters as
+outlined in Section 3.3.4).
+3.3.2
+Multi-scheme Setup
+In the multi-scheme setting, our goal was twofold: Firstly, we aimed to investigate further the effect of each subsampling
+scheme on the quality of DL-based reconstruction. Secondly and most importantly, we wanted to asses whether or not a
+DL-based model trained in a multi-scheme fashion (i.e. training measurements subsampled with multiple subsampling
+patterns) demonstrated higher reconstruction performance compared to being trained in a scheme-specific fashion (as in
+Section 3.3.1). Therefore, for each dataset a RecurrentVarNet was trained on data arbitrarily subsampled with any of
+the presented subsampling schemes in Section 3.1 and evaluated on all of them.
+8
+
+..
+..Submitted to Magnetic Resonance in Medicine
+3.3.3
+Subsampling
+In both, scheme-specific and multi-scheme settings, throughout the training phase, subsampling masks were generated
+with an acceleration factor of R = 2, 4 or 8, and were retrospectively applied onto the fully-sampled data. At validation
+and testing times, data were 2-fold, 4-fold, and 8-fold retrospectively subsampled.
+For the Cartesian masks we set racs = 0.16, 0.08, 0.04 for R = 2, 4 or 8, respectively.
+Note that for all experiments, subsampling masks during training were randomly generated to maximize the amount of
+data that each model saw, while for validation and testing they were predetermined. Additionally, the same subsampling
+mask was applied to all slices of each volume during validation and testing. It should also be highlighted that all
+individual coil data of each sample were subsampled with the same subsampling mask as this reflected clinical
+subsampling.
+3.3.4
+Model Implementation Details
+Hyper-parameter Choice
+For the RecurrentVarNets we used T = 8 time-steps. For each RecurrentVarNet Block
+we used nl = 4 alternating cascades and for the number of filters in each Conv and hidden size in each ConvGRU
+we chose nf = 256 channels. For the implementation of the RSI and SER modules we picked the same choice of
+hyper-parameters as the original paper [10].
+Training & Optimization Details
+For training and optimization we utilized PyTorch [34]. All components of the
+Recurrent Variational Network were trained end-to-end and optimization was performed utilizing the Adam algorithm
+with coefficients (β1, β2) = (0.9, 0.999), stability parameter ϵ = 1 × 10−8 and no weight decay.
+Experiments were performed utilizing NVIDIA RTX A6000, Quadro RTX 8000, or A100 GPUs. Models were trained
+to convergence with a batch size of 1 slice multi-coil k-space data. The total number of trainable parameters for each
+model amounted to approximately 27,626k.
+Training Loss Function
+At each training iteration each model was fed with subsampled multi-coil k-space measure-
+ments ˜y and produced a prediction yT of the fully-sampled reference k-space y. Loss was computed in the image
+domain using xT = RSS ◦ F−1(yT ) as the image prediction and ˆx = RSS ◦ F−1(y) as the image reference.
+As a loss function we used a combination of the mean average error (MAE) loss LMAE and the structural similarity
+index measure (SSIM) loss LSSIM:
+L(ˆx, xT ) = LMAE(ˆx, xT ) + LSSIM(ˆx, xT )
+=
+����ˆx − xT
+����
+1 +
+�
+1 − SSIM(ˆx, xT )
+�
+,
+(23)
+where SSIM is defined in Section 3.3.6.
+3.3.5
+Datasets
+To perform our experiments we employed three open source datasets, the fastMRI knee and brain datasets [35] which
+are to-date the largest publicly available MRI datasets, and the Calgary-Campinas (CC) brain dataset which was released
+as part of the Multi-Coil MRI Reconstruction Challenge [32]. All datasets consist of raw three dimensional k-space
+volumes which are multi-coil and fully-sampled conventional Cartesian acquisitions. The acquisition parameters and
+the splitting ratios we opted for in our experiments are summarized in Supporting Information Table S1.
+3.3.6
+Quality Analysis
+Metrics
+To analyse and compare our results we employed three evaluation metrics commonly used in image process-
+ing. Assuming u ∈ Rn was the ground truth image and v ∈ Rn the prediction, they are defined as follows:
+A) Structural Similarity Index Measure (SSIM)
+SSIM(u, v) = 1
+M
+M
+�
+i=1
+(2µuiµvi + c1)(2σuivi + c2)
+(µ2ui + µ2vi + c1)(σ2ui + σ2vi + c2),
+(24)
+where ui, vi, i = 1, · · · , M are image windows of size (wx, wy) = (7, 7) from u and v, respectively. The numbers
+µui, µvi, σui and σvi denote the means and standard deviations of each image window and σuivi denotes the covariance
+between ui and vi. The constants c1 = 0.01 and c2 = 0.03 are used for numerical stability.
+9
+
+Submitted to Magnetic Resonance in Medicine
+B) Peak Signal-to-Noise Ratio (pSNR)
+pSNR(u, v) = 20 log10
+�
+max(u)
+�
+1
+n
+�n
+i (ui − vi)2
+�
+(25)
+C) Normalized Mean Squared Error (NMSE)
+NMSE(u, v) = ||u − v||2
+2
+||u||2
+2
+=
+�n
+i (ui − vi)2
+�n
+i u2
+i
+(26)
+Note that the higher the computed SSIM and pSNR values are, the higher the quality of the reconstruction is, whereas
+for the NMSE values, the lower they are the lower the quality of the reconstruction is. Reported values for SSIM and
+NMSE are multiplied by 100 and 1000, respectively.
+3.3.7
+Significance Tests
+To perform significance tests we used the Almost Stochastic Order (ASO) test [36, 37] with a 95% confidence level
+(α = 0.05). Each ASO test outputs a violation error ϵmin which denotes the degree to which the hypothesis that "method
+A is always better than method B" is being violated. If ϵmin ≤ 0.5 one can claim that A is better than B, and otherwise
+if ϵmin > 0.5.
+4
+Results
+To obtain our results, data preparation, retrospective subsampling generation, and model training we used the Deep
+Image Reconstruction Toolkit (DIRECT) [38].
+4.1
+Scheme-specific Setup Results
+Figure 5 illustrates the quantitative metrics computed on the test sets in the form of violin-plots. The average metrics are
+reported in Supporting Information Table S2. Additionally, for visual assessment, in Figure 6, Figure 7, and Supporting
+Information Figure S1 we present example reconstructions of a test sample from each dataset using all methods along
+with ground truths and the retrospective subsampling mask used in each scenario.
+4.1.1
+Cartesian
+Rectilinear
+Evidently, Figure 5 indicates that all models trained on rectilinearly subsampled measurements achieved
+comparable results when evaluated on the test sets subsampled with the respective rectilinear schemes. Moreover,
+although Gaussian 1D subsampling outperformed the rest of the rectilinear schemes in the case of both fastMRI datasets,
+the opposite was noted in the case of the CC dataset. This is also visible by the example reconstructions in Figures 6, 7,
+and Supporting Information Figure S1, especially for 8-fold acceleration.
+Non-Rectilinear
+As Figure 5 and Supporting Information Table S2 indicate, models trained on data subsampled
+with either VDPD or Gaussian 2D schemes were the best-performing models when evaluated on the respective data.
+Additionally, they produced the higher average SSIM and pSNR and lower NMSE reconstructions for all combinations
+of datasets and acceleration factors.
+4.1.2
+Non-Cartesian
+As shown in Figure 5 and Supporting Information Table S2, models trained and evaluated on simulated non-Cartesian
+(radially and spirally) subsampled data yielded similar performance to Cartesian non-rectilinear schemes. In general,
+spiral schemes outperformed radial schemes for all combinations of datasets and acceleration factors, though the
+difference in performance was minor.
+4.1.3
+Comparisons
+Considering Figure 5, models trained on non-rectilinearly subsampled data, produced reconstructions of higher fidelity
+when evaluated on the respective test sets in comparison to models trained and evaluated on rectilinearly subsampled
+data for all datasets and acceleration factors. Models trained with VDPD-subsampled k-spaces obtained the best average
+quantitative evaluation results.
+10
+
+Submitted to Magnetic Resonance in Medicine
+75
+81
+87
+94
+100
+SSIM (×100)
+R = 2
+29
+36
+44
+51
+59
+pSNR
+fastMRI Knee
+0
+4
+8
+13
+17
+NMSE (×1e3)
+65
+74
+83
+91
+100
+R = 4
+27
+34
+41
+47
+54
+0
+6
+12
+18
+24
+57
+68
+78
+89
+100
+R = 8
+27
+33
+39
+45
+51
+0
+8
+17
+25
+34
+80
+85
+90
+95
+100
+SSIM (×100)
+32
+38
+45
+52
+59
+pSNR
+fastMRI Brain
+0
+3
+5
+8
+11
+NMSE (×1e3)
+74
+80
+87
+93
+100
+27
+34
+41
+49
+56
+0
+7
+15
+22
+29
+66
+74
+83
+91
+100
+23
+30
+38
+46
+54
+0
+23
+47
+70
+93
+93
+94
+96
+98
+100
+SSIM (×100)
+32
+36
+40
+43
+47
+pSNR
+Calgary Campinas
+0
+2
+5
+7
+10
+NMSE (×1e3)
+81
+86
+90
+95
+100
+25
+30
+34
+39
+43
+0
+10
+19
+29
+39
+69
+76
+84
+91
+99
+22
+27
+31
+36
+41
+0
+23
+46
+69
+92
+Subsampling
+Scheme
+Random
+Rectilinear
+Equispaced
+Rectilinear
+Equispaced+
+Rectilinear
+Gaussian 1D
+Rectilinear
+VDPD
+Gaussian 2D
+Radial
+Spiral
+Figure 5: Scheme-specific experiments quantitative results on the test sets. For each dataset-subsampling scheme pair
+a distinct model was trained (in total 24 models). For each dataset-acceleration-metric combination, pair-wise ASO
+significance tests were performed between the average best performing (VDPD) and the rest schemes. ⋆ indicates that
+VDPD was not found to be significantly better (ϵmin > 0.5). Average metrics are reported in Supporting Information
+Table S2.
+11
+
+Submitted to Magnetic Resonance in Medicine
+Ground
+Truth
+SSIM:0.9218
+pSNR:38.31
+NMSE:0.0041
+Random
+Rectilinear
+R = 2
+SSIM:0.9193
+pSNR:38.00
+NMSE:0.0044
+Equispaced
+Rectilinear
+SSIM:0.9094
+pSNR:37.41
+NMSE:0.0050
+Equispaced+
+Rectilinear
+SSIM:0.9255
+pSNR:38.59
+NMSE:0.0038
+Gaussian 1D
+Rectilinear
+SSIM:0.9350
+pSNR:39.52
+NMSE:0.0031
+VDPD
+SSIM:0.9325
+pSNR:39.26
+NMSE:0.0033
+Gaussian 2D
+SSIM:0.9243
+pSNR:38.77
+NMSE:0.0037
+Radial
+SSIM:0.9230
+pSNR:38.59
+NMSE:0.0038
+Spiral
+SSIM:0.8603
+pSNR:34.78
+NMSE:0.0092
+R = 4
+SSIM:0.8627
+pSNR:34.82
+NMSE:0.0091
+SSIM:0.8649
+pSNR:35.08
+NMSE:0.0086
+SSIM:0.8750
+pSNR:35.85
+NMSE:0.0072
+SSIM:0.8969
+pSNR:37.14
+NMSE:0.0053
+SSIM:0.8960
+pSNR:36.92
+NMSE:0.0056
+SSIM:0.8944
+pSNR:37.01
+NMSE:0.0055
+SSIM:0.8908
+pSNR:36.56
+NMSE:0.0061
+SSIM:0.8121
+pSNR:32.50
+NMSE:0.0155
+R = 8
+SSIM:0.8166
+pSNR:31.91
+NMSE:0.0178
+SSIM:0.8189
+pSNR:32.33
+NMSE:0.0161
+SSIM:0.8388
+pSNR:33.81
+NMSE:0.0115
+SSIM:0.8696
+pSNR:35.44
+NMSE:0.0079
+SSIM:0.8708
+pSNR:35.41
+NMSE:0.0079
+SSIM:0.8702
+pSNR:35.67
+NMSE:0.0075
+SSIM:0.8628
+pSNR:35.01
+NMSE:0.0087
+Figure 6: Representative reconstruction of a sample from the fastMRI knee test set obtained from the scheme-specific
+experiments. For the ground truth image the RSS method was applied on the fully sampled k-space. For each
+subsampling scheme, we retrospectively applied subsampling masks as shown in rows 2, 4, 6 (with accelerations factors
+of 2, 4, 8, respectively) onto the fully sampled k-space. Rows 1, 3, 5 illustrate the center-cropped RSS-reconstructed
+k-space output from each model. Quantitative metrics against the ground truth are inscribed on the top left of each
+reconstruction.
+As apparent from Figure 6, Figure 7, and Supporting Information Figure S1, for high acceleration factors (4 or 8)
+models trained on rectilinear schemes were more prone to reconstructing images with more errors and artifacts, in
+contrast to non-Cartesian or non-rectilinear schemes. However, for R = 2 all models evaluated closely.
+4.2
+Multi-scheme Setup Results
+Figure 8 reports the quantitative evaluation results on the test sets in the multi-scheme setup and Supporting Information
+Table S3 the corresponding average metrics. We observe that, similarly to the results of Section 4.1, reconstructions
+of non-rectilinearly subsampled measurements produced better quantitative results compared to reconstructions of
+rectilinearly subsampled data. In addition, VDPD and Gaussian 2D-subsampled reconstructions were the highest
+performing in average.
+For further investigation, in Supporting Information Figure S2, utilizing the results reported in Figure 8, we calculated
+the per-case difference in evaluation metrics change for each pattern using as reference the results obtained in the
+scheme-specific setup (Figure 5). We also report the average differences for rectilinear and non-rectilinear patterns in
+Table 1. Interestingly, models trained in the multi-scheme setting exemplified superior performance when evaluated on
+measurements subsampled with rectilinear schemes compared against the models trained on individual schemes when
+evaluated on the same data. In particular, in the case of rectilinear schemes, noticeable improvements (SSIM/pSNR
+increase, NMSE decrease) on the reconstruction performance were remarked for all datasets and acceleration factors,
+whilst for non-rectilinear patterns no change or minor deterioration was observed.
+12
+
+.Submitted to Magnetic Resonance in Medicine
+Ground
+Truth
+SSIM:0.9765
+pSNR:42.60
+NMSE:0.0032
+Random
+Rectilinear
+R = 2
+SSIM:0.9780
+pSNR:43.45
+NMSE:0.0027
+Equispaced
+Rectilinear
+SSIM:0.9732
+pSNR:42.03
+NMSE:0.0037
+Equispaced+
+Rectilinear
+SSIM:0.9801
+pSNR:44.05
+NMSE:0.0023
+Gaussian 1D
+Rectilinear
+SSIM:0.9779
+pSNR:45.96
+NMSE:0.0015
+VDPD
+SSIM:0.9815
+pSNR:45.44
+NMSE:0.0017
+Gaussian 2D
+SSIM:0.9669
+pSNR:43.87
+NMSE:0.0024
+Radial
+SSIM:0.9634
+pSNR:43.73
+NMSE:0.0025
+Spiral
+SSIM:0.9577
+pSNR:37.73
+NMSE:0.0100
+R = 4
+SSIM:0.9619
+pSNR:38.80
+NMSE:0.0078
+SSIM:0.9599
+pSNR:38.49
+NMSE:0.0084
+SSIM:0.9641
+pSNR:39.51
+NMSE:0.0066
+SSIM:0.9677
+pSNR:43.33
+NMSE:0.0027
+SSIM:0.9737
+pSNR:43.04
+NMSE:0.0029
+SSIM:0.9507
+pSNR:41.24
+NMSE:0.0044
+SSIM:0.9434
+pSNR:40.95
+NMSE:0.0047
+SSIM:0.9375
+pSNR:33.93
+NMSE:0.0239
+R = 8
+SSIM:0.9339
+pSNR:33.04
+NMSE:0.0293
+SSIM:0.9345
+pSNR:33.30
+NMSE:0.0276
+SSIM:0.9444
+pSNR:34.62
+NMSE:0.0204
+SSIM:0.9613
+pSNR:41.26
+NMSE:0.0044
+SSIM:0.9652
+pSNR:41.06
+NMSE:0.0046
+SSIM:0.9400
+pSNR:39.19
+NMSE:0.0071
+SSIM:0.9333
+pSNR:39.48
+NMSE:0.0066
+Figure 7: Representative reconstruction of a T2-weighted sample from the fastMRI brain test set obtained from
+the scheme-specific experiments. For the ground truth image the RSS method was applied on the fully sampled
+k-space. For each subsampling scheme, we retrospectively applied subsampling masks as shown in rows 2, 4, 6 (with
+accelerations factors of 2, 4, 8, respectively) onto the fully sampled k-space. Rows 1, 3, 5 illustrate the center-cropped
+RSS-reconstructed k-space output from each model. Quantitative metrics against the ground truth are inscribed on the
+top left of each reconstruction.
+In Figure 9 and Supporting Information Figure S3 we plot representative reconstructions predicted from rectilinearly-
+subsampled data obtained from both, the scheme-specific and multi-scheme setups. By visual investigation, we notice
+that in the latter reconstructed images were more faithful.
+4.3
+Results - Model Dependence
+To demonstrate that the results presented above did not rely on the choice of the model architecture we further
+experimented by employing an additional state-of-the-art deep-MRI reconstruction network: the Recurrent Inference
+Machine (RIM). More precisely, we repeated the scheme-specific experiments on the CC dataset by replacing the
+RecurrentVarNet with a RIM. The choice of hyper-parameters for each RIM was identical as in [19].
+In Supporting Information Figure S4 and Table S4 are presented the quantitative metrics for our RIM experiments.
+Results were on par with the findings of our original experiments, suggesting that our conclusions were not dependent
+on the choice of DL architecture.
+5
+Discussion
+In this work we investigated and compared various retrospective k-space subsampling patterns and we experimentally
+studied their effect on the quality of DL-based reconstructions. Since the data we utilized in our studies were Cartesian
+fully-sampled acquisitions, we retrospectively generated subsampling masks on the Cartesian grid as demonstrated in
+13
+
+.八福Submitted to Magnetic Resonance in Medicine
+75
+81
+87
+94
+100
+SSIM (×100)
+R = 2
+29
+36
+44
+51
+59
+pSNR
+fastMRI Knee
+0
+4
+9
+13
+17
+NMSE (×1e3)
+65
+74
+83
+91
+100
+R = 4
+27
+34
+41
+47
+54
+0
+6
+12
+17
+23
+57
+68
+79
+89
+100
+R = 8
+27
+33
+39
+45
+51
+0
+8
+16
+23
+31
+80
+85
+90
+95
+100
+SSIM (×100)
+32
+38
+45
+52
+59
+pSNR
+fastMRI Brain
+0
+3
+5
+8
+11
+NMSE (×1e3)
+74
+81
+87
+94
+100
+27
+34
+41
+48
+56
+0
+7
+14
+21
+28
+67
+75
+83
+92
+100
+23
+31
+38
+46
+54
+0
+20
+41
+61
+82
+94
+96
+97
+98
+99
+SSIM (×100)
+34
+37
+41
+44
+47
+pSNR
+Calgary Campinas
+0
+2
+4
+5
+7
+NMSE (×1e3)
+85
+88
+92
+95
+99
+27
+31
+35
+39
+43
+0
+8
+15
+23
+31
+76
+81
+87
+93
+98
+23
+28
+32
+36
+40
+0
+17
+34
+50
+67
+Subsampling
+Scheme
+Random
+Rectilinear
+Equispaced
+Rectilinear
+Equispaced+
+Rectilinear
+Gaussian 1D
+Rectilinear
+VDPD
+Gaussian 2D
+Radial
+Spiral
+Figure 8: Multi-scheme experiments quantitative results on the test sets. For each dataset-acceleration-metric combi-
+nation, pair-wise ASO significance tests were performed between the average best performing (VDPD) and the rest
+schemes. ⋆ indicates that VDPD was not found to be significantly better (ϵmin > 0.5). Average metrics are reported in
+Supporting Information Table S3.
+14
+
+Submitted to Magnetic Resonance in Medicine
+Section 3.1, simulating prospective Cartesian or non-Cartesian accelerated acquisitions. In particular, we generated four
+Cartesian rectilinear schemes (random, equispaced, equispaced with symmetry correction, Gaussian), non-rectilinear
+Cartesian schemes (Variable Density Poisson-disk and Gaussian 2D), and simulated (non-Cartesian) radial and spiral
+schemes.
+Our experiments consisted of utilizing a state-of-the-art DL-based accelerated MRI reconstruction method - the
+Recurrent Variational Network. Although choosing the optimal DL-based MRI reconstruction algorithm was out of the
+scope of this project, it is important to note that other DL-based models do exist with similar performance. However,
+the RecurrentVarNet was chosen as it has previously shown to produce high-fidelity reconstructions and outperform
+other models in reconstruction performance. Additionally, to demonstrate that the results were not model-dependent,
+the scheme-specific experiments on the CC dataset were repeated using the RIM model which produced on-par results.
+Experiments were performed under two setups: scheme-specific and multi-scheme setups. In the scheme-specific
+settings, we trained individual models on data retrospectively subsampled with individual subsampling patterns applying
+various acceleration factors. Quantitative and qualitative results demonstrated that the models trained on conventional
+rectilinear schemes, in contrast to other schemes, produced lower quality reconstructions with more artifacts especially
+for higher acceleration factors (4 or 8). This can be attributed to the fact that non-rectilinear Cartesian such as the
+VDPD or the Gaussian 2D, and non-Cartesian patterns such as the radial or spiral, allow for more incoherent sampling.
+This means that these schemes result in a more randomized and less correlated distribution of samples, reducing the
+dependence of the reconstruction quality on any specific pattern of missing data. Additionally, these schemes allow for
+center oversampling, which contains information such as contrast and the general shape of the reconstruction.
+In the multi-scheme setup, unified models were trained on all distributions of subsampled patterns. Although quantitative
+results were on par with the results of our scheme-specific experiments, we observed noticeable improvements (compared
+to the scheme-specific results) of the reconstruction inference performance for rectilinearly subsampled measurements.
+Additionally, the violin-plots in Figure 5 and Figure 8 show that for most dataset-metric-acceleration combinations
+quantitative results for the four rectilinear and the four non-rectilinear schemes formed similar metric distributions and
+were with similar numbers of outliers, suggesting that results were also case-dependent.
+Even though comparing prospective acquisition speeds was out of the scope of this project, we provide a brief
+discussion. While the reconstruction performance of Cartesian rectilinearly subsampled data was inferior to Cartesian
+non-rectilinear, someone could argue about the trade-off between acquisition speed and quality, as MRI scanners
+can perform rectilinear sampling in fast acquisition times [1], while strategies such as VDPD or Gaussian 2D can be
+slower due to physical limitations. For instance, in the prospective case these strategies may require large gradient
+switches in the MRI scanner due to the enlarged k-space spacing which can cause extended times due to hardware
+constraints, whilst in the retrospective settings, efficient algorithms are used to pick the samples. On the other hand, our
+results indicated that synthesized radial or spiral schemes on the Cartesian grid using the CIRCUS method provided
+similar performance to VDPD and Gaussian 2D schemes. In the original paper [30], the authors state that their method
+improved sampling efficiency over VDPD while maintaining the reconstruction performance in the prospective case.
+However, prospective non-Cartesian sampling trajectories do not sample on a Cartesian grid, and therefore samples
+closer to the center are more densely placed, and more scattered far off. As a result of this non-uniformity, for the
+application of the FFT a gridding and inverse-gridding process [39] is required to place them onto a Cartesian grid
+accumulating additional computation times. Note that gridding can be replaced with the application of the non-uniform
+FFT (NUFFT) [40].
+The main limitation of this study is the fact that experiments were performed retrospectively. In our future work, we
+will repeat our experiments using prospectively subsampled data with the different schemes we employed in this study
+Table 1: Scheme-specific-Multi-scheme experiments performance average percentage difference. Percentages were
+acquired by averaging the per-case differences (as calculated and illustrated in Supporting Information Figure S2) for
+all dataset-scheme type (rectilinear or non-rectilinear) combinations.
+Dataset
+Type of
+Subsampling
+Scheme
+Acceleration Factor (R)
+2
+4
+8
+SSIM
+pSNR
+NMSE
+SSIM
+pSNR
+NMSE
+SSIM
+pSNR
+NMSE
+FastMRI Knee
+Rectilinear
+0.1%
+0.5%
+-0.7%
+0.3%
+0.8%
+-3.2%
+0.7%
+1.4%
+-9.4%
+Non-rectilinear
+0.0%
+-0.3%
+5.2%
+0.0%
+-0.2%
+4.2%
+0.0%
+-0.1%
+3.1%
+FastMRI Brain
+Rectilinear
+0.0%
+0.3%
+-2.6%
+0.1%
+0.2%
+-1.6%
+0.3%
+0.7%
+-4.8%
+Non-rectilinear
+0.0%
+-0.2%
+1.8%
+0.0%
+-0.1%
+0.7%
+0.0%
+0.0%
+-0.6%
+Calgary Campinas
+Rectilinear
+0.3%
+1.6%
+-13.9%
+1.2%
+2.7%
+-18.5%
+3.7%
+5.0%
+-26.0%
+Non-rectilinear
+0.0%
+0.1%
+-1.4%
+0.0%
+0.0%
+0.3%
+0.0%
+0.0%
+0.2%
+15
+
+Submitted to Magnetic Resonance in Medicine
+Ground Truth
+R=2
+SSIM:0.9453
+pSNR:37.18
+NMSE:0.0034
+Unimodal
+Random Rectilinear
+SSIM:0.9481
+pSNR:37.66
+NMSE:0.0031
+Multimodal
+SSIM:0.9533
+pSNR:38.57
+NMSE:0.0025
+Unimodal
+Equispaced Rectilinear
+SSIM:0.9555
+pSNR:38.98
+NMSE:0.0023
+Multimodal
+SSIM:0.9431
+pSNR:36.25
+NMSE:0.0043
+Unimodal
+Equispaced+ Rectilinear
+SSIM:0.9497
+pSNR:37.65
+NMSE:0.0031
+Multimodal
+SSIM:0.9576
+pSNR:39.31
+NMSE:0.0021
+Unimodal
+Gaussian 1D Rectilinear
+SSIM:0.9571
+pSNR:39.27
+NMSE:0.0021
+Multimodal
+R=4
+SSIM:0.9158
+pSNR:33.71
+NMSE:0.0077
+SSIM:0.9234
+pSNR:34.65
+NMSE:0.0062
+SSIM:0.9299
+pSNR:35.23
+NMSE:0.0054
+SSIM:0.9297
+pSNR:35.23
+NMSE:0.0054
+SSIM:0.9180
+pSNR:33.55
+NMSE:0.0079
+SSIM:0.9236
+pSNR:34.34
+NMSE:0.0066
+SSIM:0.9311
+pSNR:35.29
+NMSE:0.0053
+SSIM:0.9287
+pSNR:34.96
+NMSE:0.0057
+R=8
+SSIM:0.8445
+pSNR:27.18
+NMSE:0.0344
+SSIM:0.8677
+pSNR:28.84
+NMSE:0.0235
+SSIM:0.8130
+pSNR:26.08
+NMSE:0.0443
+SSIM:0.8433
+pSNR:27.50
+NMSE:0.0319
+SSIM:0.8411
+pSNR:27.30
+NMSE:0.0334
+SSIM:0.8691
+pSNR:28.99
+NMSE:0.0227
+SSIM:0.8963
+pSNR:31.19
+NMSE:0.0137
+SSIM:0.8919
+pSNR:31.02
+NMSE:0.0142
+R=2
+SSIM:0.9575
+pSNR:39.41
+NMSE:0.0013
+SSIM:0.9629
+pSNR:40.10
+NMSE:0.0011
+SSIM:0.9716
+pSNR:41.32
+NMSE:0.0008
+SSIM:0.9778
+pSNR:42.58
+NMSE:0.0006
+SSIM:0.9684
+pSNR:40.94
+NMSE:0.0009
+SSIM:0.9731
+pSNR:41.72
+NMSE:0.0007
+SSIM:0.9649
+pSNR:40.27
+NMSE:0.0010
+SSIM:0.9702
+pSNR:41.09
+NMSE:0.0009
+R=4
+SSIM:0.9019
+pSNR:35.18
+NMSE:0.0033
+SSIM:0.9123
+pSNR:35.80
+NMSE:0.0029
+SSIM:0.8753
+pSNR:33.41
+NMSE:0.0050
+SSIM:0.9086
+pSNR:35.73
+NMSE:0.0029
+SSIM:0.9168
+pSNR:35.84
+NMSE:0.0029
+SSIM:0.9296
+pSNR:36.83
+NMSE:0.0023
+SSIM:0.9128
+pSNR:35.93
+NMSE:0.0028
+SSIM:0.9226
+pSNR:36.52
+NMSE:0.0024
+R=8
+SSIM:0.8108
+pSNR:29.76
+NMSE:0.0116
+SSIM:0.8303
+pSNR:30.91
+NMSE:0.0089
+SSIM:0.8227
+pSNR:30.54
+NMSE:0.0097
+SSIM:0.8493
+pSNR:32.12
+NMSE:0.0067
+SSIM:0.8236
+pSNR:31.27
+NMSE:0.0082
+SSIM:0.8458
+pSNR:32.36
+NMSE:0.0064
+SSIM:0.8479
+pSNR:32.65
+NMSE:0.0059
+SSIM:0.8613
+pSNR:33.30
+NMSE:0.0051
+Figure 9: Scheme-specific vs multi-scheme setup visual comparison: Representative reconstructions of two samples
+from the fastMRI dataset. Measurements were subsampled with the four rectilinear schemes obtained from the
+scheme-specific and multi-scheme experiments for three acceleration factors.
+in order to determine the extend to which our findings hold true in prospective settings. To that end it will be necessary
+to apply one of the methods discussed in the previous paragraph to handle non-Cartesian data in order to incorporate it
+into our deep learning pipeline and match the multi-scheme framework used in this study.
+16
+
+文文文文文文文文文文Submitted to Magnetic Resonance in Medicine
+6
+Conclusion
+In summary, we experimentally showed that DL-based accelerated MRI reconstruction methods trained on data
+retrospectively subsampled with Cartesian non-rectilinear or simulated non-Cartesian trajectories, compared against
+Cartesian rectilinear trajectories, can produce more robust and truthful reconstructions allowing for high acceleration
+factors.
+Acknowledgements
+This work was funded by an institutional grant from the Dutch Cancer Society and the Dutch Ministry of Health,
+Welfare and Sport.
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+19
+
+Submitted to Magnetic Resonance in Medicine
+Supporting information
+Ground
+Truth
+SSIM:0.9645
+pSNR:37.30
+NMSE:0.0022
+Random
+Rectilinear
+R = 2
+SSIM:0.9672
+pSNR:38.25
+NMSE:0.0018
+Equispaced
+Rectilinear
+SSIM:0.9604
+pSNR:36.95
+NMSE:0.0024
+Equispaced+
+Rectilinear
+SSIM:0.9674
+pSNR:37.73
+NMSE:0.0020
+Gaussian 1D
+Rectilinear
+SSIM:0.9751
+pSNR:39.97
+NMSE:0.0012
+VDPD
+SSIM:0.9729
+pSNR:39.41
+NMSE:0.0014
+Gaussian 2D
+SSIM:0.9705
+pSNR:39.32
+NMSE:0.0014
+Radial
+SSIM:0.9709
+pSNR:39.62
+NMSE:0.0013
+Spiral
+SSIM:0.8912
+pSNR:30.53
+NMSE:0.0107
+R = 4
+SSIM:0.9197
+pSNR:32.06
+NMSE:0.0075
+SSIM:0.9132
+pSNR:31.79
+NMSE:0.0080
+SSIM:0.9081
+pSNR:31.33
+NMSE:0.0089
+SSIM:0.9561
+pSNR:36.78
+NMSE:0.0025
+SSIM:0.9537
+pSNR:36.36
+NMSE:0.0028
+SSIM:0.9530
+pSNR:36.31
+NMSE:0.0028
+SSIM:0.9530
+pSNR:36.64
+NMSE:0.0026
+SSIM:0.7998
+pSNR:26.84
+NMSE:0.0250
+R = 8
+SSIM:0.8172
+pSNR:27.50
+NMSE:0.0215
+SSIM:0.8357
+pSNR:28.05
+NMSE:0.0189
+SSIM:0.7997
+pSNR:26.43
+NMSE:0.0274
+SSIM:0.9393
+pSNR:34.55
+NMSE:0.0042
+SSIM:0.9274
+pSNR:33.51
+NMSE:0.0054
+SSIM:0.9245
+pSNR:32.89
+NMSE:0.0062
+SSIM:0.9245
+pSNR:33.39
+NMSE:0.0055
+Figure S1 : Representative reconstruction of a sample from the Calgary Campinas test set obtained from the scheme-
+specific experiments. For the ground truth image the RSS method was applied on the fully sampled k-space. For each
+subsampling scheme, we retrospectively applied subsampling masks as shown in rows 2, 4, 6 (with accelerations factors
+of 2, 4, 8, respectively) onto the fully sampled k-space. Rows 1, 3, 5 illustrate the RSS reconstructed k-space output
+from each model. Quantitative metrics against the ground truth are inscribed on the top left of each reconstruction.
+Note that the right 15% portion of the k-space was not collected and therefore subsampling masks were zero-filled
+accordingly.
+20
+
+-Submitted to Magnetic Resonance in Medicine
+Table S1 : Acquisition parameters and experiment splits per dataset used in our experiments.
+Dataset
+fastMRI Knee
+fastMRI Brain
+Calgary Campinas
+Field Strength
+1.5T & 3.0T
+1.5T & 3.0T
+3.0T
+Sequence
+PD 1 with and
+without fat suppression
+T1-w 2, T1-w post
+contrast, T2-w, FLAIR
+T1-w
+Subjects
+Healthy or
+Abnormality present
+Healthy or
+Pathology present
+Presumed
+healthy
+Acquisition
+Cartesian
+Cartesian
+Cartesian
+Fully Sampled
+Yes
+Yes
+Yes
+Subsampling
+Directions
+Phase and slice
+encoding directions
+Phase and slice
+encoding directions
+Phase and slice
+encoding directions
+No. Coils
+15
+2 - 24
+12
+No. Volumes
+1172
+5846
+67
+No. Slices
+41877
+92574
+10452
+Split size
+(No. volumes /
+No. slices)
+Training
+973 / 34742
+4469 / 70748
+40 / 6240
+Validation
+100 / 3587
+686 / 10880
+14 / 2184
+Test
+99 / 3548
+691 / 10946
+13 / 2028
+Proton Density
+Weighted
+Table S2 : Scheme-specific experiments average results on the test sets. Bold numbers indicate the best across
+subsampling schemes.
+Dataset
+Subsampling
+Scheme
+Acceleration Factor (R)
+2
+4
+8
+SSIM
+pSNR
+NMSE
+SSIM
+pSNR
+NMSE
+SSIM
+pSNR
+NMSE
+fastMRI
+Knee
+Random Rect.
+0.9482
+41.79
+0.0032
+0.9099
+38.04
+0.0066
+0.8594
+34.48
+0.0138
+Equispaced Rect.
+0.9439
+41.43
+0.0037
+0.9120
+38.37
+0.0062
+0.8643
+34.17
+0.0142
+Equispaced+ Rect.
+0.9449
+41.77
+0.0033
+0.9129
+38.48
+0.0060
+0.8658
+34.23
+0.0140
+Gaussian 1D
+0.9533
+42.39
+0.0028
+0.9187
+39.07
+0.0054
+0.8862
+36.16
+0.0094
+VDPD
+0.9656
+44.76
+0.0020
+0.9378
+41.54
+0.0037
+0.9176
+39.56
+0.0053
+Gaussian 2D
+0.9594
+43.53
+0.0024
+0.9347
+40.86
+0.0042
+0.9164
+39.21
+0.0057
+Radial
+0.9556
+43.27
+0.0026
+0.9347
+41.05
+0.0040
+0.9165
+39.26
+0.0054
+Spiral
+0.9570
+43.51
+0.0026
+0.9365
+41.23
+0.0040
+0.9189
+39.52
+0.0054
+fastMRI
+Brain
+Random Rect.
+0.9698
+42.74
+0.0024
+0.9436
+37.72
+0.0075
+0.9016
+32.26
+0.0266
+Equispaced Rect.
+0.9720
+43.69
+0.0020
+0.9495
+38.97
+0.0056
+0.9121
+33.53
+0.0194
+Equispaced+ Rect.
+0.9687
+42.95
+0.0023
+0.9490
+38.90
+0.0056
+0.9111
+33.34
+0.0198
+Gaussian 1D
+0.9729
+43.60
+0.0020
+0.9544
+39.87
+0.0045
+0.9289
+35.73
+0.0114
+VDPD
+0.9794
+45.90
+0.0012
+0.9664
+42.82
+0.0023
+0.9545
+40.56
+0.0039
+Gaussian 2D
+0.9763
+44.45
+0.0016
+0.9646
+42.01
+0.0028
+0.9526
+40.03
+0.0043
+Radial
+0.9730
+44.06
+0.0018
+0.9615
+41.70
+0.0030
+0.9503
+39.27
+0.0051
+Spiral
+0.9745
+44.50
+0.0016
+0.9635
+42.14
+0.0027
+0.9526
+40.14
+0.0042
+Calgary
+Campinas
+Random Rect.
+0.9634
+37.86
+0.0039
+0.9139
+32.61
+0.0137
+0.8387
+28.53
+0.0346
+Equispaced Rect.
+0.9737
+40.47
+0.0021
+0.9280
+33.77
+0.0100
+0.8541
+29.24
+0.0276
+Equispaced+ Rect.
+0.9682
+39.13
+0.0028
+0.9249
+33.49
+0.0105
+0.8495
+29.06
+0.0290
+Gaussian 1D
+0.9713
+39.34
+0.0027
+0.9158
+32.53
+0.0130
+0.8174
+27.59
+0.0399
+VDPD
+0.9811
+42.89
+0.0012
+0.9653
+38.95
+0.0029
+0.9486
+36.27
+0.0055
+Gaussian 2D
+0.9798
+42.27
+0.0013
+0.9633
+38.47
+0.0033
+0.9384
+35.13
+0.0071
+Radial
+0.9763
+41.37
+0.0017
+0.9596
+37.79
+0.0039
+0.9331
+34.29
+0.0087
+Spiral
+0.9765
+41.53
+0.0016
+0.9602
+38.09
+0.0036
+0.9337
+34.64
+0.0080
+21
+
+Submitted to Magnetic Resonance in Medicine
+-0
+0
+1
+1
+1
+SSIM (×100)
+R = 2
+-1
+-0
+1
+2
+3
+pSNR
+fastMRI Knee
+-2
+-1
+-0
+1
+2
+NMSE (×1e3)
+-1
+0
+1
+1
+2
+R = 4
+-1
+-0
+1
+2
+3
+-2
+-1
+0
+1
+2
+-1
+-1
+0
+1
+2
+R = 8
+-1
+-0
+1
+1
+2
+-9
+-6
+-3
+0
+3
+-1
+-0
+-0
+0
+1
+SSIM (×100)
+-1
+-1
+-0
+1
+1
+pSNR
+fastMRI Brain
+-1
+-1
+-0
+0
+1
+NMSE (×1e3)
+-1
+-0
+0
+1
+1
+-2
+-1
+0
+1
+2
+-3
+-2
+-1
+1
+2
+-1
+-0
+0
+1
+2
+-2
+-1
+0
+1
+2
+-23
+-14
+-6
+3
+11
+-2
+-1
+0
+1
+3
+SSIM (×100)
+-4
+-2
+1
+3
+5
+pSNR
+Calgary Campinas
+-4
+-2
+-0
+1
+3
+NMSE (×1e3)
+-4
+-1
+1
+4
+7
+-4
+-1
+1
+3
+6
+-15
+-9
+-3
+2
+8
+-5
+-1
+3
+7
+10
+-2
+-0
+2
+3
+5
+-34
+-22
+-10
+2
+14
+Subsampling
+Scheme
+Random
+Rectilinear
+Equispaced
+Rectilinear
+Equispaced+
+Rectilinear
+Gaussian 1D
+Rectilinear
+VDPD
+Gaussian 2D
+Radial
+Spiral
+Figure S2 : Scheme-specific setup vs multi-scheme setup quantitative results comparison. Violin-plots illustrate the
+evaluation metric differences between the reconstructed output of the model trained on all subsampling schemes
+(multi-scheme) and the reconstructed output of the model trained on a single scheme (scheme-specific). Let y, ˜yS be the
+fully-sampled and subsampled (with scheme S) k-space, respectively, and let u be the RSS-reconstruction of y (ground
+truth). Then, assuming that xS
+M and xS
+U are the image predictions output from the model trained on all subsampling
+schemes and the model trained on scheme S, respectively, we compute the difference m(u, xS
+M) − m(u, xS
+U), for each
+metric m. Positive and negative differences in the case of SSIM/pSNR and NMSE, respectively, indicate that the multi-
+scheme model, performed better than then scheme-specific model. For each dataset-acceleration-metric combination,
+pair-wise ASO significance tests were performed between the multi-scheme and scheme-specific quantitative results. ⋆
+indicates that the performance difference was not found to be significant (ϵmin > 0.5)
+22
+
+Submitted to Magnetic Resonance in Medicine
+Ground Truth
+R=2
+SSIM:0.9696
+pSNR:37.90
+NMSE:0.0024
+Unimodal
+SSIM:0.9731
+pSNR:38.33
+NMSE:0.0021
+Multimodal
+Random Rectilinear
+SSIM:0.9736
+pSNR:39.02
+NMSE:0.0018
+Unimodal
+SSIM:0.9755
+pSNR:39.17
+NMSE:0.0018
+Multimodal
+Equispaced Rectilinear
+SSIM:0.9675
+pSNR:37.86
+NMSE:0.0024
+Unimodal
+SSIM:0.9698
+pSNR:38.12
+NMSE:0.0022
+Multimodal
+Equispaced+ Rectilinear
+SSIM:0.9735
+pSNR:38.55
+NMSE:0.0020
+Unimodal
+SSIM:0.9754
+pSNR:39.06
+NMSE:0.0018
+Multimodal
+Gaussian 1D Rectilinear
+R=4
+SSIM:0.9273
+pSNR:33.22
+NMSE:0.0069
+SSIM:0.9408
+pSNR:34.14
+NMSE:0.0056
+SSIM:0.9192
+pSNR:32.37
+NMSE:0.0084
+SSIM:0.9314
+pSNR:33.06
+NMSE:0.0072
+SSIM:0.9194
+pSNR:31.85
+NMSE:0.0095
+SSIM:0.9301
+pSNR:32.87
+NMSE:0.0075
+SSIM:0.9084
+pSNR:30.98
+NMSE:0.0116
+SSIM:0.9257
+pSNR:32.19
+NMSE:0.0088
+R=8
+SSIM:0.8549
+pSNR:28.54
+NMSE:0.0203
+SSIM:0.8891
+pSNR:29.91
+NMSE:0.0148
+SSIM:0.8160
+pSNR:27.31
+NMSE:0.0270
+SSIM:0.8425
+pSNR:28.15
+NMSE:0.0222
+SSIM:0.8159
+pSNR:26.68
+NMSE:0.0312
+SSIM:0.8511
+pSNR:28.23
+NMSE:0.0218
+SSIM:0.7957
+pSNR:27.05
+NMSE:0.0287
+SSIM:0.8386
+pSNR:28.42
+NMSE:0.0209
+Figure S3 : Scheme-specific vs multi-scheme setups visual comparison. Representative reconstructions of a sample
+from the Calgary-Campinas dataset subsampled with the four rectilinear schemes obtained from the scheme-specific
+and multi-scheme experiments for three acceleration factors.
+23
+
+Submitted to Magnetic Resonance in Medicine
+93
+95
+96
+98
+99
+SSIM (×100)
+R = 2
+80
+85
+89
+94
+99
+R = 4
+65
+73
+82
+90
+99
+R = 8
+32
+36
+39
+43
+47
+pSNR
+25
+30
+34
+38
+43
+20
+25
+30
+35
+40
+0
+3
+5
+8
+10
+NMSE (×1e3)
+0
+11
+21
+32
+42
+0
+25
+50
+75
+100
+Subsampling
+Scheme
+Random
+Rectilinear
+Equispaced
+Rectilinear
+Equispaced+
+Rectilinear
+Gaussian 1D
+Rectilinear
+VDPD
+Gaussian 2D
+Radial
+Spiral
+Figure S4 : Model dependence experiments quantitative results on the Calgary-Campinas dataset using RIMs instead of
+RecurrentVarNets. For each subsampling scheme a distinct RIM was trained (in total 8 models). All models were built
+with identical hyperparameters and were trained to convergence. In each case we picked to evaluate the best model
+based on the validation performance on the SSIM metric. For each acceleration-metric combination, pair-wise ASO
+significance tests were performed between the average best (VDPD) performing and the rest schemes. ⋆ indicates that
+VDPD was not found to be significantly better (ϵmin > 0.5)
+Table S3 : Multi-scheme experiments average results on the test sets. Bold numbers indicate the best across subsampling
+schemes.
+Dataset
+Subsampling
+Scheme
+Acceleration Factor (R)
+2
+4
+8
+SSIM
+pSNR
+NMSE
+SSIM
+pSNR
+NMSE
+SSIM
+pSNR
+NMSE
+fastMRI
+Knee
+Random Rect.
+0.9494
+42.04
+0.0032
+0.9126
+38.33
+0.0063
+0.8699
+34.55
+0.0134
+Equispaced Rect.
+0.9520
+42.68
+0.0029
+0.9163
+38.78
+0.0058
+0.8758
+35.04
+0.0120
+Equispaced+ Rect.
+0.9456
+41.96
+0.0034
+0.9152
+38.78
+0.0059
+0.8721
+34.75
+0.0126
+Gaussian 1D
+0.9542
+42.60
+0.0028
+0.9206
+39.29
+0.0053
+0.8905
+36.55
+0.0088
+VDPD
+0.9653
+44.54
+0.0022
+0.9376
+41.39
+0.0039
+0.9181
+39.52
+0.0054
+Gaussian 2D
+0.9591
+43.38
+0.0025
+0.9340
+40.62
+0.0045
+0.9155
+38.98
+0.0062
+Radial
+0.9555
+43.26
+0.0027
+0.9348
+41.07
+0.0041
+0.9169
+39.27
+0.0055
+Spiral
+0.9567
+43.41
+0.0027
+0.9365
+41.25
+0.0040
+0.9196
+39.60
+0.0053
+fastMRI
+Brain
+Random Rect.
+0.9708
+43.05
+0.0022
+0.9466
+38.23
+0.0067
+0.9102
+33.23
+0.0215
+Equispaced Rect.
+0.9726
+43.87
+0.0019
+0.9505
+39.09
+0.0055
+0.9150
+33.78
+0.0183
+Equispaced+ Rect.
+0.9689
+42.97
+0.0023
+0.9489
+38.79
+0.0058
+0.9126
+33.44
+0.0193
+Gaussian 1D
+0.9729
+43.57
+0.0020
+0.9534
+39.62
+0.0048
+0.9264
+35.35
+0.0125
+VDPD
+0.9790
+45.76
+0.0012
+0.9660
+42.68
+0.0024
+0.9539
+40.44
+0.0039
+Gaussian 2D
+0.9758
+44.23
+0.0017
+0.9637
+41.79
+0.0029
+0.9518
+39.94
+0.0044
+Radial
+0.9732
+44.11
+0.0018
+0.9621
+41.78
+0.0029
+0.9516
+39.43
+0.0049
+Spiral
+0.9742
+44.43
+0.0016
+0.9638
+42.24
+0.0027
+0.9538
+40.23
+0.0041
+Calgary
+Campinas
+Random Rect.
+0.9684
+38.75
+0.0031
+0.9289
+33.65
+0.0104
+0.8631
+29.56
+0.0264
+Equispaced Rect.
+0.9756
+41.02
+0.0018
+0.9364
+34.57
+0.0083
+0.8709
+30.05
+0.0230
+Equispaced+ Rect.
+0.9707
+39.72
+0.0025
+0.9311
+34.06
+0.0093
+0.8681
+29.99
+0.0235
+Gaussian 1D
+0.9719
+39.83
+0.0024
+0.9316
+33.73
+0.0099
+0.8806
+30.45
+0.0220
+VDPD
+0.9809
+42.80
+0.0012
+0.9643
+38.70
+0.0031
+0.9465
+35.97
+0.0058
+Gaussian 2D
+0.9796
+42.20
+0.0014
+0.9626
+38.34
+0.0034
+0.9371
+34.97
+0.0074
+Radial
+0.9767
+41.47
+0.0016
+0.9594
+37.75
+0.0039
+0.9324
+34.22
+0.0088
+Spiral
+0.9774
+41.82
+0.0015
+0.9621
+38.46
+0.0033
+0.9385
+35.12
+0.0071
+24
+
+Submitted to Magnetic Resonance in Medicine
+Table S4 : Model dependence experiments average quantitative results on the Calgary-Campinas dataset using RIMs
+instead of RecurrentVarNets. Bold numbers indicate the best across subsampling schemes.
+Subsampling Scheme
+Acceleration Factor (R)
+2
+4
+8
+SSIM
+pSNR
+NMSE
+SSIM
+pSNR
+NMSE
+SSIM
+pSNR
+NMSE
+Random Rect.
+0.9592
+37.25
+0.0044
+0.9044
+31.81
+0.0159
+0.8182
+27.58
+0.0424
+Equispaced Rect.
+0.9633
+38.30
+0.0034
+0.9180
+32.93
+0.0119
+0.8307
+28.09
+0.0361
+Equispaced+ Rect.
+0.9687
+39.37
+0.0027
+0.9214
+33.21
+0.0112
+0.8411
+28.45
+0.0331
+Gaussian 1D
+0.9681
+38.78
+0.0030
+0.9180
+32.47
+0.0132
+0.8277
+27.86
+0.0382
+VDPD
+0.9768
+41.87
+0.0015
+0.9620
+38.36
+0.0033
+0.9431
+35.64
+0.0063
+Gaussian 2D
+0.9755
+41.29
+0.0017
+0.9600
+37.94
+0.0037
+0.9354
+34.85
+0.0076
+Radial
+0.9719
+40.60
+0.0020
+0.9568
+37.37
+0.0043
+0.9303
+33.93
+0.0094
+Spiral
+0.9727
+40.88
+0.0019
+0.9586
+37.83
+0.0038
+0.9343
+34.51
+0.0082
+25
+
diff --git a/l9E_T4oBgHgl3EQf6hzY/content/tmp_files/load_file.txt b/l9E_T4oBgHgl3EQf6hzY/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..79841e34e12c2de9e15ddf38e57ad9aad59ccdc1
--- /dev/null
+++ b/l9E_T4oBgHgl3EQf6hzY/content/tmp_files/load_file.txt
@@ -0,0 +1,2160 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf,len=2159
+page_content='ON RETROSPECTIVE k-SPACE SUBSAMPLING SCHEMES FOR  DEEP MRI RECONSTRUCTION  George Yiasemis  Netherlands Cancer Institute,  Amsterdam, 1066 CX,  the Netherlands  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='yiasemis@nki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='nl  Clara I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Sánchez  qurAI group  University of Amsterdam,  Amsterdam, 1012 WX,  the Netherlands  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='sanchezgutierrez@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='nl  Jan-Jakob Sonke  Netherlands Cancer Institute,  Amsterdam, 1066 CX,  the Netherlands  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='sonke@nki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='nl  Jonas Teuwen  Netherlands Cancer Institute,  Amsterdam, 1066 CX,  the Netherlands  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='teuwen@nki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='nl  January 23, 2023  ABSTRACT  Purpose: The MRI k-space acquisition is time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Traditional techniques aim to acquire  accelerated data, which in conjunction with recent DL methods, aid in producing high-fidelity images  in truncated times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Conventionally, subsampling the k-space is performed by utilizing Cartesian-  rectilinear trajectories, which even with the use of DL, provide imprecise reconstructions, though, a  plethora of non-rectilinear or non-Cartesian trajectories can be implemented in modern MRI scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  This work investigates the effect of the k-space subsampling scheme on the quality of reconstructed  accelerated MRI measurements produced by trained DL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Methods: The RecurrentVarNet was used as the DL-based MRI-reconstruction architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Carte-  sian fully-sampled multi-coil k-space measurements from three datasets with different accelerations  were retrospectively subsampled using eight distinct subsampling schemes (four Cartesian-rectilinear,  two Cartesian non-rectilinear, two non-Cartesian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Experiments were conducted in two frameworks:  Scheme-specific, where a distinct model was trained and evaluated for each dataset-subsampling  scheme pair, and multi-scheme, where for each dataset a single model was trained on data randomly  subsampled by any of the eight schemes and evaluated on data subsampled by all schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Results: In the scheme-specific setting RecurrentVarNets trained and evaluated on non-rectilinearly  subsampled data demonstrated superior performance especially for high accelerations, whilst in the  multi-scheme setting, reconstruction performance on rectilinearly subsampled data improved when  compared to the scheme-specific experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Conclusion: Training DL-based MRI reconstruction algorithms on non-rectilinearly subsampled  measurements can produce more faithful reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Our findings demonstrate the potential  for using DL-based methods trained on prospective acquisitions with non-rectilinearly subsampled  measurements to optimize scan time and image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Keywords Deep MRI Reconstruction, Retrospective k-space Subsampling, Non-rectilinear Subsampling, Non-  Cartesian Subsampling, Recurrent Variational Network  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='08365v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='IV]  20 Jan 2023  Submitted to Magnetic Resonance in Medicine  1  Introduction  Magnetic Resonance Imaging (MRI) is one of the most important imaging modalities in medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' MRI’s non-invasive  nature, non-use of ionizing radiation, and ability to produce high-resolution images make it a valuable technique for a  wide range of clinical applications, including diagnosis, treatment planning, and dynamic tasks such as MR-guided  surgery or radiotherapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' However, the application of MRI to dynamic tasks has been limited by the long acquisition  times required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' MRI measurements, known as the k-space, are acquired sequentially, resulting in prolonged scanning  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Over the past two decades, several methods have been put to use in clinical practice for accelerating the MRI acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  The two most conventionally applied methods to-date are Parallel Imaging (PI) and Compressed Sensing (CS), which  are usually both incorporated in modern state-of-the-art MRI scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Compressed Sensing aims in reconstructing images from subsampled k-space measurements [1, 2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Subsampling  the k-space is, in general, a violation of the Nyquist-Shannon sampling criterion [5] and reconstructions of subsampled  data are prone to producing aliasing artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' CS reconstruction algorithms attempt to solve minimization problems  such as Total Variation (TV) optimization [6] that given a sparse low-dimensional input signals, aim to reconstruct  high-dimensional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Parallel Imaging on the other hand, employs an array of multiple - instead of one - radio-frequency receiver coils which  measure reduced sets of spatially localised k-space frequencies while maintaining the same spatial resolution [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Each independent receiver coil receives distinct measurements corresponding to their spatial location in relation to  the scanned object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Hence, for each coil a unique sensitivity profile-map exists, that encodes its spatial sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Sensitivity maps are either known or estimated by performing a pre-scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In Figure 1 we provide an example of a PI  acquisition from a MRI scanner with nc = 16 coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Figure 1: Parallel Imaging: Acquisition of k-space measurements from a 16-coil array machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The reconstructed  image can be obtained by combining the individual coil reconstructions using the root-sum-of-squares (RSS) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  With the recent advancements in Deep Learning (DL) and Computer Vision (CV), a plethora of algorithms have emerged  targeting to solve imaging inverse problems, with Accelerated MRI Reconstruction being a par excellence example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Combined with CS and PI, numerous DL-based methods involving convolutional neural networks (CNNs) have been  proposed in the literature [10, 11, 12, 13, 14, 15, 16, 17] applied to the task of Accelerated MRI Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' These  methods are usually trained in a supervised manner using retrospectively subsampled (from available fully-sampled)  k-space datasets and their target is to make a prediction of the fully-sampled k-space or its image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Rectilinear Cartesian patterns constitute the most commonly employed (prospective) sampling and subsampling  techniques applied in clinical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Subsequently, DL-based Accelerated MRI Reconstruction applications utilize  rectilinear subsampling masks to retrospectively subsample fully-sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' However, a variety of prospective and  retrospective sampling and subsampling patterns exist with the majority of them not being rectilinear Cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For  instance, non-Cartesian patterns such as radial or spiral have been shown to be applied in real-time MRI acquisitions due  to the fact that they are less susceptible to motion compared to Cartesian ones [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The authors in [19] by employing a  deep neural network architecture, namely the Recurrent Inference Machine (RIM) [13], explored the effects of training  RIMs by applying either rectilinear or radial retrospective subsampling and concluded that the RIM trained using the  latter can produce higher-fidelity reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  In this work, we aim to investigate and compare the effects of employing miscellaneous retrospective subsampling  schemes on the quality of DL-based learned image reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' To that end, we trained and tested Recurrent  Variational Networks [10] (RecurrentVarNets) on retrospectively subsampled k-space measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' We performed  experiments under either scheme-specific or multi-scheme setups, in which models were trained and evaluated on data  subsampled with either individual and multiple, respectively, subsampling schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  2  (a) Individual coil k-space measurements (b) Individual coil reconstructions  (c) Individual coil sensitivity maps  (d) Reconstruction using all coil dataSubmitted to Magnetic Resonance in Medicine  The contributions and findings of our work can be summarized as follows:  We provide a review of eight currently employed (retrospective) subsampling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  We experimentally show that DL models trained and evaluated on non-rectilinearly, compared to rectilinearly,  subsampled data output superior reconstructions, especially for high acceleration factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  We demonstrate that models trained on data subsampled with multiple instead of individual patterns, can reconstruct  rectilinearly subsampled data with higher fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  2  Background - Theory  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1  MRI Acquisition  MRI reconstruction is an inverse problem on account of the fact that MR scanners acquire MRI measurements in the  frequency domain, also known as the Fourier space, and an inversion procedure is required to produce the desired MR  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Let n = nx × ny denote the spatial size of the reconstructed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In the case of single-coil acquisition, the relationship  between the underlying (vectorized) image x ∈ Cn and the (vectorized) single-channel k-space y ∈ Cn is given by  y = F(x) + e,  (1)  where F denotes the two-dimensional (Fast) Fourier Transform (FFT) and e ∈ Cn some measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1  Parallel MRI Acquisition  In PI multiple receiver coils are placed around the subject to speed up the acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Assuming a number of nc coils,  the acquired (multi-channel) k-space measurements are given by  y =  �  y1, · · · , ync�  ∈ Cn×nc,  yk = F(Skx) + ek,  k = 1, 2, · · · , nc,  (2)  where ek denotes noise measured by the kth coil and Sk ∈ Cn×n the sensitivity map of the kth coil expressed as a  diagonal complex matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Within each coil’s reception region, these maps encode their spatial sensitivity by measuring  the relative weighting of signals acquired from various locations around the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The sensitivity maps are usually  normalized as follows  nc  �  k=1  Sk∗Sk = In,  (3)  where Sk∗ indicates the complex conjugate of Sk and In ∈ Rn×n denotes the n-rank identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Obtaining an image from multi-channel measurements y can done by either using the root-sum-of-squares (RSS) or the  SENSE methods which operate as follows:  xrss = RSS  �ˆx1, · · · , ˆxnc�  = (  nc  �  k=1  | ˆxk |2)  1  2  (4)  and,  xsense =  ��SENSE  �ˆx1, · · · , ˆxnc��� =  ��  nc  �  k=1  Sk∗ˆxk�� ,  ˆxk = F−1(yk),  k = 1, 2, · · · , nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (5)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='2  Accelerated MRI Acquisition  To accelerate the MRI acquisition, in CS settings the k-space is subsampled by collecting fewer than necessary  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The subsampling procedure can be described as the application of a subsampling operator U on the  fully-sampled k-space measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The subsampled k-space is given by  ˜yk = Uyk = UF(Skx) + ˜ek,  k = 1, 2, · · · , nc,  (6)  3  Submitted to Magnetic Resonance in Medicine  where U ∈ {0, 1}n is expressed as a binary diagonal mask, and indicates which measurements are sampled as follows:  zU := (Uz)i =  �zi,  Uii = 1  0,  Uii = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (7)  The magnitude of the acceleration is determined by an acceleration factor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For a specific R, U can be chosen such  that  n ·  �  n  �  i=1  Uii  �−1 ≈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (8)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3  Accelerated MRI Reconstruction  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1  Sensitivity Map Estimation  The sensitivity maps S = (S1, · · · , Snc) can be estimated by various methods found in the literature [20, 8, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  A common method for estimating them is by fully-sampling a small region of the center of the k-space, also known as  the autocalibration signal (ACS) which includes low frequencies [11, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Let UACS ∈ {0, 1}n denote the ACS-subsampling operator such that when applied on k-space data it outputs the  fully-sampled ACS region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' :  zacs := (UACSz)i =  �zi,  i ∈ ACS region  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (9)  Subsequently, to obtain an initial approximation of the sensitivity maps we use the root-sum-of-squares (RSS) method:  ˜Sk ≈ diag  �  xk  acs ⊘ xacs  �  ,  k = 1, 2, · · · , nc,  (10)  where ⊘ denotes the element-wise division, and  xk  acs = F−1�  UACS˜yk�  ,  xacs = RSS  ��  xk  acs  �nc  k=1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (11)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='2  Accelerated MRI Reconstruction as an Inverse Problem  Obtaining a reconstruction from accelerated multicoil k-space measurements is an inverse problem, with a forward  problem given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' We can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 6 in a more compact notation:  ˜y = AU,S(x),  AU,S := U ◦ F ◦ ES  (12)  where AU,S : Cn → Cn×nc denotes the forward operator and ES : Cn → Cn×nc is called the expand operator which  maps an image w ∈ Cn to the individual coil images using S:  ES(w) =  �  S1w, · · · , Sncw  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (13)  The backward operator of AU,S is given by  A∗  U,S := RS ◦ F−1 ◦ U : Cn×nc → Cn,  (14)  where RS : Cn×nc → Cn is called the reduce operator that combines individual coil images z ∈ Cn×nc using S as  follows:  RS(z) =  nc  �  k=1  Sk∗zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (15)  Note that the operators U, F and F−1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 12 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 14 are applied coil-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Subsampling the k-space causes obtaining a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 12 to be an ill-posed inverse problem [23, 24], and therefore,  a solution through direct inversion is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Conventionally, in CS recovering an estimation of the ground truth  image x from the subsampled MRI measurements ˜y can be formulated as a solution to a variational optimization  problem as follows:  ˆx = argmin  w  ����AU,S(w) − ˜y  ����2  2 + α G(w),  (16)  where G : Cn → R is a regularization function which can impose prior information about the solution and α > 0 is a  regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In the literature various choices of G and algorithms for solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 16 have been employed  [25, 26, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  4  Submitted to Magnetic Resonance in Medicine  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3  Deep Learning-based Accelerated MRI Reconstruction  With the advent of the involvement of DL in MRI reconstruction tasks, the need for handcrafting a specific regularization  function has been replaced with CNN-based architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' A plethora of approaches solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 16 by unrolling it into a  gradient descent iterative optimization scheme over T time-steps:  wt+1 = wt − αt+1 A∗  U,S  �  AU,S(wt) − ˜y  �  + Hθt+1(wt),  t = 0, · · · , T − 1,  (17)  where αt denotes a (trainable) learning rate and Hθt a CNN-based architecture with trainable parameters θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The initial  image w0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 17 can be chosen as a zero-filled reconstruction using ˜y as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 4 or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Sensitivity maps can be estimated as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 10 and/or can be refined using another CNN-based model Sψ with trainable  parameters ψ which takes as input the estimation ˜S = (˜S1, · · · , ˜Snc) as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 10:  S = Sψ(˜S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (18)  Optimization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 17 may alternatively be performed in the k-space domain as demonstrated by some authors [10, 11]:  yt+1 = yt − αt+1 U  �  yt − ˜y  �  +  F ◦ ES ◦ Hθt+1 ◦ RS ◦ F−1�  yt  �  ,  y0 = ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (19)  The architecture we opted for is based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='4  k-space Sampling  (a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)  Figure 2: Top: k-space sampling trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Cartesian: (a) Rectilinear: k-space is filled in a line-by-line scheme, (b)  EPI: k-space is filled in a rectilinear way but in one shot ("zig-zag").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Non-Cartesian (c) Radial: k-space is filled with  radial spokes passing through the center, (d) Spiral: k-space is filled by one or multiple hellical curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Each line in  (a), (b) and (c) and each curve in (d) represents a separate filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The dark blue arrows show the direction of each  readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Bottom: Subsampled k-space trajectories for different acceleration factors: (e) Rectilinear, R ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' (f) EPI,  R ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' (g) Radial, R ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' (h) Spiral, R ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  A sampling scheme or k-space trajectory refers to the course of filling up a complex array with k-space frequencies  acquired over a sequence of time-steps during the MRI acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' There exist a wide range of k-space trajectories  implemented in clinical settings which can be split into two groups: Cartesian and non-Cartesian trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Figure  2(a)-(d) depicts two of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Cartesian trajectories aim in collecting samples on a Cartesian or equispaced and rectangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The most common  Cartesian trajectory is the rectilinear one in which k-space samples are acquired in a line-by-line scheme as illustrated  by Figure 2(a) with resulting samples being equidistant in both axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Note that usually in the literature the Cartesian  5  ASubmitted to Magnetic Resonance in Medicine  rectilinear trajectory is referred to as simply Cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In this work we use the characterization Cartesian to refer to  any trajectory acquired on a Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Other Cartesian trajectories include the Echo-planar imaging (EPI) in which  k-space lines are acquired in a rectilinear fashion but in a "zig-zag" pattern as shown by Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Non-Cartesian trajectories include schemes such as the radial or the spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In the former the k-space signal samples are  acquired along several spokes crossing its center (see Figure 2(c)) with a result the center being sampled multiple times,  while the latter includes acquiring data on single or multiple helical curves starting from the center of the k-space (see  Figure 2(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In non-Cartesian trajectories k-space measurements are acquired on a non-Cartesian grid and are therefore  not equidistant with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For instance, in the radial filling samples closer to the center are more dense compared  to samples further on the radial spokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  To accelerate the MRI acquisition, the k-space is subsampled by an acceleration factor R, resulting in fewer measure-  ments being collected than those strictly required by the Nyquist-Shannon criterion for perfect reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' This can  lead to a degradation of the quality of the reconstruction, depending on the magnitude of the acceleration factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For  instance, for R = 2, half of the necessary k-space measurements are acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In Figure 2(e)-(h) we provide examples  of subsampled k-space trajectories: Cartesian rectilinear and EPI (Figures 2(e)-(f)), radial and spiral (Figure 2(g)-(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  3  Methods  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1  Retrospective k-space Subsampling  Figure 3: Examples of subsampling masks for Cartesian data for acceleration factor R = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' (a)-(d) Rectilinear:  generated by first selecting a fraction of ACS columns and then the rest of the columns (a) are selected uniformly at  random, (b) are equispaced with a fixed distance, (c) are equispaced but symmetric, (d) are selected from the Gaussian  distribution (e) Variable-density Poisson Disk: generated by fully-sampling a centered disk for the ACS region and then  applying the Bridson’s fast algorithm [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' (f) Gaussian 2D: samples selected from a 2D Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' (g)-(h)  Simulated non-Cartesian using the CIRCUS algorithm [30]: (g) Radial, (h) Spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  The data used in this work consisted of volumes of fully-sampled raw k-space measurements acquired on a Cartesian  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' To simulate prospective subsampling we generated subsampling masks which we retrospectively applied onto  the fully-sampled multi-coil k-space data to produce subsampled/masked measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The generated subsampling  masks were binary signifying that a sample from the fully-sampled data was masked if and only if the corresponding  mask entry was zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  As in this work we are interested in studying the role of the subsampling pattern on the quality of DL-based recon-  structions of subsampled MRI data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' we focused on the following retrospective subsampling patterns on the Cartesian  grid:  6  (a) Random Rectilinear  (b) Equispaced Rectilinear  (c) Equispaced+ Rectilinear  (d) Gaussian 1D  (e) VDPD  (f) Gaussian 2D  (g) Radial  (h) SpiralSubmitted to Magnetic Resonance in Medicine  Cartesian Subsampling  Rectilinear: Achieved by including some and omitting other horizontal (phase encoding direction) lines on the  Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For the autocalibration region we use a number of racs · ny lines, where 0 < racs < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' We used four  distinct rectilinear sampling patterns:  – Random (Figure 3(a)): Lines were included uniformly at random with possible overlap with the ACS lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  – Equispaced (Figure 3(b)): Lines were included with a fixed distancing that satisfied the desired acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  – Equispaced+ (Figure 3(c)) : Improved rectilinear equispaced pattern by exploiting the k-space symmetry [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  – Gaussian 1D (Figure 3(d)): Lines were drawn from the Gaussian distribution with mean µ = ny  2 and standard  deviation σ = 4 · √µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Variable Density Poisson Disk (VDPD, Figure 3(e)): Combines both random sampling and denser center sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  For our implementation we followed Bridson’s algorithm in [29], which is a fast algorithm of order O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Samples  were drawn with a density  1  1 + s·|r|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' inversely proportional to the k-space radius r and a slope s which were  determined by the prescribed acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For the autocalibration signal we fully-sampled a centered disk with a  radius nx·ny·racs  π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Gaussian 2D (Figure 3(f)): Samples were drawn from the Gaussian distribution with mean µ = 1  2(nx, ny) and  covariance Σ = 4 · I2 · √µT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Code for VDPD and Gaussian 2D schemes was implemented in Cython for fast and efficient sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Simulated non-Cartesian Subsampling  To simulate non-Cartesian subsampling, we applied the CIRcular Cartesian  UnderSampling (CIRCUS) [30] technique to produce the following retrospective subsampling patterns for Cartesian  data:  Radial (Figure 3(g)): Simulates radial subsampling on the Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Spiral (Figure 3(h)): Simulates spiral subsampling on the Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  For randomization CIRCUS’ offset parameter as defined in [30] can be set to produce random radial and spiral patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  CIRCUS was modified to output masks by specifying the acceleration factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Note that in contrast to the rest of the  subsampling patterns above, for the non-Cartesian case we did not sample the ACS region exclusively, as these patterns  already fully-sample a great portion of the k-space center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Therefore, for the ACS subsampling mask Uacs we calculated  the largest sampled centered disk from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='2  Deep MRI-reconstruction Model Architecture  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1  The Recurrent Variational Network  To compare and evaluate the aforementioned subsampling techniques we employed a DL-based reconstruction network,  namely the Recurrent Variational Network [10] (RecurrentVarNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The RecurrentVarNet is a DL-based inverse  problem solver previously applied on the task of Accelerated MRI Reconstruction [10] with state-of-the-art performance  (MC-MRI reconstruction challenge winning solution [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' It iteratively solves the gradient descent scheme in  the measurements domain as portrayed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 19 using convolutional recurrent neural networks (ConvRNNs) as  a regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The RecurrentVarNet takes subsampled multi-coil k-space as input and outputs a prediction of the  fully-sampled multi-coil k-space measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' It comprises of three main modules:  Recurrent Variational Block (RecurrentVarNet Block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The RecurrentVarNet consists of T RecurrentVarNet  Blocks which are the main blocks of the RecurrentVarNet each responsible for performing an unrolled gradient  descent optimization time-step as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 19 by replacing Hθt with a recurrent unit denoted as RNNθt:  wt, ht+1 = RNNθt+1  �  RS ◦ F−1�  yt  �  , ht  �  ,  yt+1 = yt − αt+1 U  �  yt − ˜y  �  + F ◦ ES  �  wt  �  ,  y0 = ˜y,  t = 0, · · · , T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (20)  Each RNNθt is consisted of a convolutional layer (Conv) with a 5 × 5 kernel followed by nl cascades of alternating  Convs with a 3 × 3 kernel and convolutional gated recurrent units (ConvGRUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' A rectified linear unit is applied after  each Conv excluding the last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' RNNθt takes as input intermediate quantities of the image projection of the refined  k-space RS ◦ F−1�  yt−1  �  and the hidden state ht−1 from the previous time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  7  Submitted to Magnetic Resonance in Medicine  Recurrent State Initializer (RSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' It produces an initialization for the first hidden state h0 to be used by RNNθ1  provided as input the SENSE reconstruction of the image projection of y0:  h0 = RSI  �  SENSE  �  F−1(y0)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (21)  Sensitivity Estimation - Refinement (SER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The RecurrentVarNet for Accelerated MRI Reconstruction also es-  timates at each iteration the coil sensitivity maps as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' 10 and refines them using a U-Net [33] with trainable  parameters ψ denoted as Sψ:  S = SER  �˜S  �  :  Sk = Sψ  � ˜  Sk�  ,  k = 1, · · · , nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  (22)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3  Experimental Setup  Recurrent  Variational  Network  Subsampled k-space  Prediction of Fully  Sampled k-space  Fully Sampled k-space  Ground Truth Image  Image Prediction  Training &  Inference  Training  Select Subsampling Scheme  Figure 4: Experiments Pipeline: For each subsampling scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' here: radial) the fully-sampled multi-coil k-space  is retrospectively subsampled and used as input to a RecurrentVarNet which outputs a prediction of the fully-sampled  k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The predicted xT and ground truth ˆx images are produced by applying the RSS ◦ F−1 operator onto yT and  y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' During training, the loss L is calculated using xT and ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  To perform our experiments, we retrospectively subsampled the fully-sampled k-space data by generating subsampling  masks as introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' We carried out two classes of experiments: Scheme-specific and Multi-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' An  overview of our experimental setup is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1  Scheme-specific Setup  To compare the individual subsampling patterns and demonstrate their effect on the quality of DL-based reconstruction,  we first performed experiments in a scheme-specific setting: for each dataset-pattern pair we ran individual experiments  by training and evaluating (twenty-four) distinct RecurrentVarNets (with the same choice of hyper-parameters as  outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='2  Multi-scheme Setup  In the multi-scheme setting, our goal was twofold: Firstly, we aimed to investigate further the effect of each subsampling  scheme on the quality of DL-based reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Secondly and most importantly, we wanted to asses whether or not a  DL-based model trained in a multi-scheme fashion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' training measurements subsampled with multiple subsampling  patterns) demonstrated higher reconstruction performance compared to being trained in a scheme-specific fashion (as in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Therefore, for each dataset a RecurrentVarNet was trained on data arbitrarily subsampled with any of  the presented subsampling schemes in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1 and evaluated on all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='.Submitted to Magnetic Resonance in Medicine  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3  Subsampling  In both, scheme-specific and multi-scheme settings, throughout the training phase, subsampling masks were generated  with an acceleration factor of R = 2, 4 or 8, and were retrospectively applied onto the fully-sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' At validation  and testing times, data were 2-fold, 4-fold, and 8-fold retrospectively subsampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  For the Cartesian masks we set racs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='04 for R = 2, 4 or 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Note that for all experiments, subsampling masks during training were randomly generated to maximize the amount of  data that each model saw, while for validation and testing they were predetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Additionally, the same subsampling  mask was applied to all slices of each volume during validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' It should also be highlighted that all  individual coil data of each sample were subsampled with the same subsampling mask as this reflected clinical  subsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='4  Model Implementation Details  Hyper-parameter Choice  For the RecurrentVarNets we used T = 8 time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each RecurrentVarNet Block  we used nl = 4 alternating cascades and for the number of filters in each Conv and hidden size in each ConvGRU  we chose nf = 256 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For the implementation of the RSI and SER modules we picked the same choice of  hyper-parameters as the original paper [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Training & Optimization Details  For training and optimization we utilized PyTorch [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' All components of the  Recurrent Variational Network were trained end-to-end and optimization was performed utilizing the Adam algorithm  with coefficients (β1, β2) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='999), stability parameter ϵ = 1 × 10−8 and no weight decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Experiments were performed utilizing NVIDIA RTX A6000, Quadro RTX 8000, or A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Models were trained  to convergence with a batch size of 1 slice multi-coil k-space data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The total number of trainable parameters for each  model amounted to approximately 27,626k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Training Loss Function  At each training iteration each model was fed with subsampled multi-coil k-space measure-  ments ˜y and produced a prediction yT of the fully-sampled reference k-space y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Loss was computed in the image  domain using xT = RSS ◦ F−1(yT ) as the image prediction and ˆx = RSS ◦ F−1(y) as the image reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  As a loss function we used a combination of the mean average error (MAE) loss LMAE and the structural similarity  index measure (SSIM) loss LSSIM:  L(ˆx, xT ) = LMAE(ˆx, xT ) + LSSIM(ˆx, xT )  =  ����ˆx − xT  ����  1 +  �  1 − SSIM(ˆx, xT )  �  ,  (23)  where SSIM is defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5  Datasets  To perform our experiments we employed three open source datasets, the fastMRI knee and brain datasets [35] which  are to-date the largest publicly available MRI datasets, and the Calgary-Campinas (CC) brain dataset which was released  as part of the Multi-Coil MRI Reconstruction Challenge [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' All datasets consist of raw three dimensional k-space  volumes which are multi-coil and fully-sampled conventional Cartesian acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The acquisition parameters and  the splitting ratios we opted for in our experiments are summarized in Supporting Information Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='6  Quality Analysis  Metrics  To analyse and compare our results we employed three evaluation metrics commonly used in image process-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Assuming u ∈ Rn was the ground truth image and v ∈ Rn the prediction, they are defined as follows:  A) Structural Similarity Index Measure (SSIM)  SSIM(u, v) = 1  M  M  �  i=1  (2µuiµvi + c1)(2σuivi + c2)  (µ2ui + µ2vi + c1)(σ2ui + σ2vi + c2),  (24)  where ui, vi, i = 1, · · · , M are image windows of size (wx, wy) = (7, 7) from u and v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The numbers  µui, µvi, σui and σvi denote the means and standard deviations of each image window and σuivi denotes the covariance  between ui and vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The constants c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='01 and c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='03 are used for numerical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  9  Submitted to Magnetic Resonance in Medicine  B) Peak Signal-to-Noise Ratio (pSNR)  pSNR(u, v) = 20 log10  �  max(u)  �  1  n  �n  i (ui − vi)2  �  (25)  C) Normalized Mean Squared Error (NMSE)  NMSE(u, v) = ||u − v||2  2  ||u||2  2  =  �n  i (ui − vi)2  �n  i u2  i  (26)  Note that the higher the computed SSIM and pSNR values are, the higher the quality of the reconstruction is, whereas  for the NMSE values, the lower they are the lower the quality of the reconstruction is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Reported values for SSIM and  NMSE are multiplied by 100 and 1000, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='7  Significance Tests  To perform significance tests we used the Almost Stochastic Order (ASO) test [36, 37] with a 95% confidence level  (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Each ASO test outputs a violation error ϵmin which denotes the degree to which the hypothesis that "method  A is always better than method B" is being violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' If ϵmin ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5 one can claim that A is better than B, and otherwise  if ϵmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  4  Results  To obtain our results, data preparation, retrospective subsampling generation, and model training we used the Deep  Image Reconstruction Toolkit (DIRECT) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1  Scheme-specific Setup Results  Figure 5 illustrates the quantitative metrics computed on the test sets in the form of violin-plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The average metrics are  reported in Supporting Information Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Additionally, for visual assessment, in Figure 6, Figure 7, and Supporting  Information Figure S1 we present example reconstructions of a test sample from each dataset using all methods along  with ground truths and the retrospective subsampling mask used in each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1  Cartesian  Rectilinear  Evidently, Figure 5 indicates that all models trained on rectilinearly subsampled measurements achieved  comparable results when evaluated on the test sets subsampled with the respective rectilinear schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Moreover,  although Gaussian 1D subsampling outperformed the rest of the rectilinear schemes in the case of both fastMRI datasets,  the opposite was noted in the case of the CC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' This is also visible by the example reconstructions in Figures 6, 7,  and Supporting Information Figure S1, especially for 8-fold acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Non-Rectilinear  As Figure 5 and Supporting Information Table S2 indicate, models trained on data subsampled  with either VDPD or Gaussian 2D schemes were the best-performing models when evaluated on the respective data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Additionally, they produced the higher average SSIM and pSNR and lower NMSE reconstructions for all combinations  of datasets and acceleration factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='2  Non-Cartesian  As shown in Figure 5 and Supporting Information Table S2, models trained and evaluated on simulated non-Cartesian  (radially and spirally) subsampled data yielded similar performance to Cartesian non-rectilinear schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In general,  spiral schemes outperformed radial schemes for all combinations of datasets and acceleration factors, though the  difference in performance was minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3  Comparisons  Considering Figure 5, models trained on non-rectilinearly subsampled data, produced reconstructions of higher fidelity  when evaluated on the respective test sets in comparison to models trained and evaluated on rectilinearly subsampled  data for all datasets and acceleration factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Models trained with VDPD-subsampled k-spaces obtained the best average  quantitative evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Submitted to Magnetic Resonance in Medicine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='81  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='87  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='94  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='SSIM (×100)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='R = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='29  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='36  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='51  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='59  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='pSNR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='fastMRI Knee  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='91  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='99  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='36  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='46  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='69  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='92  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Subsampling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Scheme  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Random  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Equispaced  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Equispaced+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Gaussian 1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='VDPD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Gaussian 2D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Radial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Spiral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Figure 5: Scheme-specific experiments quantitative results on the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each dataset-subsampling scheme pair  a distinct model was trained (in total 24 models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each dataset-acceleration-metric combination, pair-wise ASO  significance tests were performed between the average best performing (VDPD) and the rest schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' ⋆ indicates that  VDPD was not found to be significantly better (ϵmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Average metrics are reported in Supporting Information  Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  11  Submitted to Magnetic Resonance in Medicine  Ground  Truth  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9218  pSNR:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0041  Random  Rectilinear  R = 2  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0031  VDPD  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0087  Figure 6: Representative reconstruction of a sample from the fastMRI knee test set obtained from the scheme-specific  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For the ground truth image the RSS method was applied on the fully sampled k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each  subsampling scheme, we retrospectively applied subsampling masks as shown in rows 2, 4, 6 (with accelerations factors  of 2, 4, 8, respectively) onto the fully sampled k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Rows 1, 3, 5 illustrate the center-cropped RSS-reconstructed  k-space output from each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Quantitative metrics against the ground truth are inscribed on the top left of each  reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  As apparent from Figure 6, Figure 7, and Supporting Information Figure S1, for high acceleration factors (4 or 8)  models trained on rectilinear schemes were more prone to reconstructing images with more errors and artifacts, in  contrast to non-Cartesian or non-rectilinear schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' However, for R = 2 all models evaluated closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='2  Multi-scheme Setup Results  Figure 8 reports the quantitative evaluation results on the test sets in the multi-scheme setup and Supporting Information  Table S3 the corresponding average metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' We observe that, similarly to the results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1, reconstructions  of non-rectilinearly subsampled measurements produced better quantitative results compared to reconstructions of  rectilinearly subsampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In addition, VDPD and Gaussian 2D-subsampled reconstructions were the highest  performing in average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  For further investigation, in Supporting Information Figure S2, utilizing the results reported in Figure 8, we calculated  the per-case difference in evaluation metrics change for each pattern using as reference the results obtained in the  scheme-specific setup (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' We also report the average differences for rectilinear and non-rectilinear patterns in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Interestingly, models trained in the multi-scheme setting exemplified superior performance when evaluated on  measurements subsampled with rectilinear schemes compared against the models trained on individual schemes when  evaluated on the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In particular, in the case of rectilinear schemes, noticeable improvements (SSIM/pSNR  increase, NMSE decrease) on the reconstruction performance were remarked for all datasets and acceleration factors,  whilst for non-rectilinear patterns no change or minor deterioration was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Submitted to Magnetic Resonance in Medicine  Ground  Truth  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9765  pSNR:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='60  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0032  Random  Rectilinear  R = 2  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9780  pSNR:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='45  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0027  Equispaced  Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9732  pSNR:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='03  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0037  Equispaced+  Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9801  pSNR:44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='05  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0023  Gaussian 1D  Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9779  pSNR:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='96  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0015  VDPD  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9815  pSNR:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='44  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0017  Gaussian 2D  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9669  pSNR:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='87  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0024  Radial  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9634  pSNR:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='73  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0025  Spiral  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9577  pSNR:37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='73  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0100  R = 4  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9619  pSNR:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='80  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0078  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9599  pSNR:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='49  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0084  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9641  pSNR:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='51  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0066  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9677  pSNR:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='33  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0027  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9737  pSNR:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='04  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0029  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9507  pSNR:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='24  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0044  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9434  pSNR:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='95  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0047  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9375  pSNR:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='93  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0239  R = 8  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9339  pSNR:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='04  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0293  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9345  pSNR:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='30  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0276  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9444  pSNR:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='62  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0204  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9613  pSNR:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='26  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0044  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9652  pSNR:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='06  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0046  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9400  pSNR:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='19  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0071  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9333  pSNR:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='48  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0066  Figure 7: Representative reconstruction of a T2-weighted sample from the fastMRI brain test set obtained from  the scheme-specific experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For the ground truth image the RSS method was applied on the fully sampled  k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each subsampling scheme, we retrospectively applied subsampling masks as shown in rows 2, 4, 6 (with  accelerations factors of 2, 4, 8, respectively) onto the fully sampled k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Rows 1, 3, 5 illustrate the center-cropped  RSS-reconstructed k-space output from each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Quantitative metrics against the ground truth are inscribed on the  top left of each reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  In Figure 9 and Supporting Information Figure S3 we plot representative reconstructions predicted from rectilinearly-  subsampled data obtained from both, the scheme-specific and multi-scheme setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' By visual investigation, we notice  that in the latter reconstructed images were more faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3  Results - Model Dependence  To demonstrate that the results presented above did not rely on the choice of the model architecture we further  experimented by employing an additional state-of-the-art deep-MRI reconstruction network: the Recurrent Inference  Machine (RIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' More precisely, we repeated the scheme-specific experiments on the CC dataset by replacing the  RecurrentVarNet with a RIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' The choice of hyper-parameters for each RIM was identical as in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  In Supporting Information Figure S4 and Table S4 are presented the quantitative metrics for our RIM experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Results were on par with the findings of our original experiments, suggesting that our conclusions were not dependent  on the choice of DL architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  5  Discussion  In this work we investigated and compared various retrospective k-space subsampling patterns and we experimentally  studied their effect on the quality of DL-based reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Since the data we utilized in our studies were Cartesian  fully-sampled acquisitions, we retrospectively generated subsampling masks on the Cartesian grid as demonstrated in  13  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='八福Submitted to Magnetic Resonance in Medicine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='81  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='87  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='94  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='SSIM (×100)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='R = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='29  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='36  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='67  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Subsampling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Scheme  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Random  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Equispaced  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Equispaced+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Gaussian 1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='VDPD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Gaussian 2D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Radial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Spiral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Figure 8: Multi-scheme experiments quantitative results on the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each dataset-acceleration-metric combi-  nation, pair-wise ASO significance tests were performed between the average best performing (VDPD) and the rest  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' ⋆ indicates that VDPD was not found to be significantly better (ϵmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Average metrics are reported in  Supporting Information Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  14  Submitted to Magnetic Resonance in Medicine  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='1, simulating prospective Cartesian or non-Cartesian accelerated acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In particular, we generated four  Cartesian rectilinear schemes (random, equispaced, equispaced with symmetry correction, Gaussian), non-rectilinear  Cartesian schemes (Variable Density Poisson-disk and Gaussian 2D), and simulated (non-Cartesian) radial and spiral  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Our experiments consisted of utilizing a state-of-the-art DL-based accelerated MRI reconstruction method - the  Recurrent Variational Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Although choosing the optimal DL-based MRI reconstruction algorithm was out of the  scope of this project, it is important to note that other DL-based models do exist with similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' However,  the RecurrentVarNet was chosen as it has previously shown to produce high-fidelity reconstructions and outperform  other models in reconstruction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Additionally, to demonstrate that the results were not model-dependent,  the scheme-specific experiments on the CC dataset were repeated using the RIM model which produced on-par results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Experiments were performed under two setups: scheme-specific and multi-scheme setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In the scheme-specific  settings, we trained individual models on data retrospectively subsampled with individual subsampling patterns applying  various acceleration factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Quantitative and qualitative results demonstrated that the models trained on conventional  rectilinear schemes, in contrast to other schemes, produced lower quality reconstructions with more artifacts especially  for higher acceleration factors (4 or 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' This can be attributed to the fact that non-rectilinear Cartesian such as the  VDPD or the Gaussian 2D, and non-Cartesian patterns such as the radial or spiral, allow for more incoherent sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  This means that these schemes result in a more randomized and less correlated distribution of samples, reducing the  dependence of the reconstruction quality on any specific pattern of missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Additionally, these schemes allow for  center oversampling, which contains information such as contrast and the general shape of the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  In the multi-scheme setup, unified models were trained on all distributions of subsampled patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Although quantitative  results were on par with the results of our scheme-specific experiments, we observed noticeable improvements (compared  to the scheme-specific results) of the reconstruction inference performance for rectilinearly subsampled measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Additionally, the violin-plots in Figure 5 and Figure 8 show that for most dataset-metric-acceleration combinations  quantitative results for the four rectilinear and the four non-rectilinear schemes formed similar metric distributions and  were with similar numbers of outliers, suggesting that results were also case-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Even though comparing prospective acquisition speeds was out of the scope of this project, we provide a brief  discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' While the reconstruction performance of Cartesian rectilinearly subsampled data was inferior to Cartesian  non-rectilinear, someone could argue about the trade-off between acquisition speed and quality, as MRI scanners  can perform rectilinear sampling in fast acquisition times [1], while strategies such as VDPD or Gaussian 2D can be  slower due to physical limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For instance, in the prospective case these strategies may require large gradient  switches in the MRI scanner due to the enlarged k-space spacing which can cause extended times due to hardware  constraints, whilst in the retrospective settings, efficient algorithms are used to pick the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' On the other hand, our  results indicated that synthesized radial or spiral schemes on the Cartesian grid using the CIRCUS method provided  similar performance to VDPD and Gaussian 2D schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In the original paper [30], the authors state that their method  improved sampling efficiency over VDPD while maintaining the reconstruction performance in the prospective case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  However, prospective non-Cartesian sampling trajectories do not sample on a Cartesian grid, and therefore samples  closer to the center are more densely placed, and more scattered far off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' As a result of this non-uniformity, for the  application of the FFT a gridding and inverse-gridding process [39] is required to place them onto a Cartesian grid  accumulating additional computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Note that gridding can be replaced with the application of the non-uniform  FFT (NUFFT) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  The main limitation of this study is the fact that experiments were performed retrospectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In our future work, we  will repeat our experiments using prospectively subsampled data with the different schemes we employed in this study  Table 1: Scheme-specific-Multi-scheme experiments performance average percentage difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Percentages were  acquired by averaging the per-case differences (as calculated and illustrated in Supporting Information Figure S2) for  all dataset-scheme type (rectilinear or non-rectilinear) combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Dataset  Type of  Subsampling  Scheme  Acceleration Factor (R)  2  4  8  SSIM  pSNR  NMSE  SSIM  pSNR  NMSE  SSIM  pSNR  NMSE  FastMRI Knee  Rectilinear  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0051  Figure 9: Scheme-specific vs multi-scheme setup visual comparison: Representative reconstructions of two samples  from the fastMRI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Measurements were subsampled with the four rectilinear schemes obtained from the  scheme-specific and multi-scheme experiments for three acceleration factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  in order to determine the extend to which our findings hold true in prospective settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' To that end it will be necessary  to apply one of the methods discussed in the previous paragraph to handle non-Cartesian data in order to incorporate it  into our deep learning pipeline and match the multi-scheme framework used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  16  文文文文文文文文文文Submitted to Magnetic Resonance in Medicine  6  Conclusion  In summary, we experimentally showed that DL-based accelerated MRI reconstruction methods trained on data  retrospectively subsampled with Cartesian non-rectilinear or simulated non-Cartesian trajectories, compared against  Cartesian rectilinear trajectories, can produce more robust and truthful reconstructions allowing for high acceleration  factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Acknowledgements  This work was funded by an institutional grant from the Dutch Cancer Society and the Dutch Ministry of Health,  Welfare and Sport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='  18  Submitted to Magnetic Resonance in Medicine  [36] Rotem Dror, Segev Shlomov, and Roi Reichart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Deep dominance - how to properly compare deep neural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 2773–2785,  Florence, Italy, July 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  [37] Dennis Ulmer, Christian Hardmeier, and Jes Frellsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' deep-significance-easy and meaningful statistical signifi-  cance testing in the age of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='06815, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  [38] George Yiasemis, Nikita Moriakov, Dimitrios Karkalousos, Matthan Caan, and Jonas Teuwen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Direct: Deep  image reconstruction toolkit, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Jackson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Meyer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Nishimura, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Macovski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Selection of a convolution function for fourier  inversion using gridding (computerised tomography application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' IEEE Transactions on Medical Imaging,  10(3):473–478, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  [40] Karsten Fourmont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Non-equispaced fast fourier transforms with applications to tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Journal of Fourier  Analysis and Applications, 9(5):431–450, sep 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  19  Submitted to Magnetic Resonance in Medicine  Supporting information  Ground  Truth  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9645  pSNR:37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='30  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0022  Random  Rectilinear  R = 2  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9672  pSNR:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='25  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0018  Equispaced  Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0024  Equispaced+  Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0020  Gaussian 1D  Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0012  VDPD  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0014  Radial  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='0055  Figure S1 : Representative reconstruction of a sample from the Calgary Campinas test set obtained from the scheme-  specific experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For the ground truth image the RSS method was applied on the fully sampled k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each  subsampling scheme, we retrospectively applied subsampling masks as shown in rows 2, 4, 6 (with accelerations factors  of 2, 4, 8, respectively) onto the fully sampled k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Rows 1, 3, 5 illustrate the RSS reconstructed k-space output  from each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Quantitative metrics against the ground truth are inscribed on the top left of each reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Note that the right 15% portion of the k-space was not collected and therefore subsampling masks were zero-filled  accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  20  Submitted to Magnetic Resonance in Medicine  Table S1 : Acquisition parameters and experiment splits per dataset used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Dataset  fastMRI Knee  fastMRI Brain  Calgary Campinas  Field Strength  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5T & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0T  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5T & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0T  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Subsampling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Scheme  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Random  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Equispaced  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Equispaced+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Gaussian 1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='VDPD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Gaussian 2D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Radial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Spiral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Figure S2 : Scheme-specific setup vs multi-scheme setup quantitative results comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Violin-plots illustrate the  evaluation metric differences between the reconstructed output of the model trained on all subsampling schemes  (multi-scheme) and the reconstructed output of the model trained on a single scheme (scheme-specific).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Let y, ˜yS be the  fully-sampled and subsampled (with scheme S) k-space, respectively, and let u be the RSS-reconstruction of y (ground  truth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Then, assuming that xS  M and xS  U are the image predictions output from the model trained on all subsampling  schemes and the model trained on scheme S, respectively, we compute the difference m(u, xS  M) − m(u, xS  U), for each  metric m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Positive and negative differences in the case of SSIM/pSNR and NMSE, respectively, indicate that the multi-  scheme model, performed better than then scheme-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each dataset-acceleration-metric combination,  pair-wise ASO significance tests were performed between the multi-scheme and scheme-specific quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' ⋆  indicates that the performance difference was not found to be significant (ϵmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5)  22  Submitted to Magnetic Resonance in Medicine  Ground Truth  R=2  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9696  pSNR:37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='90  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0024  Unimodal  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9731  pSNR:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='33  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0021  Multimodal  Random Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9736  pSNR:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='02  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0018  Unimodal  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9755  pSNR:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='17  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0018  Multimodal  Equispaced Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9675  pSNR:37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='86  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0024  Unimodal  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9698  pSNR:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='12  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0022  Multimodal  Equispaced+ Rectilinear  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9735  pSNR:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='55  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0020  Unimodal  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9754  pSNR:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='06  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0018  Multimodal  Gaussian 1D Rectilinear  R=4  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9273  pSNR:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='22  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0069  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9408  pSNR:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='14  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0056  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9192  pSNR:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='37  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0084  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9314  pSNR:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='06  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0072  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9194  pSNR:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='85  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0095  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9301  pSNR:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='87  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0075  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9084  pSNR:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='98  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0116  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='9257  pSNR:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='19  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0088  R=8  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='8549  pSNR:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='54  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0203  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='8891  pSNR:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='91  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0148  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='8160  pSNR:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='31  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0270  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='8425  pSNR:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='15  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0222  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='8159  pSNR:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='68  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0312  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='8511  pSNR:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='7957  pSNR:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='05  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0287  SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='8386  pSNR:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='42  NMSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0209  Figure S3 : Scheme-specific vs multi-scheme setups visual comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Representative reconstructions of a sample  from the Calgary-Campinas dataset subsampled with the four rectilinear schemes obtained from the scheme-specific  and multi-scheme experiments for three acceleration factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Submitted to Magnetic Resonance in Medicine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='93  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='95  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='96  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='98  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='99  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='SSIM (×100)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='R = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='85  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='89  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='94  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='99  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='R = 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='65  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='73  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='82  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='R = 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='38  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='NMSE (×1e3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Subsampling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Scheme  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Random  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Equispaced  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Equispaced+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Gaussian 1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Rectilinear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='VDPD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Gaussian 2D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Radial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Spiral  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='Figure S4 : Model dependence experiments quantitative results on the Calgary-Campinas dataset using RIMs instead of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='RecurrentVarNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each subsampling scheme a distinct RIM was trained (in total 8 models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' All models were built  with identical hyperparameters and were trained to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' In each case we picked to evaluate the best model  based on the validation performance on the SSIM metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' For each acceleration-metric combination, pair-wise ASO  significance tests were performed between the average best (VDPD) performing and the rest schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' ⋆ indicates that  VDPD was not found to be significantly better (ϵmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='5)  Table S3 : Multi-scheme experiments average results on the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content=' Bold numbers indicate the best across subsampling  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
+page_content='  Dataset  Subsampling  Scheme  Acceleration Factor (R)  2  4  8  SSIM  pSNR  NMSE  SSIM  pSNR  NMSE  SSIM  pSNR  NMSE  fastMRI  Knee  Random Rect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E_T4oBgHgl3EQf6hzY/content/2301.08365v1.pdf'}
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diff --git a/l9FPT4oBgHgl3EQf3jX-/content/tmp_files/2301.13191v1.pdf.txt b/l9FPT4oBgHgl3EQf3jX-/content/tmp_files/2301.13191v1.pdf.txt
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+Prepared for submission to JCAP
+Minkowski Functionals in SO(3) for
+the spin–2 CMB polarisation field
+J. Carrón Duque,a,b,1 A. Carones,a,b D. Marinucci,c M.
+Migliaccio,a,b and N. Vittorioa,b
+aDipartimento di Fisica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1,
+I-00133, Roma, Italy
+bSezione INFN Roma 2, via della Ricerca Scientifica 1, I-00133, Roma, Italy
+cDipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica
+1, I-00133, Roma, Italy
+E-mail: javier.carron@roma2.infn.it, alessandro.carones@roma2.infn.it,
+marinucc@mat.uniroma2.it, marina.migliaccio@roma2.infn.it,
+nicola.vittorio@uniroma2.it
+Abstract. The study of the angular power spectrum of Cosmic Microwave Background
+(CMB) anisotropies, both in intensity and in polarisation, has led to the tightest constraints
+on cosmological parameters. However, this statistical quantity is not sensitive to any devi-
+ation from Gaussianity and statistical isotropy in the CMB data. Minkowski Functionals
+(MFs) have been adopted as one of the most powerful statistical tools to study such devi-
+ations, given that they characterise the topology and geometry of the field of interest. In
+this paper, we extend the application of MFs to CMB polarisation data by introducing a
+new formalism, where we lift the spin 2 polarisation field to a scalar function in a higher
+dimensional manifold: the group of rotations of the sphere, SO(3). Such function is defined
+as f = Q cos(2ψ) − U sin(2ψ). We analytically obtain the expected values for the MFs of f
+in the case of Gaussian isotropic polarisation maps. Furthermore, we present a new pipeline
+which estimates these MFs from input HEALPix polarisation maps. We apply it to CMB
+simulations in order to validate the theoretical results and the methodology. The pipeline is
+to be included in the publicly available Python package Pynkowski.
+Keywords: CMBR polarisation – non-gaussianity
+1Corresponding author.
+arXiv:2301.13191v1  [astro-ph.CO]  30 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Spin field as a scalar field in SO(3)
+2
+3
+Minkowski Functionals
+4
+3.1
+Definition
+4
+3.2
+Gaussian Kinematic Formula
+5
+3.3
+Theoretical predictions
+7
+4
+Implementation: Pynkowski
+10
+4.1
+General considerations
+13
+5
+Simulations
+13
+5.1
+Monochromatic maps
+13
+5.2
+Realistic angular power spectrum
+14
+6
+Results
+14
+6.1
+Normalisation constants
+14
+6.2
+Scaling relations
+14
+6.3
+Gaussian CMB simulations
+15
+7
+Conclusions
+17
+A Metric and derivatives in SO(3)
+21
+1
+Introduction
+The Cosmic Microwave Background (CMB) encodes information from the Early Universe,
+both in the intensity and polarisation of the light.
+The CMB is most commonly studied
+through the angular power spectra (equivalently, 2–points correlation functions). However,
+this tool is not sensitive to the possible presence of non–Gaussianities or departures from
+statistical isotropy of the CMB anisotropy fields.
+Non–Gaussianity is predicted by many inflationary models [1–3] and could shed a new
+light on our knowledge of the primordial Universe. There is also a growing amount of literature
+on a possible large–scale anisotropy of the Universe, with dipoles being measured in several
+observables.
+Furthermore, the CMB maps contain foregrounds contamination because of
+Galactic emission and the lensing of CMB photons due to their interaction with the Large
+Scale Structure. These effects significantly deviate from the hypothesis of Gaussianity and
+isotropy, and thus have to be carefully considered when analysing the data. These effects are
+especially important in CMB polarisation.
+Minkowski Functionals (MFs) are one of the tools adopted by the Cosmology community
+to study possible deviations from Gaussianity or statistical isotropy. These functionals encode
+geometrical and topological information of the field, not reflected in the power spectra. Other
+tools include the bispectrum and trispectrum, or, equivalently, the 3– and 4–points correlation
+– 1 –
+
+functions [4–6], the distribution of maxima and minima [7–9], or the distribution of non–
+polarised points in polarisation fields [10]. MFs present several advantages with respect to
+the bispectrum and the trispectrum, such as the computational cost, the ease of masking or
+weighting data, and the possibility of studying deviations at different thresholds. The last
+one makes MFs naturally suited to study non–Gaussianities that are not optimally expressed
+in terms of momenta expansion (fNL, gNL, . . . ); this is the case, for example, in inflationary
+models that can produce primordial black holes, such as Stochastic Inflation, as this introduces
+non–Gaussianity mostly at high values of the field [11, 12].
+The application of MFs has been mostly limited to scalar maps so far, such as CMB
+temperature [13, 14] or weak lensing [15–17]. They have also been used to study the morpho-
+logical properties of Galactic emission, like thermal dust [18], and synchrotron [19]. However,
+the CMB polarisation field is a spin 2 complex quantity and MFs have not been defined for
+this kind of maps. In polarisation studies, MFs are usually applied on the E and B scalar
+maps independently [14, 20, 21], or directly on Q and U maps, ignoring spin effects [22, 23].
+In a previous work [24], we focused on the application of MFs on the squared total
+polarised intensity of the CMB, P 2 = Q2 + U 2. We introduced the formalism and computed
+the theoretical expectations in the Gaussian isotropic case, by making use of the Gaussian
+Kinematic Formula. We also developed a Python package to estimate the MFs on arbitrary
+HEALPix scalar maps and compare them with the theoretical predictions; this software,
+called Pynkowski, is now publicly available1.
+In this work, we introduce a new theoretical framework to analyse the full information
+of polarisation data, i.e., without limiting the analysis to scalar quantities defined on the
+sphere such as P, or the E or B modes. This should provide more complete information of
+the statistics of CMB polarisation data, while also avoiding leakage contamination in the E
+and B modes decomposition of masked maps.
+The paper has the following structure. In Section 2 we introduce the field on which
+MFs will be computed and we explain some of the technical aspects needed to perform such
+computations. In Section 3 we obtain the theoretical expectations of MFs for Gaussian and
+statistically isotropic spin maps. In Section 4 we present the pipeline to estimate the MFs
+on arbitrary HEALPix spin maps from the Q and U data. In Section 5 we introduce the
+simulations we use to verify the formalism and the pipeline, while in Section 6 we present the
+results of applying this framework and software to such CMB polarisation simulated maps.
+Finally, in Section 7 we summarise our conclusions.
+2
+Spin field as a scalar field in SO(3)
+The CMB temperature anisotropies map can be seen as a real scalar field defined on the
+sphere.
+Thus, its statistical properties can be analysed with plenty of tools such as the
+MFs formalism, firstly introduced in the Cosmological literature in [13]. This tool is used to
+describe several characteristics of the excursion sets of the fields at different thresholds, which
+define their geometry and topology. However, CMB polarisation has a different geometrical
+structure: it constitutes a spin 2 complex field on the sphere [25], for which excursion sets
+cannot be directly defined.
+In order to overcome this issue, we lift the field to a higher
+dimensional space where it can be seen as a scalar field, following the framework introduced
+in [26]. See also [27] for further mathematical discussion on spin random fields.
+1https://github.com/javicarron/pynkowski
+– 2 –
+
+Let Q(φ, θ) and U(φ, θ) be the maps of Stokes parameters for linear polarisation in the
+usual base. We define f(φ, θ, ψ) : SO(3) → R as:
+f(φ, θ, ψ) = Q(φ, θ) cos(2ψ) − U(φ, θ) sin(2ψ)
+(2.1)
+which can be understood as the linear polarisation observed at the point (φ, θ) on the sky in
+the polarisation direction ψ (in the system used to define Q and U). The field f is now a
+three–dimensional scalar field, for which excursion sets and MFs are properly defined, as we
+will see in the next section.
+The domain of this function, SO(3), requires further technical discussion, to which we
+dedicate the remainder of this section. Keeping in mind the interpretation of the function
+variables (φ, θ, ψ) as the position and the polarisation direction, it can be seen that the domain
+of f must cover all points of the sphere and all possible polarisation directions. The domain
+must then be contained in a three–dimensional hypersphere, S3. Since the CMB polarisation
+is a spin 2 field, we can perform the identification (φ, θ, ψ) ∼= (φ, θ, ψ + π). Therefore, the
+domain space can actually be seen as half a 3-sphere; this space is diffeomorphic to SO(3),
+the set of rotations of the sphere. The coordinates are the longitude φ ∈ [0, 2π], the latitude
+θ ∈ [− π
+2 , π
+2 ], and the polarisation angle ψ ∈ [0, π], noting that this parameterisation fails at a
+(zero–measure) number of singular points.
+By following this framework, we can lift the complex spin field on the sphere to a
+complex scalar field on SO(3). However, the real and imaginary parts of this field are just
+translations of each other, so it is enough to study only the real part in order to characterise
+the geometry and topology of the polarisation field. This real part is what we have called f
+in equation (2.1).
+An important consequence of this construction is that if we consider an isotropic spin 2
+field on the sphere, it does not constitute an isotropic field on S3 nor SO(3). Physically, this
+can be seen as a consequence of the different behaviour of the polarisation coordinate and
+the sky coordinates. Mathematically, an isotropic random field on S3 has to be invariant in
+law to the action of any isometry of S3; it can be proven that this is only satisfied if every
+multipole component of the field is a sum of fields with spin s = −ℓ, −ℓ + 1, . . . , ℓ − 1, ℓ, each
+spin component with equal power. This can never be the case for fields produced by lifting a
+spin field, such as CMB polarisation, given that all multipoles are constituted only by s = ±2.
+The details of this construction and the consequence on random fields can be found in [26].
+Although f is not isotropic on SO(3), we still have enough information to produce
+accurate predictions for its MFs. It is enough to assume isotropy on the sphere (S2) and
+the fact that we have a spin 2 field (i.e., knowledge of the behaviour of the polarisation
+coordinate). We will exploit these factors in the next section.
+In order to correctly compute the derivatives of the field needed to estimate the MFs, we
+have to take into account the geometry of the space. The metric of SO(3) in this framework
+is given by:
+gµν =
+�
+�
+2
+0 2 sin(θ)
+0
+2
+0
+2 sin(θ) 0
+2
+�
+�
+(2.2)
+where the order is (φ, θ, ψ), see Appendix A for details on the computation. With this metric,
+– 3 –
+
+the volume element and total volume of SO(3) are:
+dV = cos(θ) dφ dθ dψ
+(2.3)
+� 2π
+0
+�
+π
+2
+− π
+2
+� π
+0
+cos(θ) dφ dθ dψ = 4π2
+(2.4)
+One consequence of this metric is that the base of the tangential space given by the
+derivatives
+�
+∂
+∂φ, ∂
+∂θ, ∂
+∂ψ
+�
+is not orthonormal. It will be useful to work in the orthonormal
+basis {e1, e2, e3} given by the following expression:
+e1 =
+√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+(2.5a)
+e2 =
+1
+√
+2
+∂
+∂θ
+(2.5b)
+e3 =
+√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+(2.5c)
+In this basis, the gradient and the Hessian of a function f can be expressed as
+∇f =
+�
+�
+e1
+e2
+e3
+�
+� f
+(2.6)
+Hij(f) = ∇2f(ei, ej) = eiejf − (∇eiej)f
+(2.7)
+The expressions of these operators in terms of the usual spatial derivatives
+�
+∂
+∂φ, ∂
+∂θ, ∂
+∂ψ
+�
+can be found in Appendix A, together with the computations to obtain them. These results
+are needed to compute the MFs of arbitrary polarisation maps, as we will see in Section 4.
+3
+Minkowski Functionals
+Minkowski Functionals are statistics that quantify the morphology produced by a scalar
+function. They are higher order statistics, meaning that their value cannot be fully predicted
+from any n–points correlation function, rendering MFs a useful complementary tool to the
+angular power spectrum. Although typically employed on the sphere, they can be used on
+any manifold, as we will show in this section for the case of SO(3).
+3.1
+Definition
+Let f : SO(3) → R be a C2 function, and u ∈ R. The excursion set of f at threshold u, called
+Au(f, SO(3)), is defined as the collection of all points in the domain where the function is
+larger than (or equal to) u:
+Au(f, SO(3)) = {x ∈ SO(3) : f(x) ≥ u} .
+(3.1)
+In the following, we will refer to it simply as Au to simplify the notation. We will use the
+standard notation ∂Au to refer to the boundary of Au, corresponding to the points with a
+value of exactly u.
+– 4 –
+
+In the case of the sphere, or any 2–dimensional manifold, we can define 3 independent
+MFs to fully characterise the morphology of the excursion set, as shown in [13]. In a 3–
+dimensional manifold, like SO(3), we can define 4 independent MFs:
+V0(Au) =
+�
+Au
+dx
+(3.2a)
+V1(Au) = 1
+6
+�
+∂Au
+da
+(3.2b)
+V2(Au) = 1
+6π
+�
+∂Au
+H(a) da
+(3.2c)
+V3(Au) = 1
+4π
+�
+∂Au
+K(a) da
+(3.2d)
+where da denotes an area element along the boundary surface ∂Au, H(a) and K(a) are,
+respectively, the mean curvature (average of the two principal curvatures) and the Gaussian
+curvature (product of the two principal curvatures) at a point a on the boundary surface.
+In general, in a n-dimensional space, we can define V0 as the hypervolume of the excursion
+set, and n independent MFs as integrals on the boundary of a combination of the principal
+curvatures, see [13] for more details. We follow the normalisation convention in this reference,
+noting that other normalisations have been used before, more notably in [28, 29].
+MFs can be interpreted as geometrical descriptors of the excursion set as a function of
+threshold and, therefore, they represent a statistical characterisation of the original function.
+V0 is the total volume of the excursion set. V1 is the area of its boundary. To our knowledge, V2
+does not have a further interpretation beyond the average mean curvature of the boundary
+of the excursion set.
+Finally, V3 is connected to the Euler–Poincaré characteristic χ due
+to an extension of the Gauss–Bonnet theorem known as the Chern–Gauss–Bonnet theorem
+[originally proven in 30] which states that:
+χ(Au) = V3(Au) + 1
+4V1(Au)
+(3.3)
+Similarly to [24], we introduce the Lipschitz–Killing curvatures, as the theoretical pre-
+dictions are obtained for these quantities (see Section 3.3). Given a tube of width ρ built
+around the manifold Au, its volume can be exactly expressed as a finite Taylor expansion on
+ρ, whose coefficients correspond to the Lipschitz–Killing curvatures of Au (see [28, 31] for a
+detailed description). These quantities are connected to the MFs in the following way:
+L3(Au) = V0(Au)
+(3.4a)
+L2(Au) = 3V1(Au)
+(3.4b)
+L1(Au) = 6V2(Au)
+(3.4c)
+L0(Au) = V3(Au)
+(3.4d)
+In order to compute the theoretical predictions for a Gaussian field we have to invoke
+the Gaussian Kinematic Formula, as we shall see in the following section.
+3.2
+Gaussian Kinematic Formula
+The theoretical expected values of the Lipschitz–Killing curvatures (and therefore of MFs) for
+a Gaussian field can be computed with the Gaussian Kinematic Formula, as done in [24] for
+the polarised intensity of the CMB.
+– 5 –
+
+In our case, we want to study the function f = Q(φ, θ) cos(2ψ) − U(φ, θ) sin(2ψ). This
+function is not fully isotropic, as the behaviour in the θ and φ directions is different than the
+behaviour in the ψ direction. This is connected to the fact that spin s functions on the sphere
+correspond to functions on SO(3) with a single spin component s, but isotropic functions
+of SO(3) must have components at all spins, and with the same angular power spectrum,
+as explained in Section 2. We shall see that this anisotropy introduces some non–negligible
+changes in the computations of the expected values for the MFs.
+Without loss of generality, we normalise the function f to have unit variance; if Q
+and U have the same variance (as required by isotropy on the sphere), this is equivalent to
+normalising both of them to unit variance. Again as a consequence of spherical isotropy, the
+variance of all first derivatives are equal, and the covariance between orthogonal directions is
+zero:
+E
+��∂Q
+∂θ
+�2�
+= E
+��∂U
+∂θ
+�2�
+= E
+��∂Q
+∂φ
+�2�
+= E
+��∂U
+∂φ
+�2�
+= µ
+(3.5a)
+E
+�∂Q
+∂θ · ∂Q
+∂φ
+�
+= E
+�∂U
+∂θ · ∂U
+∂φ
+�
+= 0.
+(3.5b)
+The value of µ can be directly computed in the following way:
+µ =
+�
+ℓ
+2ℓ + 1
+4π
+(ℓ − s)(ℓ + s + 1)
+2
+Cℓ ,
+(3.6)
+where s = 2 and:
+Cℓ = 1
+2
+�
+CEE
+ℓ
++ CBB
+ℓ
+�
+,
+with CEE
+ℓ
+and CBB
+ℓ
+the E and B angular power spectra, respectively, computed from Q and U
+maps. We note that µ ≫ 1, as a direct consequence of normalising the field to unit variance.
+The last quantity we need to introduce is the covariance matrix of the derivatives of
+f, which can be defined as Σij = E
+�
+∂f
+∂ei · ∂f
+∂ej
+�
+, using the orthonormal basis {e1, e2, e3},
+introduced in equation (2.5). Specifically, its determinant is:
+|Σ|
+1
+2 =
+5
+2
+√
+2 µ.
+(3.7)
+We note that, in a fully isotropic case, |Σ|
+1
+2 would scale as µ3/2. Thus, the anisotropy of f
+will introduce different µ–scaling factors in the prediction of the MFs.
+The Gaussian Kinematic Formula can be formulated for non–isotropic Gaussian func-
+tions with a caveat: the Lipschitz–Killing curvatures must be computed not with the usual
+metric of the ambient manifold (see [13, 24]), but with a metric given by the covariance matrix
+Σ introduced above. We denote Lf
+i (Au) the Lipschitz–Killing curvatures computed with this
+metric. For the Gaussian function f : M → R, the Gaussian Kinematic Formula takes the
+following form:
+E
+�
+Lf
+i (Au)
+�
+=
+dim(M)−i
+�
+k=0
+�k + i
+k
+�
+ρk(u)Lf
+k+i(M)
+(3.8)
+where the flag coefficients are [28]:
+� k + j
+k
+�
+= ωk+j
+ωkωj
+� k + j
+k
+�
+, ωj =
+πj/2
+Γ( j
+2 + 1)
+,
+– 6 –
+
+with ωj representing the volume of the j–dimensional unit ball, while the functions ρk(u) are
+defined as:
+ρk(u) =
+1
+(2π)k/2
+1
+√
+2π exp
+�
+− u2
+2
+�
+Hk−1(u) ,
+H−1(u) = 1 − Φ(u) ,
+H0(u) = 1 ,
+H1(u) = u ,
+H2(u) = u2 − 1 ,
+Hk(u) = (−1)k exp
+�
+u2
+2
+�
+dk
+duk exp
+�
+− u2
+2
+�
+.
+The function Φ represents the cumulative normal distribution and Hk are the Hermite poly-
+nomials.
+In the case of an isotropic function with the normalisation described above, it can be
+proven that Lf
+j (M) = Lj(M)µj/2, with 0 ≤ j ≤ dim(M). This identity simplifies equa-
+tion (3.8) to the Gaussian Kinematic Formula described in [24].
+In the non–isotropic case, the relation between the Lipschitz–Killing curvatures com-
+puted with both metrics is not trivial, but there are two important observations. First, L0
+corresponds to the Euler–Poincaré characteristic, which is a topological invariant. This means
+that this quantity depends only on the topological structure and not on the metric. There-
+fore, Lf
+0(M) = L0(M). Second, there is a useful identity for Ldim(M)(M): this quantity
+corresponds to the total volume of the manifold M or, more technically, to its Hausdorff
+measure. Computing it with a different metric is analogous to a change of variable in the
+integral of the measure. Therefore, Lf
+dim(M)(M) = |Σ|1/2Ldim(M)(M), which, thus, in our
+case can be rephrased by: Lf
+3(M) =
+�
+5
+2
+√
+2 µ
+�
+L3(M).
+Note that if we ignore the anisotropy of f, we would obtain that the factor is ∝ µ3/2
+instead of ∝ µ, yielding an incorrect behaviour with µ and therefore with the angular power
+spectrum of the polarisation maps. This is one of the most remarkable advantages of adopting
+this general formalism.
+Finally, the cases of Lf
+1(M) and Lf
+2(M) are more complicated, as they are related to the
+change of integrating submanifolds with lower dimensionality when the global (non-isotropic)
+metric changes. Our conjecture is that the scaling relations should be Lf
+1(M) ∝ L1(M) and
+Lf
+2(M) ∝ L2(M)µ1/2. In Section 6.2 we will see that these guesses are highly compatible
+with simulations, and we will compute the proportionality factors.
+3.3
+Theoretical predictions
+We are now in the position to explicitly compute the theoretical predictions for the MFs
+of the excursion set. In our case, the global manifold is M = SO(3). We note that the
+right hand side of the Gaussian Kinematic Formula, equation (3.8), always contains the term
+proportional to L3(SO(3)) as the leading term.
+As mentioned above, the last Lipschitz–
+Killing curvature is equal to the Hausdorff measure (or volume) of the manifold, which in this
+case is Vol(SO(3)) = 4π2, as shown in equation (2.4).
+– 7 –
+
+Only L3 is needed to compute all MFs at leading order but we report the rest of the
+Lipschitz–Killing curvatures of SO(3) for completeness:
+L3(SO(3)) = 4π2
+(3.9a)
+L2(SO(3)) = 0
+(3.9b)
+L1(SO(3)) = 6π
+(3.9c)
+L0(SO(3)) = 0.
+(3.9d)
+For the topology–inclined readers, the last line implies that the Euler–Poincaré characteristic
+vanishes: χ(SO(3)) = 0; this is a consequence of the 3-sphere being a (2-fold) cover of SO(3)
+and χ(S3) = 0.
+We can now compute the Lipschitz–Killing curvatures of the excursion set of f using the
+Gaussian Kinematic Formula, equation (3.8). We assume no mask in these computations, but
+they can be readily introduced by modifying the Lipschitz–Killing curvatures of the global
+manifold, most notably by multiplying the total volume by the sky fraction.
+Volume of the excursion set, V0.
+We use the Gaussian Kinematic Formula with j = 3
+to compute the expected value of the volume of the excursion set, along with the expression
+Lf
+3(M) =
+�
+5
+2
+√
+2 µ
+�
+L3(M) derived above:
+E [L3(Au)] = 2
+√
+2
+5µ E
+�
+Lf
+3(Au)
+�
+=
+= 2
+√
+2
+5µ
+3−3
+�
+k=0
+�k + 3
+k
+�
+ρk(u)Lf
+k+3(SO(3)) =
+= 2
+√
+2
+5µ
+�3
+0
+�
+ρ0(u)
+� 5
+2
+√
+2 µL3(SO(3))
+�
+= 4π2 [1 − Φ(u)]
+(3.10)
+where Φ(u) is the cumulative distribution function of the standard normal distribution.
+It can be seen that V0 is equal to the total volume at low thresholds (since the field in all
+points is greater than u), and it is 0 at high thresholds (since the field in all points is lower
+than u). Interestingly, the definition of f implies that at u = 0, L3(A0) = 2π2 with no
+scattering whatsoever. We note that the factor µ cancels out and the theoretical expectation
+depends only on the threshold u, not on the field itself (as long as it is normalised to have
+unit variance). The summation in equation (3.10) has exactly one term, corresponding to the
+volume of SO(3), so no approximation is needed for this prediction.
+Area of the boundary of the excursion set, V1.
+We can compute the theoretical ex-
+pectation by setting j = 2 in the Gaussian Kinematic Formula and assuming the scaling
+relation explained before: Lf
+2(M) = K−1
+1 µ1/2L2(M), where we have introduced the unknown
+– 8 –
+
+constant K1 (as it is related to V1). Thus:
+E [L2(Au)] = K1µ−1/2E
+�
+Lf
+2(Au)
+�
+=
+= K1µ−1/2
+3−2
+�
+k=0
+�k + 2
+k
+�
+ρk(u)Lf
+k+2(SO(3)) =
+= K1µ−1/2
+�3
+1
+�
+ρ1(u)
+� 5
+2
+√
+2 µL3(SO(3))
+�
+=
+= K1
+5
+√
+2
+√µ 4π2
+� 1
+2π exp
+�−u2
+2
+��
+(3.11)
+where the square bracket in the fourth line corresponds to ρ1(u) and the term k = 0 in the
+summation is zero because of equation (3.9).
+Mean curvature of the boundary of the excursion set, V2.
+In a similar way, we
+can compute the theoretical expectation of V2 by setting j = 1 in the Gaussian Kinematic
+Formula and assuming the scaling relation explained before Lf
+1(M) = K−1
+2 L1(M), where we
+again introduce an unknown constant K2. Thus:
+E [L1(Au)] = K2E
+�
+Lf
+1(Au)
+�
+=
+= K2
+3−1
+�
+k=0
+�k + 1
+k
+�
+ρk(u)Lf
+k+1(SO(3)) =
+= K2
+�3
+2
+�
+ρ2(u)
+� 5
+2
+√
+2 µL3(SO(3))
+�
++ O(Lf
+1(SO(3))) =
+= K2
+5
+√
+2µ 4π2
+�
+u
+(2π)3/2 exp
+�−u2
+2
+��
++ O(µ0)
+(3.12)
+where the square bracket in the fourth line corresponds to ρ2(u). Only two terms in the
+summation are not zero, one of order µ and another of order 1. Given that µ ≫ 1 (in the
+case of cosmological fields, we typically have µ ∼ 105), we neglect the second order terms in
+the computations.
+Euler–Poincaré characteristic of the excursion set, V3.
+Finally, we compute the the-
+oretical prediction for the Euler–Poincaré characteristic of excursion sets of f. As explained
+before, Lf
+0(M) = L0(M), since it is a topological invariant. Thus:
+E [L0(Au)] = E
+�
+Lf
+0(Au)
+�
+=
+=
+3−0
+�
+k=0
+�k + 0
+k
+�
+ρk(u)Lf
+k+0(SO(3)) =
+=
+�3
+3
+�
+ρ3(u)
+� 5
+2
+√
+2 µL3(SO(3))
+�
++
+�1
+1
+�
+ρ1(u)
+�
+Lf
+1(SO(3))
+�
+=
+5
+2
+√
+2 µ 4π2
+�(u2 − 1)
+(2π)2
+exp
+�
+−u2
+2
+��
++ O(µ0)
+(3.13)
+where the square bracket in the fourth line corresponds to ρ3(u). Again, only two terms in
+the summation are not zero, one of order µ and another of order 1. The latter, the term
+corresponding to Lf
+1(SO(3)), can be safely ignored.
+– 9 –
+
+We note that neglecting the anisotropy of f would yield an incorrect exponent for µ in
+the the theoretical prediction for L0 (i.e., the Euler-Poincaré characteristic).
+Predictions for the MFs
+We can now convert the predictions for Lipschitz–Killing curva-
+tures into MFs. Additionally, in order to ease the comparison with data and the interpretation,
+we work with normalised MFs, where the quantities are divided over the volume (in our case
+this is 4π2, possibly multiplied by the sky fraction fsky if we impose a mask):
+vi = Vi
+4π2
+(3.14)
+The theoretical predictions of the normalised MFs are as follows:
+E [v0] = 1 − Φ(u)
+(3.15a)
+E [v1] = K1
+5
+6π
+√
+2 µ1/2 exp
+�−u2
+2
+�
+(3.15b)
+E [v2] = K2
+5
+24π3/2 µ u exp
+�−u2
+2
+�
+(3.15c)
+E [v3] =
+5
+8
+√
+2 π2 µ (u2 − 1) exp
+�−u2
+2
+�
+(3.15d)
+We remind that the first two formulae are exact, while the last two are correct at the leading
+order in µ. One of the main advantages of this formalism is that the expected value of the
+normalised MFs do not depend on the use of masks. These theoretical predictions show the
+power of this approach. We are able to predict a topological feature of the field knowing
+only the parameter µ, which can be computed from the angular power spectra of E- and B-
+modes. Alternatively, this parameter can also be computed directly on the maps, which may
+be recommended when the maps are masked in order to avoid leaking effects which distort
+the estimation of the polarisation angular power spectra.
+4
+Implementation: Pynkowski
+We implement both the theoretical predictions (presented in Section 3) and the computation
+of the MFs of f on data (to be explained in this section). Both aspects are to be included
+in the publicly available Python package called Pynkowski, which can be found in https:
+//github.com/javicarron/pynkowski. It was first introduced in [24] in the context of MFs
+for scalar maps like T and P 2 = Q2 + U 2. It has been designed to compute MFs on different
+kinds of data, as well as the theoretical predictions for specific types of fields.
+We develop the software in order to compute all the necessary quantities related to f,
+including its derivatives in all directions, and the covariant gradient and Hessian. We use this
+code on Gaussian simulations in order to compare the results with the theoretical predictions.
+After such a validation, it can be used on arbitrary maps in order to assess any deviation
+from statistical isotropy and Gaussianity. The actual implementation of the computation of
+these MFs on maps is described in this section.
+For a map f, we store the values of Q and U in the HEALPix pixelisation scheme
+[32]. The variable ψ is not pixelised, it is computed exactly for each pixel ξi as f(ξi, ψ) =
+Q(ξi) cos(2ψ) − U(ξi) sin(2ψ).
+– 10 –
+
+The first and second spatial derivatives of Q and U with respect to θ and φ are also
+computed when needed, with the help of the healpy2 function alm2map_der, which performs
+such computation in harmonic space.
+The first and second spatial derivatives of f with
+respect to its three variables are simplified analytically and expressed as exact functions of
+the polarisation maps and their derivatives. This means that the only pixelisation is due to
+the original pixelisation of Q and U and their derivatives, while the treatment in ψ is always
+exact in order to avoid numerical artefacts.
+The first step in our procedure after loading the map is to normalise f in order to have
+unit variance. We do not assume that Q and U have similar statistical properties, but if they
+do, this step is equivalent to normalising both of them to have unit variance. By normalising
+f to unit variance, we can express the MFs as a function of an adimensional threshold u. For
+convenience, we compute the MFs divided by the total volume of SO(3) in order to have the
+normalised MFs (vi =
+Vi
+4π2 ).
+Before explaining the specific computations for each MF, we shall introduce some com-
+mon notation. Let P(ξi) =
+�
+Q(ξi)2 + U(ξi)2, i.e., the maximum value of f in each pixel
+ξi. For a fixed ξi, we note that f has a simple sinusoidal behaviour in its variable ψ: it is
+always under u if u > P(ξi), always over u if u < −P(ξi), and over u in a single segment of
+ψ otherwise. We use this observation to split all the integrals involved in the computation
+of the MFs. In the case where f(ξi) > u only in a range of ψ, let ψM(ξi) be the angle at
+which f(ξi, ψ) is maximum, while ψ1(ξi) and ψ2(ξi) the angles for which f(ξi) = u; they can
+be computed as:
+ψM(ξi) = 1
+2 arctan
+�
+−U(ξi)
+Q(ξi)
+�
+(4.1a)
+ψ1(ξi) = ψM(ξi) − 1
+2 arccos
+�
+u
+P(ξi)
+�
+(4.1b)
+ψ2(ξi) = ψM(ξi) + 1
+2 arccos
+�
+u
+P(ξi)
+�
+(4.1c)
+where the arctan in the first line is defined to be in the quadrant corresponding to the
+coordinate (Q(ξi), U(ξi)). The polarisation angles ψ and their differences are always defined
+between 0 and π, due to the geometry of SO(3).
+First MF, v0
+It can be seen as the volume fraction of the manifold for which f > u:
+v0(u) =
+1
+4π2
+�
+SO(3)
+Θ(f(x) − u) dx
+(4.2)
+where Θ(r) is the Heaviside function (1 where r ≥ 0; 0 otherwise). We can split the integrand
+by pixel with the cases explained above and integrate with respect to ψ first. The integrand
+will be 0 for pixels where f is always smaller than the threshold (u > P(ξi)) and it will be 1
+if u < −P(ξi). For all the other pixels, it will be 1 as long as ψ is between ψ1 and ψ2 and 0
+otherwise, so the integral in ψ will yield the length of the interval between these two angles,
+i.e., arccos
+�
+u
+P(ξi)
+�
+1
+π.
+Therefore, in order to compute equation (4.2), we just need the length of this interval
+for every pixel and average it over all pixels.
+2https://github.com/healpy/healpy
+– 11 –
+
+Second MF, v1
+It can be computed as the area of the boundary of the excursion sets, i.e.,
+the manifold defined by f = u:
+v1(u) =
+1
+4π2
+1
+6
+�
+∂Au
+da
+=
+1
+4π2
+1
+6
+�
+SO(3)
+δ(f(x) − u) · |∇f(x)| dx
+(4.3)
+where the second equality comes from a change of coordinates from the surface element of
+the boundary da to a volume element on SO(3), dx. The gradient of f is denoted by ∇f and
+can be computed with equation (2.5); δ is the Dirac delta.
+In a fully pixelised field (such as T or P), one typically has to bin the threshold and
+approximate the delta function (see [24] for a discussion in that case). However, in this case
+we can exactly determine the points where f(ξi, ψ) = u for each pixel (if they exist) as (ξi, ψ1)
+and (ξi, ψ2).
+In order to compute the integral, it suffices to evaluate |∇f(x)| in these points (adding
+the values corresponding to both ψ1 and ψ2 for each pixel), fill with 0 the rest of the pixels,
+and compute the average value of this quantity over the entire map. This quantity is then
+divided over 6, the normalisation factor.
+Third MF, v2
+It is the average mean curvature (H) of the boundary of the excursion sets
+(i.e., the average mean curvature of the manifold defined by f = u):
+v2(u) =
+1
+4π2
+1
+6π
+�
+∂Au
+H(a)da
+=
+1
+4π2
+1
+6π
+�
+SO(3)
+δ(f(x) − u) · H(x) · |∇f(x)| dx
+(4.4)
+where we use the same change of coordinates as in the previous case, which introduces the
+δ and the factor |∇f(x)|. The integral with the delta can be computed by evaluating the
+integrand, H(x) · |∇f(x)|, on the points where f = u, exactly as in the previous case.
+In general, evaluating the mean curvature of a surface is far from trivial. In this case,
+however, the surface is defined implicitly by the function F(x) := f(x) − u = 0, so we can use
+the expression for the mean curvature in a implicitly-defined surface [33]:
+H = ∇F H(F) ∇F T − |∇F|2 Tr(H(F))
+2|∇F|3
+(4.5)
+where all quantities are functions of x.
+As before, we compute the integral by evaluating H(x) · |∇f(x)| in the points where
+f = u (adding for each pixel the values corresponding to both ψ1 and ψ2), fill with 0 the rest
+of the pixels, and compute the average value of this quantity over all pixels. This quantity is
+then divided over 6π, the normalisation factor.
+Fourth MF, v3
+It can be computed as the average Gaussian curvature (K) of the boundary
+of the excursion sets (i.e., the manifold defined by f = u):
+v2(u) =
+1
+4π2
+1
+4π
+�
+∂Au
+K(a)da
+=
+1
+4π2
+1
+4π
+�
+SO(3)
+δ(f(x) − u) · K(x) · |∇f(x)| dx
+(4.6)
+– 12 –
+
+where we use the same strategy as in the previous case, now computing the Gaussian curvature
+K(x) instead of the mean curvature H(x). In the case of an implicitly defined surface (again
+F(x) := f(x) − u = 0), the Gaussian curvature can be computed as [33]:
+K = −
+����
+H(F) ∇F T
+∇F
+0
+����
+|∇F|4
+= −
+��������
+Fxx Fxy Fxz Fx
+Fxy Fyy Fyz Fy
+Fxz Fyz Fzz Fz
+Fx
+Fy
+Fz
+0
+��������
+|∇F|4
+(4.7)
+We then compute the integral by evaluating K(x) · |∇f(x)| in the points where f = u
+(adding for each pixel the values corresponding to both ψ1 and ψ2), fill with 0 the rest of the
+pixels, and compute the average value of this quantity over the whole map. This quantity is
+then divided over 4π, the normalisation factor.
+4.1
+General considerations
+All these computations are performed without pixelising the variable ψ, just θ and φ. However,
+numerical issues can arise, especially on v3 and, to a lesser extent, v2, due to the computation
+of the second spatial derivatives in pixelised maps.
+In order to avoid artefacts, we must
+work with smooth maps, where the value of the field does not change abruptly between
+neighbouring pixels; in other words, the derivatives of the map must exist and be reasonably
+smooth. This can be seen as a requirement on the maps Q and U to have negligible power at
+high multipoles or on the needed pixel resolution (the Nside parameter in HEALPix maps).
+This is not only a computational limitation, but it is, in fact, a mathematical requirement.
+Indeed, the Gaussian Kinematic Formula holds if the boundaries of the excursion sets are
+twice differentiable (see [28]). In practice, this means that the angular power spectrum must
+decay fast enough. For a study where this condition does not hold, see, e.g., [34].
+Nevertheless, we note that the theoretical predictions for v2 and v3 are correct up to
+leading order in µ. Given that µ increases with the considered range of multipoles, these
+approximations are increasingly accurate at higher multipoles. Therefore, it may be conve-
+nient to apply this formalism on needlet components. Indeed, needlets naturally filter out
+the low and high multipoles, thus reducing both the pixelisation effects and the error in the
+approximation in µ. MFs in needlet domain were studied for scalar fields on the sphere in,
+e.g., [14, 29].
+5
+Simulations
+In this section we introduce the different polarisation fields that we analyse with MFs.
+5.1
+Monochromatic maps
+We discussed the theoretical prediction of the MFs in Section 3.3. However, the exact value
+of v1 and v2 depend on the unknown normalisation factors K1 and K2, respectively.
+We generate simulated maps in order to verify the the predicted expectation values
+for the four MFs, and compute the two unknown normalisation factors.
+We create 1600
+monochromatic maps, i.e., maps with power in a single multipole ℓm:
+CEE
+ℓ
+= CBB
+ℓ
+=
+�
+1
+ℓ = ℓm
+0
+otherwise
+(5.1)
+– 13 –
+
+We simulate 8 sets of 200 maps, each set with a different value of ℓm = {100, 485, 675, 825, 950,
+1060, 1165, 1255}. These values of ℓm are chosen in order to have a set of approximately
+equally–spaced values of µm = (ℓm − s)(ℓm + s + 1)/2, where s = 2.
+5.2
+Realistic angular power spectrum
+In order to validate the theoretical predictions and the implementation of the computation
+of MFs, we generate 300 Gaussian isotropic CMB maps with a realistic (i.e., cosmological)
+angular power spectrum. We use best fit angular power spectrum for CMB polarisation (E
+and B modes) reported by [35].
+We use a map resolution of Nside = 1024.
+We apply a
+smoothing of the maps with a Gaussian beam of FWHM= 15′ in order to avoid pixelisation
+effects in the computation of the spatial derivatives.
+In this way, we are able to simultaneously achieve three goals: assess the accuracy of the
+theoretical formulae obtained in Section 3.3, verify the implementation of MFs computations
+on simulated maps, and check that the normalisation constants extend correctly to non–
+monochromatic maps.
+6
+Results
+6.1
+Normalisation constants
+In this section we assess the normalisation constants for v1 and v2, i.e., K1 and K2. As
+mentioned in Section 3.3, the exact values for the predicted MFs can only be computed for v0
+and v3, while the prediction for v1 and v2 is correct at leading order in µ except for a global
+normalisation factor related to the anisotropic ψ direction, see equation (3.15). We note that
+both K1 and K2 are truly constants and therefore do not depend on threshold, mask, or value
+of µ.
+In order to estimate the values of these constants we compute the MFs on the monochro-
+matic maps introduced in Section 5.1 with the software presented in Section 4 and soon to be
+included in Pynkowski. Then, K1 and K2 are obtained as the ratio between computation on
+the maps and theoretical predictions, averaging over all simulations and all thresholds. We
+find that the constants are (average and standard deviation):
+K1 = 0.31912 ± 0.00001
+(6.1a)
+K2 = 0.7088 ± 0.0002
+(6.1b)
+We have verified that these values do not significantly vary with threshold or value of µ
+(i.e., multipole). We have also checked that the constant are valid for non–monochromatic
+maps, as we will see in Section 6.3.
+Lastly, we note that the theoretical predictions yield the correct normalisation factor for
+v0 and v3, as expected. If we repeat the process with additional multiplicative factors, Kv0
+and Kv3, for these MFs, we obtain that both of them are compatible with 1:
+Kv0 = 1.0000 ± 0.0001
+(6.2a)
+Kv3 = 1.001 ± 0.003
+(6.2b)
+6.2
+Scaling relations
+The theoretical expectation of the MFs in Section 3 assume a particular dependence of µ for
+the Lipschitz–Killing computed with different metrics (see the last paragraph in Section 3.2).
+In this section we show that this scaling is highly favoured by simulations.
+– 14 –
+
+0
+10
+20
+30
+40
+50
+60
+70
+80
+( 104)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+vi
+0.5
+v1
+v2
+Figure 1. Trend with µ for the normalised MFs v1 (blue dots) and v2 (red dots), defined in the text.
+We can see in dashed lines the trends µ1/2 (blue) and µ (red), corresponding to the expected scaling
+relations.
+The assumed µ dependence have a direct impact on the predictions for v1 and v2. If
+the assumption was wrong, these MFs would present a different scaling with the parameter
+µ. We test for deviations by using the MFs computed on monochromatic maps with a wide
+range of values of µ.
+In order to assess the dependence of v1 and v2 with µ, we consider vi =
+�
+vi
+vi(ℓm=950)
+�
+,
+where ⟨·⟩ represents the average over the thresholds between u = −3 and u = 3. By consider-
+ing this quantity, we remove the dependence on threshold and constant factors, leaving only
+the dependence on µ.
+In Figure 1 we show the trend of v1 and v2 for different values of µ. We observe that they
+are perfectly compatible with the scaling assumed in the theory: µ1/2 and µ, respectively.
+Additionally, we have verified with the same procedure that v0 and v3 scale as µ0 and
+µ, respectively, as predicted by the theory. We have again verified that all scaling relations
+remain unchanged by considering different thresholds and non–monochromatic maps, as ex-
+pected.
+6.3
+Gaussian CMB simulations
+In this section we verify that the theoretical prediction for the MFs (see Section 3.3) agree
+with the results computed on CMB Gaussian isotropic maps with a realistic angular power
+spectrum. We use the 300 simulations introduced in Section 5.2, which are generated with the
+Planck best–fit polarisation angular power spectra; we use the value of µ computed from this
+angular power spectrum for the theoretical predictions. We compute the MFs (see Section 4)
+at thresholds between u = −4 and u = 4, with a spacing of ∆u = 0.2. We use the values of
+the normalisation factors K1 and K2 obtained in monochromatic maps, i.e., equation (6.1).
+This means that there are no free parameters in the analysis of this section.
+Figure 2 presents the comparison between MFs computed on these maps and the corre-
+sponding theoretical predictions. We find full compatibility between theory and simulations
+for all the MFs.
+In all cases the residuals are well within the 1σ region, where σ is the
+dispersion of the MFs among the 300 simulations.
+– 15 –
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+V0
+Simulation
+Theory
+5
+0
+5
+V0( 105)
+4
+3
+2
+1
+0
+1
+2
+3
+4
+u
+2
+0
+2
+V0/
+m
+0
+5
+10
+15
+20
+25
+30
+V1
+Simulation
+Theory
+0.025
+0.000
+0.025
+V1
+4
+3
+2
+1
+0
+1
+2
+3
+4
+u
+2
+0
+2
+V1/
+m
+600
+400
+200
+0
+200
+400
+600
+V2
+Simulation
+Theory
+1
+0
+1
+V2
+4
+3
+2
+1
+0
+1
+2
+3
+4
+u
+2
+0
+2
+V2/
+m
+5000
+4000
+3000
+2000
+1000
+0
+1000
+2000
+V3
+Simulation
+Theory
+200
+0
+200
+V3
+4
+3
+2
+1
+0
+1
+2
+3
+4
+u
+2
+0
+2
+V3/
+m
+Figure 2. MFs for CMB polarisation simulations generated with the Planck best–fit angular power
+spectrum. Top left: v0. Top right: V1. Bottom left: V2. Bottom right: V3. In each panel we show,
+from top to bottom: the average value in simulations (orange) and theoretical predictions (black), the
+difference between them compared with the standard deviation, σ, and the standard deviation of the
+mean, σm (see text).
+Additionally, in order to explore any possible systematic deviation, we compare the
+average residual on simulations with respect to the theoretical expectation. This is done by
+considering the standard deviation of the mean (σmean = σsims
+√
+300). Such comparison is shown in
+the bottom rows of each panel in Figure 2, in terms of ∆vi/σm i.e., deviations over σmean. It
+can be seen that no point deviates more than 2σmean, and there are no significant systematic
+effects.
+The four MFs are all perfectly compatible with the theory, both in the case of individual
+simulations and of the average behaviour. We verify that the results hold for arbitrary angular
+power spectra, as long as there is no significant power in the scales corresponding to the pixel
+size.
+We also note the low statistical variation of these curves: the relative uncertainty of
+every point is below 1 part in 1000 for v0, below 1% for v1 and v2 and below 10% for v3.
+These values of the statistical standard deviation depend on the angular power spectrum
+of the studied map: we observe that the variance decreases when increasing the maximum
+– 16 –
+
+multipole considered, ℓmax (i.e., when increasing the resolution of the maps). This is also
+known to be the case for temperature maps, as quantitatively studied by [29].
+These results provide a double validation: on the one hand, they verify the mathematical
+theory used to predict the expected values of the MFs computed on f; on the other hand, they
+validate our implementation to determine these statistical quantities on polarisation data.
+We have verified that masking the maps has a negligible effect on these results beyond
+slightly increasing the noise due to the smaller sky fraction.
+This robustness is expected
+because, unlike the angular power spectrum, MFs are purely local quantities. Thus, the effect
+of masking is not propagated.
+7
+Conclusions
+The study of the angular power spectrum of CMB anisotropies has led to the tightest con-
+straints on the cosmological parameters [36]. However, this statistical tool is not sensitive
+to important information which could be present at some level in CMB data such as non-
+Gaussianities or deviations from statistical isotropy. The detection of such effects would reveal
+us physical information about the Early Universe.
+In the past, MFs, applied mainly to CMB temperature anisotropies and, more in general,
+to scalar fields, have been used to search for such signatures in Cosmological data [13, 14, 24].
+In this work, we have extended the MFs formalism to the polarisation field by lifting it to
+the higher dimensional manifold SO(3), where the complex part is just a translation of the real
+one. Therefore, without loss of generality, we take just the real part, f(φ, θ, ψ) : SO(3) → R
+as:
+f(φ, θ, ψ) = Q(φ, θ) cos(2ψ) − U(φ, θ) sin(2ψ)
+(7.1)
+which is a scalar field that can be understood as the linear polarisation observed at the point
+(φ, θ) on the sky in the polarisation direction ψ.
+To summarise:
+• We introduce a formalism to predict the values of MFs for f in the case of Gaussian
+Stokes parameters through the use of the Gaussian Kinematic Formula for anisotropic
+fields (see Sects. 3.2 and 3.3).
+• We implement a code to compute MFs of f from input Q and U maps pixelised with
+the HEALPix convention and to be released in the already publicly available Python
+package Pynkowski.
+• We find that theoretical predictions are compatible with the computations on Gaussian
+CMB maps generated with the Planck best-fit angular power spectra.
+The importance of the application of statistical tools (such as MFs formalism introduced
+in this paper) on CMB polarisation data will grow in the future with upcoming much more
+sensitive experiments either from ground: ACT [37], SPT [38], Simons Observatory [39]; or
+from space: LiteBIRD [40], PICO [41].
+We are currently applying this formalism to the CMB polarisation data from Planck and
+simulated future experiments. The corresponding results will be presented in an upcoming
+paper.
+Furthermore, this kind of analysis can help to blindly detect the presence of non-Gaussian
+contaminants in the maps like residual foregrounds contamination. It can also be used to
+– 17 –
+
+characterise the morphology of the galactic emission, as recently done in [19] for synchrotron
+radiation.
+The application proposed in this paper is versatile and can be performed on other spin–2
+quantities.
+Acknowledgements
+The authors thank Michele Stecconi and Maurizia Rossi for insightful discussions. MM and
+NV acknowledge support by ASI/COSMOS grant n. 2016-24-H.0 and ASI/LiteBIRD grant
+n. 2020-9-HH.0. Part of this work was also supported by the InDark INFN project. DM
+acknowledges support from the MIUR Excellence Project awarded to the Department of
+Mathematics, Università di Roma Tor Vergata, CUP E83C18000100006. DM is also grateful
+to the Department of Excellence Programme MatModTov for support.
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+Groups and their Applications, Courier Dover Publications (1963).
+– 20 –
+
+A
+Metric and derivatives in SO(3)
+In this paper, a key part of the argument is based on defining the field f(φ, θ, ψ) = Q(φ, θ) cos(2ψ)−
+U(φ, θ) sin(2ψ), with domain on SO(3). The geometry of this manifold is not trivial. In par-
+ticular, we were unable to find in the literature the expressions for the orthonormal basis and
+the Hessian of a function defined on this manifold. We reproduce these computations here in
+case they are useful for the community.
+SO(3) can be seen as the group of rotations of the sphere, which can be parametrised
+by the Euler angles. These admit many different definitions, depending on the order of the
+axis selected to perform the rotations. In order to make the angles compatible with the usual
+coordinates on the sphere, we must select the definition in the zyx convention.
+Then, a
+generic rotation of the sphere can be expressed as [see, e.g., 42]:
+R(φ, θ, ψ) = Rx(ψ)Ry(θ)Rz(φ),
+where:
+Rx(ψ) =
+�
+�
+1
+0
+0
+0 cos (ψ) − sin (ψ)
+0 sin (ψ)
+cos (ψ)
+�
+�
+Ry(θ) =
+�
+�
+cos (θ) 0 sin (θ)
+0
+1
+0
+− sin (θ) 0 cos (θ)
+�
+�
+Rz(φ) =
+�
+�
+cos (φ) − sin (φ) 0
+sin (φ)
+cos (φ) 0
+0
+0
+1
+�
+�
+It is then possible to compute the derivative of a random rotation with respect to the
+three parameters. Now, taking the Frobenius inner product between matrices (the sum of the
+element–wise multiplication of matrices), we can compute the metric:
+gij = ⟨∂iR, ∂jR⟩ = 2
+�
+�
+1
+0 sin (θ)
+0
+1
+0
+sin (θ) 0
+1
+�
+�
+in the order (φ, θ, ψ). We can now determine the Christoffel symbols of this manifold:
+Γφ
+θψ = Γφ
+ψθ = Γψ
+φθ = Γψ
+θφ =
+1
+2 cos(θ)
+Γφ
+θφ = Γφ
+φθ = Γψ
+ψθ = Γψ
+θψ = − tan(θ)
+2
+Γθ
+φψ = Γθ
+ψφ = − cos(θ)
+2
+with the remaining ones equal to 0. With the Chistoffel symbols, the covariant derivatives
+are just:
+∇∂a∂b = Γc
+ab∂c
+– 21 –
+
+which, in our case implies:
+∇∂φ∂φ = 0
+∇∂θ∂θ = 0
+∇∂ψ∂ψ = 0
+∇∂φ∂θ = ∇∂θ∂φ = −tan(θ)
+2
+∂φ +
+1
+2 cos(θ)∂ψ
+∇∂ψ∂θ = ∇∂θ∂ψ =
+1
+2 cos(θ)∂ψ − tan(θ)
+2
+∂φ
+∇∂φ∂ψ = ∇∂ψ∂φ = −cos(θ)
+2
+∂θ
+These covariant derivatives can be then employed to compute the component of the
+Hessians according to:
+Hij(f) = ∇2f(ei, ej) = eiejf − ∇eiejf
+For the first part of each term we obtain:
+e1e1 =
+�√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+�√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+=
+=
+1
+4 cos2 (θ)
+�
+(1 + cos (θ)) ∂2
+∂φ2 + (1 − cos (θ)) ∂2
+∂ψ2 − 2 sin (θ)
+∂2
+∂ψ∂φ
+�
+e2e2 =
+�
+1
+√
+2
+∂
+∂θ
+��
+1
+√
+2
+∂
+∂θ
+�
+=
+∂2
+2∂θ2
+e3e3 =
+�√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+�√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+=
+=
+1
+4 cos2 (θ)
+�
+(1 − cos (θ)) ∂2
+∂φ2 + (1 + cos (θ)) ∂2
+∂ψ2 − 2 sin (θ)
+∂2
+∂ψ∂φ
+�
+e1e2 = e2e1 =
+�√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+��
+1
+√
+2
+∂
+∂θ
+�
+=
+=
+1
+4 cos (θ)
+� ��
+1 − sin (θ) −
+�
+sin (θ) + 1
+�
+∂2
+∂θ∂ψ +
+��
+1 − sin (θ) +
+�
+sin (θ) + 1
+�
+∂2
+∂θ∂φ
+�
+– 22 –
+
+e1e3 = e3e1 =
+�√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+�√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+=
+=
+1
+4 cos2 (θ)
+�
+− sin (θ) ∂2
+∂φ2 − sin (θ) ∂2
+∂θ2 + 2
+∂2
+∂φ∂ψ
+�
+e2e3 = e2e3 =
+�
+1
+√
+2
+∂
+∂θ
+��√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ +
+√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+=
+=
+1
+4 cos (θ)
+� ��
+1 − sin (θ) −
+�
+sin (θ) + 1
+�
+∂2
+∂θ∂ψ +
+��
+1 − sin (θ) +
+�
+sin (θ) + 1
+�
+∂2
+∂θ∂φ
+�
+while the second part of each term is given by:
+∇e1e1 =∇ √1−sin θ+√sin θ+1
+2
+√
+2 cos θ
+∂
+∂φ+
+√1−sin θ−√sin θ+1
+2
+√
+2 cos θ
+∂
+∂ψ
+�√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ+
++
+√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+= sin (2θ)
+8 cos2 θ
+∂
+∂θ
+∇e2e2 =∇ 1
+√
+2
+∂
+∂θ
+�
+1
+√
+2
+∂
+∂θ
+�
+= 0
+∇e3e3 =∇ √1−sin θ−√sin θ+1
+2
+√
+2 cos θ
+∂
+∂φ+
+√1−sin θ+√sin θ+1
+2
+√
+2 cos θ
+∂
+∂ψ
+�√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ+
++
+√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+= sin (2θ)
+8 cos2 θ
+∂
+∂θ
+∇e1e2 = ∇e2e1 =∇ √1−sin θ+√sin θ+1
+2
+√
+2 cos θ
+∂
+∂φ+
+√1−sin θ−√sin θ+1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+1
+√
+2
+∂
+∂θ
+�
+=
+=
+1
+8 cos2 (θ)
+��
+P(θ) sin (θ) − M(θ)
+� ∂
+∂φ +
+�
+M(θ) sin (θ) − P(θ)
+� ∂
+∂ψ
+�
+∇e1e3 = ∇e3e1 =∇ √1−sin θ+√sin θ+1
+2
+√
+2 cos θ
+∂
+∂φ+
+√1−sin θ−√sin θ+1
+2
+√
+2 cos θ
+∂
+∂ψ
+�√
+1 − sin θ −
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂φ+
++
+√
+1 − sin θ +
+√
+sin θ + 1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+= −
+1
+4 cos (θ)
+∂
+∂θ
+∇e2e3 = ∇e3e2 =∇� √1−sin θ−√sin θ+1
+2
+√
+2 cos θ
+∂
+∂φ+
+√1−sin θ+√sin θ+1
+2
+√
+2 cos θ
+∂
+∂ψ
+�
+�
+1
+√
+2
+∂
+∂θ
+�
+=
+=
+1
+8 cos2 (θ)
+��
+M(θ) sin (θ) − P(θ)
+� ∂
+∂φ +
+�
+P(θ) sin (θ) − M(θ)
+� ∂
+∂ψ
+�
+– 23 –
+
+where:
+P(θ) :=
+�
+1 − sin (θ) +
+�
+sin (θ) + 1
+M(θ) :=
+�
+1 − sin (θ) −
+�
+sin (θ) + 1
+Therefore, the terms of the Hessian are the following:
+H11 =
+1
+4 cos2 (θ)
+�
+(1 + cos (θ)) ∂2
+∂φ2 + (1 − cos (θ)) ∂2
+∂ψ2 − 2 sin (θ)
+∂2
+∂ψ∂φ − sin (2θ)
+2
+∂
+∂θ
+�
+H22 = ∂2
+2∂θ2
+H33 =
+1
+4 cos2 (θ)
+�
+(1 − cos (θ)) ∂2
+∂φ2 + (1 + cos (θ)) ∂2
+∂ψ2 − 2 sin (θ)
+∂2
+∂ψ∂φ − sin (2θ)
+2
+∂
+∂θ
+�
+H12 = H21 =
+1
+8 cos2 (θ)
+�
+2 cos (θ)
+�
+M(θ) ∂2
+∂θ∂ψ + P(θ) ∂2
+∂θ∂φ
+�
++ (P(θ) sin (θ) − M(θ)) ∂
+∂φ+
++ (M(θ) sin (θ) − P(θ)) ∂
+∂ψ
+�
+H13 = H31 =
+1
+4 cos2 (θ)
+�
+− sin (θ) ∂2
+∂φ2 − sin (θ) ∂2
+∂ψ2 + 2
+∂2
+∂ψ∂φ + cos (θ) ∂
+∂θ
+�
+H23 = H32 =
+1
+8 cos2 (θ)
+�
+2 cos (θ)
+�
+M(θ) ∂2
+∂θ∂φ + P(θ) ∂2
+∂θ∂ψ
+�
++
+�
+M(θ) sin (θ) − P(θ)
+� ∂
+∂φ+
++
+�
+P(θ) sin (θ) − M(θ)
+� ∂
+∂ψ
+�
+– 24 –
+
diff --git a/l9FPT4oBgHgl3EQf3jX-/content/tmp_files/load_file.txt b/l9FPT4oBgHgl3EQf3jX-/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d1ff6da4572d25f666d1ea5dc580e330551f61f7
--- /dev/null
+++ b/l9FPT4oBgHgl3EQf3jX-/content/tmp_files/load_file.txt
@@ -0,0 +1,1376 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf,len=1375
+page_content='Prepared for submission to JCAP  Minkowski Functionals in SO(3) for  the spin–2 CMB polarisation field  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Carrón Duque,a,b,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Carones,a,b D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Marinucci,c M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Migliaccio,a,b and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Vittorioa,b  aDipartimento di Fisica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1,  I-00133, Roma, Italy  bSezione INFN Roma 2, via della Ricerca Scientifica 1, I-00133, Roma, Italy  cDipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica  1, I-00133, Roma, Italy  E-mail: javier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='carron@roma2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='it, alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='carones@roma2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='it,  marinucc@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='uniroma2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='it, marina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='migliaccio@roma2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='it,  nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='vittorio@uniroma2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='it  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The study of the angular power spectrum of Cosmic Microwave Background  (CMB) anisotropies, both in intensity and in polarisation, has led to the tightest constraints  on cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However, this statistical quantity is not sensitive to any devi-  ation from Gaussianity and statistical isotropy in the CMB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Minkowski Functionals  (MFs) have been adopted as one of the most powerful statistical tools to study such devi-  ations, given that they characterise the topology and geometry of the field of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In  this paper, we extend the application of MFs to CMB polarisation data by introducing a  new formalism, where we lift the spin 2 polarisation field to a scalar function in a higher  dimensional manifold: the group of rotations of the sphere, SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Such function is defined  as f = Q cos(2ψ) − U sin(2ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We analytically obtain the expected values for the MFs of f  in the case of Gaussian isotropic polarisation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Furthermore, we present a new pipeline  which estimates these MFs from input HEALPix polarisation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We apply it to CMB  simulations in order to validate the theoretical results and the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The pipeline is  to be included in the publicly available Python package Pynkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Keywords: CMBR polarisation – non-gaussianity  1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='13191v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='CO]  30 Jan 2023  Contents  1  Introduction  1  2  Spin field as a scalar field in SO(3)  2  3  Minkowski Functionals  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  Definition  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  Gaussian Kinematic Formula  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3  Theoretical predictions  7  4  Implementation: Pynkowski  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  General considerations  13  5  Simulations  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  Monochromatic maps  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  Realistic angular power spectrum  14  6  Results  14  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  Normalisation constants  14  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  Scaling relations  14  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3  Gaussian CMB simulations  15  7  Conclusions  17  A Metric and derivatives in SO(3)  21  1  Introduction  The Cosmic Microwave Background (CMB) encodes information from the Early Universe,  both in the intensity and polarisation of the light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The CMB is most commonly studied  through the angular power spectra (equivalently, 2–points correlation functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However,  this tool is not sensitive to the possible presence of non–Gaussianities or departures from  statistical isotropy of the CMB anisotropy fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Non–Gaussianity is predicted by many inflationary models [1–3] and could shed a new  light on our knowledge of the primordial Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' There is also a growing amount of literature  on a possible large–scale anisotropy of the Universe, with dipoles being measured in several  observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Furthermore, the CMB maps contain foregrounds contamination because of  Galactic emission and the lensing of CMB photons due to their interaction with the Large  Scale Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' These effects significantly deviate from the hypothesis of Gaussianity and  isotropy, and thus have to be carefully considered when analysing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' These effects are  especially important in CMB polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Minkowski Functionals (MFs) are one of the tools adopted by the Cosmology community  to study possible deviations from Gaussianity or statistical isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' These functionals encode  geometrical and topological information of the field, not reflected in the power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Other  tools include the bispectrum and trispectrum, or, equivalently, the 3– and 4–points correlation  – 1 –  functions [4–6], the distribution of maxima and minima [7–9], or the distribution of non–  polarised points in polarisation fields [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' MFs present several advantages with respect to  the bispectrum and the trispectrum, such as the computational cost, the ease of masking or  weighting data, and the possibility of studying deviations at different thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The last  one makes MFs naturally suited to study non–Gaussianities that are not optimally expressed  in terms of momenta expansion (fNL, gNL, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' this is the case, for example, in inflationary  models that can produce primordial black holes, such as Stochastic Inflation, as this introduces  non–Gaussianity mostly at high values of the field [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The application of MFs has been mostly limited to scalar maps so far, such as CMB  temperature [13, 14] or weak lensing [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' They have also been used to study the morpho-  logical properties of Galactic emission, like thermal dust [18], and synchrotron [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However,  the CMB polarisation field is a spin 2 complex quantity and MFs have not been defined for  this kind of maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In polarisation studies, MFs are usually applied on the E and B scalar  maps independently [14, 20, 21], or directly on Q and U maps, ignoring spin effects [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In a previous work [24], we focused on the application of MFs on the squared total  polarised intensity of the CMB, P 2 = Q2 + U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We introduced the formalism and computed  the theoretical expectations in the Gaussian isotropic case, by making use of the Gaussian  Kinematic Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We also developed a Python package to estimate the MFs on arbitrary  HEALPix scalar maps and compare them with the theoretical predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' this software,  called Pynkowski, is now publicly available1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In this work, we introduce a new theoretical framework to analyse the full information  of polarisation data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', without limiting the analysis to scalar quantities defined on the  sphere such as P, or the E or B modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This should provide more complete information of  the statistics of CMB polarisation data, while also avoiding leakage contamination in the E  and B modes decomposition of masked maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The paper has the following structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In Section 2 we introduce the field on which  MFs will be computed and we explain some of the technical aspects needed to perform such  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In Section 3 we obtain the theoretical expectations of MFs for Gaussian and  statistically isotropic spin maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In Section 4 we present the pipeline to estimate the MFs  on arbitrary HEALPix spin maps from the Q and U data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In Section 5 we introduce the  simulations we use to verify the formalism and the pipeline, while in Section 6 we present the  results of applying this framework and software to such CMB polarisation simulated maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Finally, in Section 7 we summarise our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  2  Spin field as a scalar field in SO(3)  The CMB temperature anisotropies map can be seen as a real scalar field defined on the  sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Thus, its statistical properties can be analysed with plenty of tools such as the  MFs formalism, firstly introduced in the Cosmological literature in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This tool is used to  describe several characteristics of the excursion sets of the fields at different thresholds, which  define their geometry and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However, CMB polarisation has a different geometrical  structure: it constitutes a spin 2 complex field on the sphere [25], for which excursion sets  cannot be directly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In order to overcome this issue, we lift the field to a higher  dimensional space where it can be seen as a scalar field, following the framework introduced  in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' See also [27] for further mathematical discussion on spin random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='com/javicarron/pynkowski  – 2 –  Let Q(φ, θ) and U(φ, θ) be the maps of Stokes parameters for linear polarisation in the  usual base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We define f(φ, θ, ψ) : SO(3) → R as:  f(φ, θ, ψ) = Q(φ, θ) cos(2ψ) − U(φ, θ) sin(2ψ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1)  which can be understood as the linear polarisation observed at the point (φ, θ) on the sky in  the polarisation direction ψ (in the system used to define Q and U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The field f is now a  three–dimensional scalar field, for which excursion sets and MFs are properly defined, as we  will see in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The domain of this function, SO(3), requires further technical discussion, to which we  dedicate the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Keeping in mind the interpretation of the function  variables (φ, θ, ψ) as the position and the polarisation direction, it can be seen that the domain  of f must cover all points of the sphere and all possible polarisation directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The domain  must then be contained in a three–dimensional hypersphere, S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Since the CMB polarisation  is a spin 2 field, we can perform the identification (φ, θ, ψ) ∼= (φ, θ, ψ + π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Therefore, the  domain space can actually be seen as half a 3-sphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' this space is diffeomorphic to SO(3),  the set of rotations of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The coordinates are the longitude φ ∈ [0, 2π], the latitude  θ ∈ [− π  2 , π  2 ], and the polarisation angle ψ ∈ [0, π], noting that this parameterisation fails at a  (zero–measure) number of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  By following this framework, we can lift the complex spin field on the sphere to a  complex scalar field on SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However, the real and imaginary parts of this field are just  translations of each other, so it is enough to study only the real part in order to characterise  the geometry and topology of the polarisation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This real part is what we have called f  in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  An important consequence of this construction is that if we consider an isotropic spin 2  field on the sphere, it does not constitute an isotropic field on S3 nor SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Physically, this  can be seen as a consequence of the different behaviour of the polarisation coordinate and  the sky coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Mathematically, an isotropic random field on S3 has to be invariant in  law to the action of any isometry of S3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' it can be proven that this is only satisfied if every  multipole component of the field is a sum of fields with spin s = −ℓ, −ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' , ℓ − 1, ℓ, each  spin component with equal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This can never be the case for fields produced by lifting a  spin field, such as CMB polarisation, given that all multipoles are constituted only by s = ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The details of this construction and the consequence on random fields can be found in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Although f is not isotropic on SO(3), we still have enough information to produce  accurate predictions for its MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' It is enough to assume isotropy on the sphere (S2) and  the fact that we have a spin 2 field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', knowledge of the behaviour of the polarisation  coordinate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We will exploit these factors in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In order to correctly compute the derivatives of the field needed to estimate the MFs, we  have to take into account the geometry of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The metric of SO(3) in this framework  is given by:  gµν =  �  �  2  0 2 sin(θ)  0  2  0  2 sin(θ) 0  2  �  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2)  where the order is (φ, θ, ψ), see Appendix A for details on the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' With this metric,  – 3 –  the volume element and total volume of SO(3) are:  dV = cos(θ) dφ dθ dψ  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3)  � 2π  0  �  π  2  − π  2  � π  0  cos(θ) dφ dθ dψ = 4π2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4)  One consequence of this metric is that the base of the tangential space given by the  derivatives  �  ∂  ∂φ, ∂  ∂θ, ∂  ∂ψ  �  is not orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' It will be useful to work in the orthonormal  basis {e1, e2, e3} given by the following expression:  e1 =  √  1 − sin θ +  √  sin θ + 1  2  √  2 cos θ  ∂  ∂φ +  √  1 − sin θ −  √  sin θ + 1  2  √  2 cos θ  ∂  ∂ψ  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5a)  e2 =  1  √  2  ∂  ∂θ  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5b)  e3 =  √  1 − sin θ −  √  sin θ + 1  2  √  2 cos θ  ∂  ∂φ +  √  1 − sin θ +  √  sin θ + 1  2  √  2 cos θ  ∂  ∂ψ  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5c)  In this basis, the gradient and the Hessian of a function f can be expressed as  ∇f =  �  �  e1  e2  e3  �  � f  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='6)  Hij(f) = ∇2f(ei, ej) = eiejf − (∇eiej)f  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='7)  The expressions of these operators in terms of the usual spatial derivatives  �  ∂  ∂φ, ∂  ∂θ, ∂  ∂ψ  �  can be found in Appendix A, together with the computations to obtain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' These results  are needed to compute the MFs of arbitrary polarisation maps, as we will see in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  3  Minkowski Functionals  Minkowski Functionals are statistics that quantify the morphology produced by a scalar  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' They are higher order statistics, meaning that their value cannot be fully predicted  from any n–points correlation function, rendering MFs a useful complementary tool to the  angular power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Although typically employed on the sphere, they can be used on  any manifold, as we will show in this section for the case of SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  Definition  Let f : SO(3) → R be a C2 function, and u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The excursion set of f at threshold u, called  Au(f, SO(3)), is defined as the collection of all points in the domain where the function is  larger than (or equal to) u:  Au(f, SO(3)) = {x ∈ SO(3) : f(x) ≥ u} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1)  In the following, we will refer to it simply as Au to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We will use the  standard notation ∂Au to refer to the boundary of Au, corresponding to the points with a  value of exactly u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  – 4 –  In the case of the sphere, or any 2–dimensional manifold, we can define 3 independent  MFs to fully characterise the morphology of the excursion set, as shown in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In a 3–  dimensional manifold, like SO(3), we can define 4 independent MFs:  V0(Au) =  �  Au  dx  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2a)  V1(Au) = 1  6  �  ∂Au  da  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2b)  V2(Au) = 1  6π  �  ∂Au  H(a) da  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2c)  V3(Au) = 1  4π  �  ∂Au  K(a) da  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2d)  where da denotes an area element along the boundary surface ∂Au, H(a) and K(a) are,  respectively, the mean curvature (average of the two principal curvatures) and the Gaussian  curvature (product of the two principal curvatures) at a point a on the boundary surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In general, in a n-dimensional space, we can define V0 as the hypervolume of the excursion  set, and n independent MFs as integrals on the boundary of a combination of the principal  curvatures, see [13] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We follow the normalisation convention in this reference,  noting that other normalisations have been used before, more notably in [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  MFs can be interpreted as geometrical descriptors of the excursion set as a function of  threshold and, therefore, they represent a statistical characterisation of the original function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  V0 is the total volume of the excursion set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' V1 is the area of its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' To our knowledge, V2  does not have a further interpretation beyond the average mean curvature of the boundary  of the excursion set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Finally, V3 is connected to the Euler–Poincaré characteristic χ due  to an extension of the Gauss–Bonnet theorem known as the Chern–Gauss–Bonnet theorem  [originally proven in 30] which states that:  χ(Au) = V3(Au) + 1  4V1(Au)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3)  Similarly to [24], we introduce the Lipschitz–Killing curvatures, as the theoretical pre-  dictions are obtained for these quantities (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Given a tube of width ρ built  around the manifold Au, its volume can be exactly expressed as a finite Taylor expansion on  ρ, whose coefficients correspond to the Lipschitz–Killing curvatures of Au (see [28, 31] for a  detailed description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' These quantities are connected to the MFs in the following way:  L3(Au) = V0(Au)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4a)  L2(Au) = 3V1(Au)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4b)  L1(Au) = 6V2(Au)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4c)  L0(Au) = V3(Au)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4d)  In order to compute the theoretical predictions for a Gaussian field we have to invoke  the Gaussian Kinematic Formula, as we shall see in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  Gaussian Kinematic Formula  The theoretical expected values of the Lipschitz–Killing curvatures (and therefore of MFs) for  a Gaussian field can be computed with the Gaussian Kinematic Formula, as done in [24] for  the polarised intensity of the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  – 5 –  In our case, we want to study the function f = Q(φ, θ) cos(2ψ) − U(φ, θ) sin(2ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This  function is not fully isotropic, as the behaviour in the θ and φ directions is different than the  behaviour in the ψ direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This is connected to the fact that spin s functions on the sphere  correspond to functions on SO(3) with a single spin component s, but isotropic functions  of SO(3) must have components at all spins, and with the same angular power spectrum,  as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We shall see that this anisotropy introduces some non–negligible  changes in the computations of the expected values for the MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Without loss of generality, we normalise the function f to have unit variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' if Q  and U have the same variance (as required by isotropy on the sphere), this is equivalent to  normalising both of them to unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Again as a consequence of spherical isotropy, the  variance of all first derivatives are equal, and the covariance between orthogonal directions is  zero:  E  ��∂Q  ∂θ  �2�  = E  ��∂U  ∂θ  �2�  = E  ��∂Q  ∂φ  �2�  = E  ��∂U  ∂φ  �2�  = µ  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5a)  E  �∂Q  ∂θ · ∂Q  ∂φ  �  = E  �∂U  ∂θ · ∂U  ∂φ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5b)  The value of µ can be directly computed in the following way:  µ =  �  ℓ  2ℓ + 1  4π  (ℓ − s)(ℓ + s + 1)  2  Cℓ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='6)  where s = 2 and:  Cℓ = 1  2  �  CEE  ℓ  + CBB  ℓ  �  ,  with CEE  ℓ  and CBB  ℓ  the E and B angular power spectra, respectively, computed from Q and U  maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We note that µ ≫ 1, as a direct consequence of normalising the field to unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The last quantity we need to introduce is the covariance matrix of the derivatives of  f, which can be defined as Σij = E  �  ∂f  ∂ei · ∂f  ∂ej  �  , using the orthonormal basis {e1, e2, e3},  introduced in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Specifically, its determinant is:  |Σ|  1  2 =  5  2  √  2 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='7)  We note that, in a fully isotropic case, |Σ|  1  2 would scale as µ3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Thus, the anisotropy of f  will introduce different µ–scaling factors in the prediction of the MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The Gaussian Kinematic Formula can be formulated for non–isotropic Gaussian func-  tions with a caveat: the Lipschitz–Killing curvatures must be computed not with the usual  metric of the ambient manifold (see [13, 24]), but with a metric given by the covariance matrix  Σ introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We denote Lf  i (Au) the Lipschitz–Killing curvatures computed with this  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' For the Gaussian function f : M → R, the Gaussian Kinematic Formula takes the  following form:  E  �  Lf  i (Au)  �  =  dim(M)−i  �  k=0  �k + i  k  �  ρk(u)Lf  k+i(M)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8)  where the flag coefficients are [28]:  � k + j  k  �  = ωk+j  ωkωj  � k + j  k  �  , ωj =  πj/2  Γ( j  2 + 1)  ,  – 6 –  with ωj representing the volume of the j–dimensional unit ball, while the functions ρk(u) are  defined as:  ρk(u) =  1  (2π)k/2  1  √  2π exp  �  − u2  2  �  Hk−1(u) ,  H−1(u) = 1 − Φ(u) ,  H0(u) = 1 ,  H1(u) = u ,  H2(u) = u2 − 1 ,  Hk(u) = (−1)k exp  �  u2  2  �  dk  duk exp  �  − u2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The function Φ represents the cumulative normal distribution and Hk are the Hermite poly-  nomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In the case of an isotropic function with the normalisation described above, it can be  proven that Lf  j (M) = Lj(M)µj/2, with 0 ≤ j ≤ dim(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This identity simplifies equa-  tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8) to the Gaussian Kinematic Formula described in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In the non–isotropic case, the relation between the Lipschitz–Killing curvatures com-  puted with both metrics is not trivial, but there are two important observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' First, L0  corresponds to the Euler–Poincaré characteristic, which is a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This means  that this quantity depends only on the topological structure and not on the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' There-  fore, Lf  0(M) = L0(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Second, there is a useful identity for Ldim(M)(M): this quantity  corresponds to the total volume of the manifold M or, more technically, to its Hausdorff  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Computing it with a different metric is analogous to a change of variable in the  integral of the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Therefore, Lf  dim(M)(M) = |Σ|1/2Ldim(M)(M), which, thus, in our  case can be rephrased by: Lf  3(M) =  �  5  2  √  2 µ  �  L3(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Note that if we ignore the anisotropy of f, we would obtain that the factor is ∝ µ3/2  instead of ∝ µ, yielding an incorrect behaviour with µ and therefore with the angular power  spectrum of the polarisation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This is one of the most remarkable advantages of adopting  this general formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Finally, the cases of Lf  1(M) and Lf  2(M) are more complicated, as they are related to the  change of integrating submanifolds with lower dimensionality when the global (non-isotropic)  metric changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Our conjecture is that the scaling relations should be Lf  1(M) ∝ L1(M) and  Lf  2(M) ∝ L2(M)µ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 we will see that these guesses are highly compatible  with simulations, and we will compute the proportionality factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3  Theoretical predictions  We are now in the position to explicitly compute the theoretical predictions for the MFs  of the excursion set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In our case, the global manifold is M = SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We note that the  right hand side of the Gaussian Kinematic Formula, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8), always contains the term  proportional to L3(SO(3)) as the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  As mentioned above, the last Lipschitz–  Killing curvature is equal to the Hausdorff measure (or volume) of the manifold, which in this  case is Vol(SO(3)) = 4π2, as shown in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  – 7 –  Only L3 is needed to compute all MFs at leading order but we report the rest of the  Lipschitz–Killing curvatures of SO(3) for completeness:  L3(SO(3)) = 4π2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='9a)  L2(SO(3)) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='9b)  L1(SO(3)) = 6π  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='9c)  L0(SO(3)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='9d)  For the topology–inclined readers, the last line implies that the Euler–Poincaré characteristic  vanishes: χ(SO(3)) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' this is a consequence of the 3-sphere being a (2-fold) cover of SO(3)  and χ(S3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We can now compute the Lipschitz–Killing curvatures of the excursion set of f using the  Gaussian Kinematic Formula, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We assume no mask in these computations, but  they can be readily introduced by modifying the Lipschitz–Killing curvatures of the global  manifold, most notably by multiplying the total volume by the sky fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Volume of the excursion set, V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We use the Gaussian Kinematic Formula with j = 3  to compute the expected value of the volume of the excursion set, along with the expression  Lf  3(M) =  �  5  2  √  2 µ  �  L3(M) derived above:  E [L3(Au)] = 2  √  2  5µ E  �  Lf  3(Au)  �  =  = 2  √  2  5µ  3−3  �  k=0  �k + 3  k  �  ρk(u)Lf  k+3(SO(3)) =  = 2  √  2  5µ  �3  0  �  ρ0(u)  � 5  2  √  2 µL3(SO(3))  �  = 4π2 [1 − Φ(u)]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='10)  where Φ(u) is the cumulative distribution function of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  It can be seen that V0 is equal to the total volume at low thresholds (since the field in all  points is greater than u), and it is 0 at high thresholds (since the field in all points is lower  than u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Interestingly, the definition of f implies that at u = 0, L3(A0) = 2π2 with no  scattering whatsoever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We note that the factor µ cancels out and the theoretical expectation  depends only on the threshold u, not on the field itself (as long as it is normalised to have  unit variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The summation in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='10) has exactly one term, corresponding to the  volume of SO(3), so no approximation is needed for this prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Area of the boundary of the excursion set, V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We can compute the theoretical ex-  pectation by setting j = 2 in the Gaussian Kinematic Formula and assuming the scaling  relation explained before: Lf  2(M) = K−1  1 µ1/2L2(M), where we have introduced the unknown  – 8 –  constant K1 (as it is related to V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Thus:  E [L2(Au)] = K1µ−1/2E  �  Lf  2(Au)  �  =  = K1µ−1/2  3−2  �  k=0  �k + 2  k  �  ρk(u)Lf  k+2(SO(3)) =  = K1µ−1/2  �3  1  �  ρ1(u)  � 5  2  √  2 µL3(SO(3))  �  =  = K1  5  √  2  √µ 4π2  � 1  2π exp  �−u2  2  ��  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='11)  where the square bracket in the fourth line corresponds to ρ1(u) and the term k = 0 in the  summation is zero because of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Mean curvature of the boundary of the excursion set, V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In a similar way, we  can compute the theoretical expectation of V2 by setting j = 1 in the Gaussian Kinematic  Formula and assuming the scaling relation explained before Lf  1(M) = K−1  2 L1(M), where we  again introduce an unknown constant K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Thus:  E [L1(Au)] = K2E  �  Lf  1(Au)  �  =  = K2  3−1  �  k=0  �k + 1  k  �  ρk(u)Lf  k+1(SO(3)) =  = K2  �3  2  �  ρ2(u)  � 5  2  √  2 µL3(SO(3))  �  + O(Lf  1(SO(3))) =  = K2  5  √  2µ 4π2  �  u  (2π)3/2 exp  �−u2  2  ��  + O(µ0)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='12)  where the square bracket in the fourth line corresponds to ρ2(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Only two terms in the  summation are not zero, one of order µ and another of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Given that µ ≫ 1 (in the  case of cosmological fields, we typically have µ ∼ 105), we neglect the second order terms in  the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Euler–Poincaré characteristic of the excursion set, V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Finally, we compute the the-  oretical prediction for the Euler–Poincaré characteristic of excursion sets of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' As explained  before, Lf  0(M) = L0(M), since it is a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Thus:  E [L0(Au)] = E  �  Lf  0(Au)  �  =  =  3−0  �  k=0  �k + 0  k  �  ρk(u)Lf  k+0(SO(3)) =  =  �3  3  �  ρ3(u)  � 5  2  √  2 µL3(SO(3))  �  +  �1  1  �  ρ1(u)  �  Lf  1(SO(3))  �  =  5  2  √  2 µ 4π2  �(u2 − 1)  (2π)2  exp  �  −u2  2  ��  + O(µ0)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='13)  where the square bracket in the fourth line corresponds to ρ3(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Again, only two terms in  the summation are not zero, one of order µ and another of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The latter, the term  corresponding to Lf  1(SO(3)), can be safely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  – 9 –  We note that neglecting the anisotropy of f would yield an incorrect exponent for µ in  the the theoretical prediction for L0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', the Euler-Poincaré characteristic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Predictions for the MFs  We can now convert the predictions for Lipschitz–Killing curva-  tures into MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Additionally, in order to ease the comparison with data and the interpretation,  we work with normalised MFs, where the quantities are divided over the volume (in our case  this is 4π2, possibly multiplied by the sky fraction fsky if we impose a mask):  vi = Vi  4π2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='14)  The theoretical predictions of the normalised MFs are as follows:  E [v0] = 1 − Φ(u)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='15a)  E [v1] = K1  5  6π  √  2 µ1/2 exp  �−u2  2  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='15b)  E [v2] = K2  5  24π3/2 µ u exp  �−u2  2  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='15c)  E [v3] =  5  8  √  2 π2 µ (u2 − 1) exp  �−u2  2  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='15d)  We remind that the first two formulae are exact, while the last two are correct at the leading  order in µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' One of the main advantages of this formalism is that the expected value of the  normalised MFs do not depend on the use of masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' These theoretical predictions show the  power of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We are able to predict a topological feature of the field knowing  only the parameter µ, which can be computed from the angular power spectra of E- and B-  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Alternatively, this parameter can also be computed directly on the maps, which may  be recommended when the maps are masked in order to avoid leaking effects which distort  the estimation of the polarisation angular power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  4  Implementation: Pynkowski  We implement both the theoretical predictions (presented in Section 3) and the computation  of the MFs of f on data (to be explained in this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Both aspects are to be included  in the publicly available Python package called Pynkowski, which can be found in https:  //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='com/javicarron/pynkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' It was first introduced in [24] in the context of MFs  for scalar maps like T and P 2 = Q2 + U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' It has been designed to compute MFs on different  kinds of data, as well as the theoretical predictions for specific types of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We develop the software in order to compute all the necessary quantities related to f,  including its derivatives in all directions, and the covariant gradient and Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We use this  code on Gaussian simulations in order to compare the results with the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  After such a validation, it can be used on arbitrary maps in order to assess any deviation  from statistical isotropy and Gaussianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The actual implementation of the computation of  these MFs on maps is described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  For a map f, we store the values of Q and U in the HEALPix pixelisation scheme  [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The variable ψ is not pixelised, it is computed exactly for each pixel ξi as f(ξi, ψ) =  Q(ξi) cos(2ψ) − U(ξi) sin(2ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  – 10 –  The first and second spatial derivatives of Q and U with respect to θ and φ are also  computed when needed, with the help of the healpy2 function alm2map_der, which performs  such computation in harmonic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The first and second spatial derivatives of f with  respect to its three variables are simplified analytically and expressed as exact functions of  the polarisation maps and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This means that the only pixelisation is due to  the original pixelisation of Q and U and their derivatives, while the treatment in ψ is always  exact in order to avoid numerical artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The first step in our procedure after loading the map is to normalise f in order to have  unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We do not assume that Q and U have similar statistical properties, but if they  do, this step is equivalent to normalising both of them to have unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' By normalising  f to unit variance, we can express the MFs as a function of an adimensional threshold u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' For  convenience, we compute the MFs divided by the total volume of SO(3) in order to have the  normalised MFs (vi =  Vi  4π2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Before explaining the specific computations for each MF, we shall introduce some com-  mon notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Let P(ξi) =  �  Q(ξi)2 + U(ξi)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', the maximum value of f in each pixel  ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' For a fixed ξi, we note that f has a simple sinusoidal behaviour in its variable ψ: it is  always under u if u > P(ξi), always over u if u < −P(ξi), and over u in a single segment of  ψ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We use this observation to split all the integrals involved in the computation  of the MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In the case where f(ξi) > u only in a range of ψ, let ψM(ξi) be the angle at  which f(ξi, ψ) is maximum, while ψ1(ξi) and ψ2(ξi) the angles for which f(ξi) = u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' they can  be computed as:  ψM(ξi) = 1  2 arctan  �  −U(ξi)  Q(ξi)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1a)  ψ1(ξi) = ψM(ξi) − 1  2 arccos  �  u  P(ξi)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1b)  ψ2(ξi) = ψM(ξi) + 1  2 arccos  �  u  P(ξi)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1c)  where the arctan in the first line is defined to be in the quadrant corresponding to the  coordinate (Q(ξi), U(ξi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The polarisation angles ψ and their differences are always defined  between 0 and π, due to the geometry of SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  First MF, v0  It can be seen as the volume fraction of the manifold for which f > u:  v0(u) =  1  4π2  �  SO(3)  Θ(f(x) − u) dx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2)  where Θ(r) is the Heaviside function (1 where r ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' 0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We can split the integrand  by pixel with the cases explained above and integrate with respect to ψ first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The integrand  will be 0 for pixels where f is always smaller than the threshold (u > P(ξi)) and it will be 1  if u < −P(ξi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' For all the other pixels, it will be 1 as long as ψ is between ψ1 and ψ2 and 0  otherwise, so the integral in ψ will yield the length of the interval between these two angles,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', arccos  �  u  P(ξi)  �  1  π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Therefore, in order to compute equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2), we just need the length of this interval  for every pixel and average it over all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='com/healpy/healpy  – 11 –  Second MF, v1  It can be computed as the area of the boundary of the excursion sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=',  the manifold defined by f = u:  v1(u) =  1  4π2  1  6  �  ∂Au  da  =  1  4π2  1  6  �  SO(3)  δ(f(x) − u) · |∇f(x)| dx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3)  where the second equality comes from a change of coordinates from the surface element of  the boundary da to a volume element on SO(3), dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The gradient of f is denoted by ∇f and  can be computed with equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' δ is the Dirac delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In a fully pixelised field (such as T or P), one typically has to bin the threshold and  approximate the delta function (see [24] for a discussion in that case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However, in this case  we can exactly determine the points where f(ξi, ψ) = u for each pixel (if they exist) as (ξi, ψ1)  and (ξi, ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In order to compute the integral, it suffices to evaluate |∇f(x)| in these points (adding  the values corresponding to both ψ1 and ψ2 for each pixel), fill with 0 the rest of the pixels,  and compute the average value of this quantity over the entire map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This quantity is then  divided over 6, the normalisation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Third MF, v2  It is the average mean curvature (H) of the boundary of the excursion sets  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', the average mean curvature of the manifold defined by f = u):  v2(u) =  1  4π2  1  6π  �  ∂Au  H(a)da  =  1  4π2  1  6π  �  SO(3)  δ(f(x) − u) · H(x) · |∇f(x)| dx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4)  where we use the same change of coordinates as in the previous case, which introduces the  δ and the factor |∇f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The integral with the delta can be computed by evaluating the  integrand, H(x) · |∇f(x)|, on the points where f = u, exactly as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In general, evaluating the mean curvature of a surface is far from trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In this case,  however, the surface is defined implicitly by the function F(x) := f(x) − u = 0, so we can use  the expression for the mean curvature in a implicitly-defined surface [33]:  H = ∇F H(F) ∇F T − |∇F|2 Tr(H(F))  2|∇F|3  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5)  where all quantities are functions of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  As before, we compute the integral by evaluating H(x) · |∇f(x)| in the points where  f = u (adding for each pixel the values corresponding to both ψ1 and ψ2), fill with 0 the rest  of the pixels, and compute the average value of this quantity over all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This quantity is  then divided over 6π, the normalisation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Fourth MF, v3  It can be computed as the average Gaussian curvature (K) of the boundary  of the excursion sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', the manifold defined by f = u):  v2(u) =  1  4π2  1  4π  �  ∂Au  K(a)da  =  1  4π2  1  4π  �  SO(3)  δ(f(x) − u) · K(x) · |∇f(x)| dx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='6)  – 12 –  where we use the same strategy as in the previous case, now computing the Gaussian curvature  K(x) instead of the mean curvature H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In the case of an implicitly defined surface (again  F(x) := f(x) − u = 0), the Gaussian curvature can be computed as [33]:  K = −  ����  H(F) ∇F T  ∇F  0  ����  |∇F|4  = −  ��������  Fxx Fxy Fxz Fx  Fxy Fyy Fyz Fy  Fxz Fyz Fzz Fz  Fx  Fy  Fz  0  ��������  |∇F|4  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='7)  We then compute the integral by evaluating K(x) · |∇f(x)| in the points where f = u  (adding for each pixel the values corresponding to both ψ1 and ψ2), fill with 0 the rest of the  pixels, and compute the average value of this quantity over the whole map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This quantity is  then divided over 4π, the normalisation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  General considerations  All these computations are performed without pixelising the variable ψ, just θ and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However,  numerical issues can arise, especially on v3 and, to a lesser extent, v2, due to the computation  of the second spatial derivatives in pixelised maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In order to avoid artefacts, we must  work with smooth maps, where the value of the field does not change abruptly between  neighbouring pixels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' in other words, the derivatives of the map must exist and be reasonably  smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This can be seen as a requirement on the maps Q and U to have negligible power at  high multipoles or on the needed pixel resolution (the Nside parameter in HEALPix maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  This is not only a computational limitation, but it is, in fact, a mathematical requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Indeed, the Gaussian Kinematic Formula holds if the boundaries of the excursion sets are  twice differentiable (see [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In practice, this means that the angular power spectrum must  decay fast enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' For a study where this condition does not hold, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Nevertheless, we note that the theoretical predictions for v2 and v3 are correct up to  leading order in µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Given that µ increases with the considered range of multipoles, these  approximations are increasingly accurate at higher multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Therefore, it may be conve-  nient to apply this formalism on needlet components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Indeed, needlets naturally filter out  the low and high multipoles, thus reducing both the pixelisation effects and the error in the  approximation in µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' MFs in needlet domain were studied for scalar fields on the sphere in,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', [14, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  5  Simulations  In this section we introduce the different polarisation fields that we analyse with MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  Monochromatic maps  We discussed the theoretical prediction of the MFs in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However, the exact value  of v1 and v2 depend on the unknown normalisation factors K1 and K2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We generate simulated maps in order to verify the the predicted expectation values  for the four MFs, and compute the two unknown normalisation factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We create 1600  monochromatic maps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', maps with power in a single multipole ℓm:  CEE  ℓ  = CBB  ℓ  =  �  1  ℓ = ℓm  0  otherwise  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1)  – 13 –  We simulate 8 sets of 200 maps, each set with a different value of ℓm = {100, 485, 675, 825, 950,  1060, 1165, 1255}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' These values of ℓm are chosen in order to have a set of approximately  equally–spaced values of µm = (ℓm − s)(ℓm + s + 1)/2, where s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  Realistic angular power spectrum  In order to validate the theoretical predictions and the implementation of the computation  of MFs, we generate 300 Gaussian isotropic CMB maps with a realistic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', cosmological)  angular power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We use best fit angular power spectrum for CMB polarisation (E  and B modes) reported by [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We use a map resolution of Nside = 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We apply a  smoothing of the maps with a Gaussian beam of FWHM= 15′ in order to avoid pixelisation  effects in the computation of the spatial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In this way, we are able to simultaneously achieve three goals: assess the accuracy of the  theoretical formulae obtained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3, verify the implementation of MFs computations  on simulated maps, and check that the normalisation constants extend correctly to non–  monochromatic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  6  Results  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  Normalisation constants  In this section we assess the normalisation constants for v1 and v2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', K1 and K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' As  mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3, the exact values for the predicted MFs can only be computed for v0  and v3, while the prediction for v1 and v2 is correct at leading order in µ except for a global  normalisation factor related to the anisotropic ψ direction, see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We note that  both K1 and K2 are truly constants and therefore do not depend on threshold, mask, or value  of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In order to estimate the values of these constants we compute the MFs on the monochro-  matic maps introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 with the software presented in Section 4 and soon to be  included in Pynkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Then, K1 and K2 are obtained as the ratio between computation on  the maps and theoretical predictions, averaging over all simulations and all thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We  find that the constants are (average and standard deviation):  K1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='31912 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='00001  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1a)  K2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='7088 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='0002  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1b)  We have verified that these values do not significantly vary with threshold or value of µ  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', multipole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We have also checked that the constant are valid for non–monochromatic  maps, as we will see in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Lastly, we note that the theoretical predictions yield the correct normalisation factor for  v0 and v3, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' If we repeat the process with additional multiplicative factors, Kv0  and Kv3, for these MFs, we obtain that both of them are compatible with 1:  Kv0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='0000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='0001  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2a)  Kv3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='003  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2b)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  Scaling relations  The theoretical expectation of the MFs in Section 3 assume a particular dependence of µ for  the Lipschitz–Killing computed with different metrics (see the last paragraph in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In this section we show that this scaling is highly favoured by simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  – 14 –  0  10  20  30  40  50  60  70  80  ( 104)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='75  vi  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='5  v1  v2  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Trend with µ for the normalised MFs v1 (blue dots) and v2 (red dots), defined in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We can see in dashed lines the trends µ1/2 (blue) and µ (red), corresponding to the expected scaling  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The assumed µ dependence have a direct impact on the predictions for v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' If  the assumption was wrong, these MFs would present a different scaling with the parameter  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We test for deviations by using the MFs computed on monochromatic maps with a wide  range of values of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In order to assess the dependence of v1 and v2 with µ, we consider vi =  �  vi  vi(ℓm=950)  �  ,  where ⟨·⟩ represents the average over the thresholds between u = −3 and u = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' By consider-  ing this quantity, we remove the dependence on threshold and constant factors, leaving only  the dependence on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In Figure 1 we show the trend of v1 and v2 for different values of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We observe that they  are perfectly compatible with the scaling assumed in the theory: µ1/2 and µ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Additionally, we have verified with the same procedure that v0 and v3 scale as µ0 and  µ, respectively, as predicted by the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We have again verified that all scaling relations  remain unchanged by considering different thresholds and non–monochromatic maps, as ex-  pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3  Gaussian CMB simulations  In this section we verify that the theoretical prediction for the MFs (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3) agree  with the results computed on CMB Gaussian isotropic maps with a realistic angular power  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We use the 300 simulations introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2, which are generated with the  Planck best–fit polarisation angular power spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' we use the value of µ computed from this  angular power spectrum for the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We compute the MFs (see Section 4)  at thresholds between u = −4 and u = 4, with a spacing of ∆u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We use the values of  the normalisation factors K1 and K2 obtained in monochromatic maps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  This means that there are no free parameters in the analysis of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Figure 2 presents the comparison between MFs computed on these maps and the corre-  sponding theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We find full compatibility between theory and simulations  for all the MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In all cases the residuals are well within the 1σ region, where σ is the  dispersion of the MFs among the 300 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  – 15 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='0  V0  Simulation  Theory  5  0  5  V0( 105)  4  3  2  1  0  1  2  3  4  u  2  0  2  V0/  m  0  5  10  15  20  25  30  V1  Simulation  Theory  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='025  V1  4  3  2  1  0  1  2  3  4  u  2  0  2  V1/  m  600  400  200  0  200  400  600  V2  Simulation  Theory  1  0  1  V2  4  3  2  1  0  1  2  3  4  u  2  0  2  V2/  m  5000  4000  3000  2000  1000  0  1000  2000  V3  Simulation  Theory  200  0  200  V3  4  3  2  1  0  1  2  3  4  u  2  0  2  V3/  m  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' MFs for CMB polarisation simulations generated with the Planck best–fit angular power  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Top left: v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Top right: V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Bottom left: V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Bottom right: V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In each panel we show,  from top to bottom: the average value in simulations (orange) and theoretical predictions (black), the  difference between them compared with the standard deviation, σ, and the standard deviation of the  mean, σm (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Additionally, in order to explore any possible systematic deviation, we compare the  average residual on simulations with respect to the theoretical expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This is done by  considering the standard deviation of the mean (σmean = σsims  √  300).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Such comparison is shown in  the bottom rows of each panel in Figure 2, in terms of ∆vi/σm i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', deviations over σmean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' It  can be seen that no point deviates more than 2σmean, and there are no significant systematic  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The four MFs are all perfectly compatible with the theory, both in the case of individual  simulations and of the average behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We verify that the results hold for arbitrary angular  power spectra, as long as there is no significant power in the scales corresponding to the pixel  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We also note the low statistical variation of these curves: the relative uncertainty of  every point is below 1 part in 1000 for v0, below 1% for v1 and v2 and below 10% for v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  These values of the statistical standard deviation depend on the angular power spectrum  of the studied map: we observe that the variance decreases when increasing the maximum  – 16 –  multipole considered, ℓmax (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', when increasing the resolution of the maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' This is also  known to be the case for temperature maps, as quantitatively studied by [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  These results provide a double validation: on the one hand, they verify the mathematical  theory used to predict the expected values of the MFs computed on f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' on the other hand, they  validate our implementation to determine these statistical quantities on polarisation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We have verified that masking the maps has a negligible effect on these results beyond  slightly increasing the noise due to the smaller sky fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  This robustness is expected  because, unlike the angular power spectrum, MFs are purely local quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Thus, the effect  of masking is not propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  7  Conclusions  The study of the angular power spectrum of CMB anisotropies has led to the tightest con-  straints on the cosmological parameters [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' However, this statistical tool is not sensitive  to important information which could be present at some level in CMB data such as non-  Gaussianities or deviations from statistical isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The detection of such effects would reveal  us physical information about the Early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In the past, MFs, applied mainly to CMB temperature anisotropies and, more in general,  to scalar fields, have been used to search for such signatures in Cosmological data [13, 14, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  In this work, we have extended the MFs formalism to the polarisation field by lifting it to  the higher dimensional manifold SO(3), where the complex part is just a translation of the real  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Therefore, without loss of generality, we take just the real part, f(φ, θ, ψ) : SO(3) → R  as:  f(φ, θ, ψ) = Q(φ, θ) cos(2ψ) − U(φ, θ) sin(2ψ)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1)  which is a scalar field that can be understood as the linear polarisation observed at the point  (φ, θ) on the sky in the polarisation direction ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  To summarise:  We introduce a formalism to predict the values of MFs for f in the case of Gaussian  Stokes parameters through the use of the Gaussian Kinematic Formula for anisotropic  fields (see Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We implement a code to compute MFs of f from input Q and U maps pixelised with  the HEALPix convention and to be released in the already publicly available Python  package Pynkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We find that theoretical predictions are compatible with the computations on Gaussian  CMB maps generated with the Planck best-fit angular power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The importance of the application of statistical tools (such as MFs formalism introduced  in this paper) on CMB polarisation data will grow in the future with upcoming much more  sensitive experiments either from ground: ACT [37], SPT [38], Simons Observatory [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' or  from space: LiteBIRD [40], PICO [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  We are currently applying this formalism to the CMB polarisation data from Planck and  simulated future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The corresponding results will be presented in an upcoming  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Furthermore, this kind of analysis can help to blindly detect the presence of non-Gaussian  contaminants in the maps like residual foregrounds contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' It can also be used to  – 17 –  characterise the morphology of the galactic emission, as recently done in [19] for synchrotron  radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  The application proposed in this paper is versatile and can be performed on other spin–2  quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Acknowledgements  The authors thank Michele Stecconi and Maurizia Rossi for insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' MM and  NV acknowledge support by ASI/COSMOS grant n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' 2016-24-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='0 and ASI/LiteBIRD grant  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' 2020-9-HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Part of this work was also supported by the InDark INFN project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' DM  acknowledges support from the MIUR Excellence Project awarded to the Department of  Mathematics, Università di Roma Tor Vergata, CUP E83C18000100006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' DM is also grateful  to the Department of Excellence Programme MatModTov for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content=' Naess, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Abitbol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', The Atacama  Cosmology Telescope: DR4 maps and cosmological parameters, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Cosmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  2020 (2020) 047 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='07288].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Sayre, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Reichardt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Henning, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Ade, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Austermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=',  Measurements of B -mode polarization of the cosmic microwave background from 500 square  degrees of SPTpol data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' D 101 (2020) 122003 [1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='05748].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  [39] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Ade, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Aguirre, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Ahmed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Aiola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Ali, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', The Simons Observatory:  science goals and forecasts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Cosmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' 2019 (2019) 056 [1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='07445].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  [40] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Matsumura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Akiba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Borrill, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Chinone, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Dobbs, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Fuke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', Mission Design of  LiteBIRD, Journal of Low Temperature Physics 176 (2014) 733 [1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2847].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Hanany, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Alvarez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Artis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Ashton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Aumont, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Aurlien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', PICO: Probe of  Inflation and Cosmic Origins, arXiv e-prints (2019) [1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='10541].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  [42] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Gelfand, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Minlos and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Shapiro, Representations of the Rotation and Lorentz  Groups and their Applications, Courier Dover Publications (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  – 20 –  A  Metric and derivatives in SO(3)  In this paper, a key part of the argument is based on defining the field f(φ, θ, ψ) = Q(φ, θ) cos(2ψ)−  U(φ, θ) sin(2ψ), with domain on SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' The geometry of this manifold is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In par-  ticular, we were unable to find in the literature the expressions for the orthonormal basis and  the Hessian of a function defined on this manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We reproduce these computations here in  case they are useful for the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  SO(3) can be seen as the group of rotations of the sphere, which can be parametrised  by the Euler angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' These admit many different definitions, depending on the order of the  axis selected to perform the rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' In order to make the angles compatible with the usual  coordinates on the sphere, we must select the definition in the zyx convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='  Then, a  generic rotation of the sphere can be expressed as [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=', 42]:  R(φ, θ, ψ) = Rx(ψ)Ry(θ)Rz(φ),  where:  Rx(ψ) =  �  �  1  0  0  0 cos (ψ) − sin (ψ)  0 sin (ψ)  cos (ψ)  �  �  Ry(θ) =  �  �  cos (θ) 0 sin (θ)  0  1  0  − sin (θ) 0 cos (θ)  �  �  Rz(φ) =  �  �  cos (φ) − sin (φ) 0  sin (φ)  cos (φ) 0  0  0  1  �  �  It is then possible to compute the derivative of a random rotation with respect to the  three parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' Now, taking the Frobenius inner product between matrices (the sum of the  element–wise multiplication of matrices), we can compute the metric:  gij = ⟨∂iR, ∂jR⟩ = 2  �  �  1  0 sin (θ)  0  1  0  sin (θ) 0  1  �  �  in the order (φ, θ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' We can now determine the Christoffel symbols of this manifold:  Γφ  θψ = Γφ  ψθ = Γψ  φθ = Γψ  θφ =  1  2 cos(θ)  Γφ  θφ = Γφ  φθ = Γψ  ψθ = Γψ  θψ = − tan(θ)  2  Γθ  φψ = Γθ  ψφ = − cos(θ)  2  with the remaining ones equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' With the Chistoffel symbols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' the covariant derivatives  are just:  ∇∂a∂b = Γc  ab∂c  – 21 –  which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' in our case implies:  ∇∂φ∂φ = 0  ∇∂θ∂θ = 0  ∇∂ψ∂ψ = 0  ∇∂φ∂θ = ∇∂θ∂φ = −tan(θ)  2  ∂φ +  1  2 cos(θ)∂ψ  ∇∂ψ∂θ = ∇∂θ∂ψ =  1  2 cos(θ)∂ψ − tan(θ)  2  ∂φ  ∇∂φ∂ψ = ∇∂ψ∂φ = −cos(θ)  2  ∂θ  These covariant derivatives can be then employed to compute the component of the  Hessians according to:  Hij(f) = ∇2f(ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' ej) = eiejf − ∇eiejf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='For the first part of each term we obtain:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e1e1 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin θ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin θ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin θ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin θ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin θ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin θ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='sin (θ) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='�√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4 cos2 (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='− sin (θ) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='e2e3 = e2e3 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin (θ) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='1 − sin (θ) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin (θ) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='while the second part of each term is given by:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin θ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='= sin (2θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8 cos2 θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='1 − sin θ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin θ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin θ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='= sin (2θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8 cos2 θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∇e1e2 = ∇e2e1 =∇ √1−sin θ+√sin θ+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='√1−sin θ−√sin θ+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8 cos2 (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='P(θ) sin (θ) − M(θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='M(θ) sin (θ) − P(θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∇e1e3 = ∇e3e1 =∇ √1−sin θ+√sin θ+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
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+page_content='√1−sin θ−√sin θ+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin θ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin θ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin θ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin θ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='= −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4 cos (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∇e2e3 = ∇e3e2 =∇� √1−sin θ−√sin θ+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√1−sin θ+√sin θ+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8 cos2 (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='M(θ) sin (θ) − P(θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='P(θ) sin (θ) − M(θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='– 23 –  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='where:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='P(θ) :=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin (θ) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin (θ) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='M(θ) :=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1 − sin (θ) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='sin (θ) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content=' the terms of the Hessian are the following:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='H11 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4 cos2 (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='(1 + cos (θ)) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ2 + (1 − cos (θ)) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ2 − 2 sin (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ∂φ − sin (2θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='H22 = ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2∂θ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='H33 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4 cos2 (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='(1 − cos (θ)) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ2 + (1 + cos (θ)) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ2 − 2 sin (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ∂φ − sin (2θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='H12 = H21 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8 cos2 (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='M(θ) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ∂ψ + P(θ) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ∂φ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='+ (P(θ) sin (θ) − M(θ)) ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='+ (M(θ) sin (θ) − P(θ)) ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='H13 = H31 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='4 cos2 (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='− sin (θ) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ2 − sin (θ) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ2 + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ∂φ + cos (θ) ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='H23 = H32 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='8 cos2 (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='2 cos (θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='M(θ) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ∂φ + P(θ) ∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂θ∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='M(θ) sin (θ) − P(θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂φ+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='P(θ) sin (θ) − M(θ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='∂ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
+page_content='– 24 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf'}
diff --git a/ldAyT4oBgHgl3EQfYfc8/content/tmp_files/2301.00203v1.pdf.txt b/ldAyT4oBgHgl3EQfYfc8/content/tmp_files/2301.00203v1.pdf.txt
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@@ -0,0 +1,1419 @@
+MNRAS 000, 1–8 (2023)
+Preprint 3 January 2023
+Compiled using MNRAS LATEX style file v3.0
+The vertex coordinates of the Galaxy’s stellar systems according to the
+Gaia DR3 catalogue
+A. M. Dmytrenko,1★ P. N. Fedorov,1† V. S. Akhmetov,1,2‡ A. B. Velichko1,3 and S. I. Denyshchenko1
+1Institute of astronomy of V.N.Karazin Kharkiv national university, Svobody sq. 4, 61022, Kharkiv, Ukraine
+2INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, Pino Torinese, Turin, I-10025, Italy
+3Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We present the results of determining the coordinates of the vertices of various stellar systems, the centroids of which are located
+in the Galactic plane. To do this, the positions, parallaxes, proper motions, and radial velocities of red giants and subgiants
+contained in the 𝐺𝑎𝑖𝑎 DR3 catalogue have been used. When determining the components of the deformation velocity tensors in
+local coordinate systems, we found the coordinates of the vertices of the stellar systems under study. It turned out that there is a
+complex dependence of vertex deviations 𝑙𝑥𝑦 in Galactocentric cylindrical (𝑅, 𝜃) and Galactic rectangular (𝑋,𝑌) coordinates.
+Based on the approach proposed in this paper, heliocentric distances to vertices have been determined for the first time. The
+results obtained show that in addition to the fact that the spherical coordinates of the Galactic center and the vertices of stellar
+systems do not coincide, their heliocentric distances do not coincide as well. This indicates that there are structures in the Galaxy
+that noticeably affect its axisymmetry.
+Key words: methods: data analysis–proper motions–stars: kinematics and dynamics–Galaxy: kinematics and dynamics–solar
+neighbourhood.
+1 INTRODUCTION
+The third release of the 𝐺𝑎𝑖𝑎 mission, 𝐺𝑎𝑖𝑎 DR3 data (Gaia collab-
+oration et al. (2016, 2022)), made new data available for studying the
+stellar kinematics not only in the Solar neighborhood, but also in a
+significant part of the Milky Way. The availability of high-precision
+data of millions of stars in the releases of the 𝐺𝑎𝑖𝑎 mission makes it
+possible to obtain new information about the kinematics of stars.
+Particularly valuable for kinematic studies is the availability of
+data on radial velocities and parallaxes, which, together with the stel-
+lar proper motions, make it possible to analyze the three-dimensional
+velocity field V(r). This entails the emergence of new possibilities
+for determining some global kinematic parameters, as shown in the
+works by Fedorov et al. (2021, 2023). In this work, we present the
+results of determining the kinematic centers of rotation of various
+stellar systems. And although there are certain difficulties in inter-
+preting the results obtained, caused, for example, by the discrepancy
+between the distances to sources determined from the 𝐺𝑎𝑖𝑎 paral-
+laxes and using the Bayesian method (Bailer-Jones et al. (2021)),
+usage of different estimates of the distances 𝑅⊙ to the Galactic cen-
+ter, or relatively small number of stars with known radial velocities
+(∼33 million), the relevance of such works is beyond doubt.
+In this paper, we determine the vertex coordinates of stellar sam-
+ples whose centroids are located in the Galactic plane. One way to
+determine the coordinates of the vertex is to analyze the strain rate
+★ E-mail: astronom.karazin007@gmail.com (AMD)
+† E-mail: pnfedorov@gmail.com (PNF)
+‡ E-mail: akhmetovvs@gmail.com (VSA)
+tensor of the stellar system (Bobylev (2004); Bobylev & Bajkova
+(2020)).
+The paper is structured as follows. In section 2, we describe the ba-
+sic steps for organizing the selection of stellar systems in a rectangular
+coordinate system from giants and subgiants, which are contained in
+the 𝐺𝑎𝑖𝑎 DR3 catalog. In section 3 we present the formulas that
+are used in this paper to calculate the angle 𝑙𝑥𝑦. Section 4 contains
+the progress of solving the problem and analysis of the results of
+determining the coordinates of the vertices.
+2 THE SAMPLE
+It is well known, that the Galactic rectangular coordinate system with
+the origin at the barycenter of the Solar System is defined by a right-
+handed triple of mutually orthogonal unit vectors (i, j, k) directed
+as follows: the X axis from the observer towards the galactic center
+𝐿 = 0◦, 𝐵 = 0◦, the axis Y in the direction of Galactic rotation
+𝐿 = 90◦, 𝐵 = 0◦, the Z axis is parallel to the direction to the
+North Pole of the Galaxy 𝐵 = 90◦. As noted in (Fedorov et al. (2021,
+2023)), a similar coordinate system can be introduced at any arbitrary
+point on the Galactic plane, provided that the spatial coordinates and
+components of the spatial velocity are known for each star. The
+transition from the Galactic Cartesian coordinate system with the
+origin at the barycenter of the Solar System (𝑋𝑌𝑍) to such a local
+Cartesian system (𝑥𝑦𝑧) with the origin at the chosen point (𝑥𝑦𝑧) is
+equivalent to moving a fictitious observer from the barycenter of the
+Solar System to the point specified by the coordinates of the chosen
+origin of the system. In the local Cartesian coordinate system, as
+© 2023 The Authors
+arXiv:2301.00203v1  [astro-ph.GA]  31 Dec 2022
+
+2
+Dmytrenko et al.
+Figure 1. Selection of red giants and subgiants.
+well as in the Galactic Cartesian system, the x-axis (𝑙 = 𝐿 = 0◦,
+𝑏 = 𝐵 = 0◦) is always directed from a particular centroid to the
+center of the Galaxy, the 𝑂𝑦-axis (𝑙 = 𝐿 = 90◦ , 𝑏 = 𝐵 = 0◦) in
+the direction of galactic rotation and perpendicular to 𝑂𝑥, while the
+𝑂𝑧-axis (𝑏 = 𝐵 = 90◦) is always perpendicular to the plane of the
+Galaxy. The orientation of the x and y axes of the local coordinate
+systems was specified using the value R⊙ = 8.28 kpc. (GRAVITY
+Collaboration et al. (2021)). In a particular case, for an observer
+who is in the Sun, the local coordinate system will coincide with the
+rectangular Galactic coordinate system.
+In this work, 33 million stars from 𝐺𝑎𝑖𝑎 DR3 were selected for
+which the radial velocities are known. From this sample, as in our
+previous work (Fedorov et al. (2023)), were excluded those stars for
+which the following conditions are not satisfied (Lindegren et al.
+(2018)):
+����
+����
+𝑅𝑈𝑊𝐸 > 1.4,
+𝜋/𝜎𝜋 > 5,
+(𝜇𝛼/𝜎𝜇𝛼)2 + (𝜇𝛿/𝜎𝜇𝛿)2 > 25.
+By cutting off the main sequence on the 𝑀𝐺 − (𝐵𝑃 − 𝑅𝑃)
+Hertzsprung-Russell diagram with two linear functions, as shown
+in Fig. 1, from the approximately 30 million stars (remaining after
+applying the Lindegren criteria), giants and subgiants were selected.
+Finally, a sample of approximately 15 million giants and subgiants
+was used further in kinematic studies. The points from which, fic-
+titious observations were made by a fictitious observer, were given
+as follows. Firstly, we single out spherical regions with a radius of
+1 kpc, whose centers are located at the nodes of a rectangular grid
+coinciding with the Galactic plane. Coincidence with the Galactic
+plane is provided by setting the condition 𝑍 = 0 for the coordinates
+of any node. Thus, the position of each node is uniquely specified by
+a pair of coordinates 𝑋 and 𝑌. The distance between adjacent nodes
+along both coordinates was set equal to 250 pc. Each sphere circum-
+scribed around a given node includes stars located at distances not
+exceeding 1 kpc from it. Thus, the nodes of the rectangular grid are
+centroids whose velocities are equal to the average velocity of the
+stars located within the corresponding spheres.
+3 DETERMINING THE COORDINATES OF THE VERTEX.
+In this paper, deformation velocity tensors were computed for all
+stellar systems whose centers are located at the nodes of a rectangular
+grid, as indicated above (see also Fedorov et al. (2023)). If in the local
+Cartesian coordinate system the radius-vector of stars is denoted as
+r = (𝑥, 𝑦, 𝑧) = 𝑞𝑖, and their velocities are V = (𝑉𝑥,𝑉𝑦,𝑉𝑧) = 𝑉𝑖,
+then from the expansion of the velocity field V(r) in the vicinity of
+the node (centroid), we get:
+𝑉𝑖(𝑑r) = 𝑉𝑖(0) + 1
+2
+� 𝜕𝑉𝑖
+𝜕𝑞𝑘
+− 𝜕𝑉𝑘
+𝜕𝑞𝑖
+�
+0
+𝑑𝑞𝑘 + 1
+2
+� 𝜕𝑉𝑖
+𝜕𝑞𝑘
++ 𝜕𝑉𝑘
+𝜕𝑞𝑖
+�
+0
+𝑑𝑞𝑘,
+(1)
+where
+1
+2
+� 𝜕𝑉𝑖
+𝜕𝑞𝑘
+− 𝜕𝑉𝑘
+𝜕𝑞𝑖
+�
+0
+= 𝜔𝑖𝑘 = −𝜔𝑘𝑖 = 𝑀−
+are components of antisymmetric and
+1
+2
+� 𝜕𝑉𝑖
+𝜕𝑞𝑘
++ 𝜕𝑉𝑘
+𝜕𝑞𝑖
+�
+0
+= 𝑚+
+𝑖𝑘 = 𝑚+
+𝑘𝑖 = 𝑀+
+symmetric tensors respectively. 𝑖, 𝑘 = 1, 2, 3.
+By making the appropriate substitutions, this equation can also be
+written in Galactic spherical coordinates. In this form, it is known
+in the literature as the Ogorodnikov–Milne (O–M) kinematic model
+(Ogorodnikov (1932, 1965)). The second rank symmetric tensor 𝑀+
+is called the deformation velocity tensor. The matrix of this tensor
+has the form:
+𝑀+ = ��
+�
+𝑚+
+11
+𝑚+
+12
+𝑚+
+13
+𝑚+
+21
+𝑚+
+22
+𝑚+
+23
+𝑚+
+31
+𝑚+
+32
+𝑚+
+33
+��
+�
+.
+(2)
+The 𝑀+ tensor completely determines the rate of deformation
+motion in the stellar system under consideration. In the rectangular
+Galactic coordinate system, it has 9 components, 6 of which are in-
+dependent. As is known, the kinematic interpretation of the diagonal
+components of the tensor 𝑚+
+11, 𝑚+
+22, 𝑚+
+33 is that these quantities are
+the velocities of relative elongation (contraction/expansion) along
+the axes of the coordinate system, and the non-diagonal components
+𝑚+
+12 = 𝑚+
+21, 𝑚+
+13, = 𝑚+
+31, 𝑚+
+23 = 𝑚+
+32, characterize the velocities of
+angular deformation in the planes (𝑥𝑂𝑦), (𝑦𝑂𝑧) and (𝑥𝑂𝑧), respec-
+tively. The velocity of angular deformation is understood as a change
+in the right angle in these planes as a result of deformation.
+It is convenient to present the results of calculations in the cylin-
+drical Galactocentric coordinate system 𝑅, 𝜃, 𝑍, since its unit vectors
+are parallel to the unit vectors of local rectangular coordinate systems
+in each centroid. For example, a node centered on the Sun will have
+the following coordinates: 𝑅 = 𝑅⊙ = 8.28 kpc, 𝜃 = 180◦, 𝑍 = 0 kpc.
+Fig. 2 shows the dependencies of the diagonal components of
+the deformation velocity tensors 𝑚+
+11, 𝑚+
+22, 𝑚+
+33 in the Galactocen-
+tric cylindrical coordinates 𝑅. These dependencies are additionally
+color-coded for different values of the coordinate 𝜃. Fig. 3 shows
+the dependencies of the non-diagonal components of the velocity
+deformation tensors 𝑚+
+12, 𝑚+
+13, 𝑚+
+23 on 𝑅 and 𝜃. It is known from
+continuum mechanics that no matter how a particle of a continuous
+medium moves, all its deformation can be reduced to the simplest
+- expansion (contraction) along three mutually perpendicular direc-
+tions, which are the main axes 𝑥′𝑦′𝑧′ of the tensor. In the system of
+its main axes 𝑥′𝑦′𝑧′, the velocity deformation tensor 𝑀+ will have
+the following matrix:
+𝑀+ = ��
+�
+𝜆1
+0
+0
+0
+𝜆2
+0
+0
+0
+𝜆3
+��
+�
+.
+(3)
+where the diagonal components 𝜆1, 𝜆2, 𝜆3 are the principal values of
+MNRAS 000, 1–8 (2023)
+
+-2
+0
+MG, mag
+2
+4
+6
+8
+0
+0.5
+1
+1.5
+2
+2.5
+3
+BP-RP, magThe vertex coordinates
+3
+Figure 2. Diagonal components of the 𝑀+ tensor as a function of Galacto-
+centric cylindrical coordinates.
+the tensor, which are determined by solving the characteristic cubic
+equation:
+𝑀+ = ��
+�
+𝑚+
+11 − 𝜆1
+𝑚+
+12
+𝑚+
+13
+𝑚+
+21
+𝑚+
+22 − 𝜆2
+𝑚+
+23
+𝑚+
+31
+𝑚+
+32
+𝑚+
+33 − 𝜆3
+��
+�
+.
+(4)
+If 𝜆1, 𝜆2, 𝜆3 are positive, then the tensor surface is an ellipsoid, if
+𝜆1, 𝜆2, 𝜆3 have different signs, the tensor surface is a hyperboloid.
+Fig. 4 shows the dependencies of the principal values of the tensor
+in the Galactocentric cylindrical coordinates 𝑅 and 𝜃. As can be
+seen from the figures, 𝜆1 and 𝜆2 have opposite signs and different
+behavior, while the value of 𝜆3 is almost independent of 𝑅 and 𝜃
+and is close to zero on average. Only at the Galactocentric distance
+greater than 11 kpc do we see a slight systematic deviation of 𝜆3 from
+zero, which we neglected in this work. The numerical values of the
+Figure 3. Non-diagonal components of the 𝑀+ tensor as a function of Galac-
+tocentric cylindrical coordinates.
+parameters 𝜆1, 𝜆2, 𝜆3 indicate that the velocity deformation tensor
+in the principal axes is almost independent of the 𝑍 coordinate and,
+therefore, is very close to a flat (two-dimensional) tensor.
+The kinematic interpretation of the parameters 𝜆1, 𝜆2, 𝜆3 in the
+system of principal axes 𝑥′𝑦′𝑧′ is identical to that of 𝑚+
+11, 𝑚+
+22,
+𝑚+
+33 in the system of axes 𝑥𝑦𝑧. These quantities are the velocities
+of relative elongations (contractions/expansions) along the principal
+axes of the tensor. As can be seen from the figures, the transition
+from the Galactic coordinate system to the system of principal axes
+practically does not change the 𝑚+
+33 component. Indeed, it turned out
+that 𝑚+
+33 almost coincides with 𝜆3. However, it is clearly seen that
+the components 𝑚+
+11 and 𝑚+
+22 were not completely transformed into
+𝜆2 and 𝜆1. This is due to the fact that the 𝑚+
+12 component, being
+non-zero, contributes to 𝜆2 and 𝜆1. Considering the components 𝜆2
+and 𝜆1 in the range of 4-12 kpc, one can see that the dependence of
+MNRAS 000, 1–8 (2023)
+
+m
+11
+30
+270
+25
+20
+240
+15
+10
+210
+5
+0
+180%
+-5
+, UI
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpcm
+30
+270
+25
+20
+240
+15
+10
+210
+5
+0
+180%
+-5
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpcm
+33
+30
+270
+25
+20
+240
+15
+kpc*
+10
+210
+5
+0
+180%
+-5
+, U
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpcm
+30
+270
+25
+20
+240
+15
+kpc*
+10
+210
+5
+0
+180%
+-5
+, UI
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpcm
+23
+30
+270
+25
+20
+240
+15
+kpc-
+10
+210
+5
+0
+180%
+-5
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpcm
+23
+30
+270
+25
+20
+240
+15
+kpc-
+10
+210
+5
+0
+180%
+-5
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpc4
+Dmytrenko et al.
+Figure 4. Eigenvalues of the 𝑀+ tensor as a function of Galactocentric
+cylindrical coordinates.
+𝑚+
+11 on 𝑅 and 𝜃, similar to a sinusoidal one, has been transformed
+into an almost analogous dependence of the component 𝜆2. And the
+dependence 𝑚+
+22 on 𝑅 and 𝜃 has been transformed into a similar
+dependence 𝜆1. These facts indicate that the main axes 𝑂𝑥′ and 𝑂𝑦′
+of the deformation velocity tensors are located almost in the Galactic
+plane, and the 𝑂𝑧′ axis practically coincides with the 𝑂𝑧 axis. This
+is also confirmed by the behavior of the 𝑧-dependent parameters
+𝑚+
+33, 𝑚+
+13 and 𝑚+
+23, which were presented above. The values of these
+parameters in the entire range of 𝑅 and 𝜃 are close to zero and
+indicate that the motion of stars is predominantly parallel to the
+Galactic plane. Therefore, in the further analysis, we consider the
+tensor 𝑀+ as flat, i.e. having only four components 𝑚+
+11, 𝑚+
+12, 𝑚+
+21,
+𝑚+
+22, three of which are independent (𝑚+
+12 = 𝑚+
+21):
+𝑀+ =
+�𝑚+
+11
+𝑚+
+12
+𝑚+
+21
+𝑚+
+22
+�
+.
+(5)
+The components of this tensor are related to the gradients of the veloc-
+ity components in the local rectangular Galactic (𝑉𝑥, 𝑉𝑦) and in the
+Galactocentric cylindrical (𝑉𝑅, 𝑉𝜃) coordinate systems along their
+coordinate axes by the following relations (Chandrasekhar (1945);
+Ogorodnikov (1965)):
+𝐴 = 𝑚+
+12 = 1
+2
+� 𝜕𝑉𝑥
+𝜕𝑦 + 𝜕𝑉𝑦
+𝜕𝑥
+�
+= 1
+2
+�
+1
+𝑅
+𝜕𝑉𝑅
+𝜕𝜃 − 𝑉𝜃
+𝑅 + 𝜕𝑉𝜃
+𝜕𝑅
+�
+𝐵 = 𝜔3 = 1
+2
+� 𝜕𝑉𝑦
+𝜕𝑥 − 𝜕𝑉𝑥
+𝜕𝑦
+�
+= 1
+2
+� 𝜕𝑉𝜃
+𝜕𝑅 − 1
+𝑅
+𝜕𝑉𝑅
+𝜕𝜃 + 𝑉𝜃
+𝑅
+�
+𝐶 = 𝑚+
+11−𝑚+
+22
+2
+= 1
+2
+� 𝜕𝑉𝑥
+𝜕𝑥 − 𝜕𝑉𝑦
+𝜕𝑦
+�
+= 1
+2
+� 𝜕𝑉𝑅
+𝜕𝑅 − 1
+𝑅
+𝜕𝑉𝜃
+𝜕𝜃 − 𝑉𝑅
+𝑅
+�
+𝐾 = 𝑚+
+11+𝑚+
+22
+2
+= 1
+2
+� 𝜕𝑉𝑥
+𝜕𝑥 + 𝜕𝑉𝑦
+𝜕𝑦
+�
+= 1
+2
+� 𝜕𝑉𝑅
+𝜕𝑅 + 1
+𝑅
+𝜕𝑉𝜃
+𝜕𝜃 + 𝑉𝑅
+𝑅
+�
+(6)
+where 𝐴, 𝐵, 𝐶 and 𝐾 are parameters similar to the generalized
+Oort constants that can be found for each region of stars under study.
+In continuum mechanics, it is shown that for a two-dimensional
+tensor 𝑀+, the angles 𝛽1 and 𝛽2, which form the main axes of the
+tensor 𝑀+ with the axes 𝑂𝑥 and 𝑂𝑦 of the coordinate system used,
+are found from the expression:
+𝑡𝑔(2𝛽) = 𝑡𝑔(2𝛽1) = 𝑡𝑔(2𝛽2) =
+2𝑚+
+12
+𝑚+
+11 − 𝑚+
+22
+,
+(7)
+where 𝛽1 is the angle between 𝑂𝑥 and 𝑂𝑥′, and 𝛽2 is between 𝑂𝑦
+and 𝑂𝑦′.
+If the components of the 𝑀+ tensor are expressed in terms of the
+parameters 𝐴, 𝐵, 𝐶, 𝐾 then the matrix 5 will have the following form:
+𝑀+ =
+�𝐾 + 𝐶
+𝐴
+𝐴
+𝐾 − 𝐶
+�
+.
+(8)
+and formula 7 will be accordingly transformed to the form:
+𝑡𝑔(2𝛽) =
+2𝐴
+(𝐾 + 𝐶) − (𝐾 − 𝐶) = 𝐴
+𝐶 .
+(9)
+In the case of a purely Oort rotation, there is only the rotational
+component 𝑉𝜃, while the components 𝑉𝑧 and 𝑉𝑅 = 0. In this case,
+𝑉𝜃 does not depend on 𝜃, i.e.
+𝜕𝑉𝑅
+𝜕𝑅 = 0,
+𝑉𝑅 + 𝜕𝑉𝜃
+𝜕𝜃 = 0.
+(10)
+As a result, the terms by which the parameters𝐶 and 𝐾 are determined
+in a cylindrical Galactocentric coordinate system, are equal to zero:
+𝑉𝑅 = 0,
+𝜕𝑉𝜃
+𝜕𝜃 = 0,
+(11)
+and hence 𝐶 and 𝐾 are also equal to zero, and the tensor 𝑀+ will
+have zero diagonal components:
+𝑀+ =
+�0
+𝐴
+𝐴
+0
+�
+.
+(12)
+Thus, for a purely Oort rotation, it follows from formula 9 that the
+main axes of the deformation velocity tensor will be directed at an
+angle 𝛽 = 45◦ to the axes of the local rectangular Galactic coordinate
+system. In this case, the angle 𝛽, for any non-zero values of 𝐴, will
+be equal to 45◦. If 𝐶 or 𝐾 is non-zero, this means that the rotation is
+not Oort (axisymmetric). In this case, the angle 𝛽 ≠ 45◦.
+MNRAS 000, 1–8 (2023)
+
+30
+270
+25
+20
+240
+15
+10
+210
+5
+0
+180%
+-5
+3
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpc30
+270
+25
+20
+240
+15
+10
+210
+kpc'
+5
+0
+180%
+-5
+2
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpc30
+270
+25
+20
+240
+15
+10
+210
+kpc'
+5
+0
+180%
+-5
+-10
+150
+-15
+-20
+120
+-25
+-30
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpcThe vertex coordinates
+5
+Figure 5. The values of the angle between the axes of local Galactic coordinate
+systems and the main axes of the tensor 𝑀+ depending on Galactocentric
+cylindrical coordinates.
+Figure 6. Vertex deviations depending on Galactocentric cylindrical coordi-
+nates (𝑅, 𝜃).
+4 SOLUTION OF THE PROBLEM AND ANALYSIS OF THE
+OBTAINED RESULTS.
+The term vertex is commonly referred to as a point on the sky,
+relative to which the stellar system under consideration rotates. To
+determine the coordinates of the vertex, we used the deformation
+velocity tensors derived by expanding the velocity field of various
+subsamples (stellar systems) described in Section 3.
+Fig. 5 shows the angles 𝛽 between the principal axes 𝑥′ of the
+deformation velocity tensors of the stellar systems under study and
+the 𝑥 axes of the local rectangular Galactic coordinate systems in
+which these tensors have been computed. Fig. 5 shows the changes in
+the angle 𝛽 depending on the cylindrical Galactocentric coordinates
+𝑅 and 𝜃. The angles have been calculated from the formula 7. As
+can be seen from Fig. 5, the values of the angles 𝛽 depending on 𝑅
+and 𝜃 differ markedly. As shown above, with axisymmetric (Oort)
+rotation, the angle 𝑙𝑥𝑦 = (𝛽 − 45◦), called the deviation of the vertex
+longitudes from the direction to the Galactic center, is exactly equal
+to zero and does not depend on the coordinate angle 𝜃.
+In Fig. 6, we present the dependencies of the vertex deviations
+𝑙𝑥𝑦 on the cylindrical coordinates 𝑅, 𝜃. And Fig. 7 shows the same
+values, but in the form of a map, where the rectangular Galactic
+coordinates 𝑋 and 𝑌 are plotted along the axes, and the 𝑙𝑥𝑦 values
+Figure 7. Vertex deviations depending on rectangular Galactic coordinates
+(𝑋, 𝑌 ).
+are displayed in color. The advantage of such a graphical presentation
+of the results is that one can immediately see the behavior of 𝑙𝑥𝑦 in
+the entire range of rectangular Galactic coordinates 𝑋 and 𝑌, where
+the kinematic analysis has been carried out.
+In Fig. 6 it is clearly seen that the angle 𝑙𝑥𝑦 is not equal to zero,
+and it changes with 𝑅 and 𝜃. One can clearly see the “stratification
+and intertwining” of the dependencies 𝑙𝑥𝑦(𝑅) corresponding to var-
+ious fixed values of the angle 𝜃. This behavior of the dependencies
+𝑙𝑥𝑦(𝑅, 𝜃) is probably due to the difference in the deformation veloci-
+ties in different parts of the Galaxy. This assumption is confirmed by
+Fig. 7, which demonstrates the differences in the orientations of the
+tensor surfaces due to the difference in the deformation velocities in
+different parts of the Galaxy. However, the similarity of the depen-
+dencies 𝑙𝑥𝑦(𝑅) in Figs. 6, at different values of the angle 𝜃, as well
+as a noticeable predominance of red color in the upper part of Fig. 7,
+and blue at the bottom, suggests that "stratification and intertwining"
+may not be due to kinematic causes alone.
+One of these reasons may be the incorrect value of the accepted
+Galactocentric distance of the Sun 𝑅⊙ = 8.28 kpc. In this case, the
+use of the accepted value of 𝑅⊙ to determine the rotation angles of
+the 𝑥 axes of local coordinate systems in the direction of the Galactic
+center will cause an inaccuracy in determining these angles. As a
+result, the behavior of 𝑙𝑥𝑦(𝑅, 𝜃) will be determined not only by
+kinematic differences, but also by the orientation inaccuracy of the
+axes of local coordinate systems.
+Checking this assumption, we found that the maximum conver-
+gence of the functions 𝑙𝑥𝑦(𝑅, 𝜃) is realized at a certain value 𝑅V,
+which differs noticeably from the accepted one. In fact, when using
+the vertex coordinates to set the orientation of local coordinate sys-
+tems, the dependencies 𝑙𝑥𝑦(𝑅, 𝜃) become closest and practically turn
+into one, “least stratified” function 𝑙𝑥𝑦(𝑅), which weakly depends
+on 𝜃.
+It also turned out that the use of one vertex does not provide
+the best convergence of the functions 𝑙𝑥𝑦(𝑅, 𝜃) in the entire range
+of distances 𝑅 used. Thus, achieving the best convergence of the
+functions 𝑙𝑥𝑦(𝑅, 𝜃) in the range of 5-–10 kpc results in to a noticeable
+deterioration in convergence within the range of 10–15 kpc (see Fig. 9
+below). This result means that the vertices of different star systems
+are at different distances from the Sun.
+MNRAS 000, 1–8 (2023)
+
+βo
+90
+270
+75
+60
+240
+45
+30
+210
+15
+p
+0
+180%
+-15
+-30
+150
+-45
+-60
+120
+-75
+-90
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpc90
+270
+75
+60
+240
+45
+30
+210
+15
+0
+0
+180%
+-15
+-30
+150
+-45
+-60
+120
+-75
+-90
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpc30
+25
+20
+15
+10
+2
+5
+0
+Y
+-5
+-2
+-10
+.4
+-15
+-20
+-6
+-25
+-8
+-30
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+X, kpc6
+Dmytrenko et al.
+Amendt & Cuddeford (1991); Kuijken & Gilmore (1991); Smith et
+al. (2012) showed that in a stationary, axisymmetric disk galaxy, the
+axes of the stellar velocity ellipsoid of any local stellar system ideally
+coincide with the galactic coordinate axes (e.g., Binney & Tremaine
+(2008); Smith et al. (2012)). The main axes of the deformation ve-
+locity tensor for the Oort (axisymmetric) rotation, as shown above,
+rotate relative to the local axes by an angle of 45◦. It was shown
+by Dehnen (2000); Minchev & Famaey (2010); Vorobyov & Theis
+(2008); Saha et al. (2013) that structures which are not axisymmetric
+can have a noticeable effect on the observed orientation of the stellar
+velocity ellipsoids. They will have a similar effect on the orientation
+of the principal axes of the deformation velocity tensor.
+Therefore, in the general case, when a non-axisymmetric rotation
+is realized, the direction to the Galactic center (a point on the celestial
+sphere with coordinates 𝛼𝐺𝐶 = 266◦,40499, 𝛿𝐺𝐶 = −28◦,93617,
+accepted by the Hipparcos consortium Perryman et al. (1997)) and
+the direction to the vertex (the point of the celestial sphere relative
+to which the stellar system rotates) do not coincide. Our result shows
+that not only the spherical coordinates of the Galactic center and the
+vertices of stellar systems, but also their distances from the Sun do
+not coincide.
+In addition, it turned out that the values of the 𝑅V distance can
+be estimated using the approach we proposed. Assuming that the
+accepted value 𝑅⊙=8.28 kpc is correct, the 𝑥 axes of local coordinate
+systems will always be directed to the same point – the Galactic center
+with coordinates 𝐿 = 0◦, 𝐵 = 0◦, 𝑅⊙=8.28 kpc. In this case, the main
+axis 𝑥′ of the deformation velocity tensor, which is calculated in the
+local coordinate system 𝑋𝑂𝑌, being rotated by 45 degrees, will be
+directed to the vertex. In other words, the longitude of the vertex in
+the local coordinate system will be numerically equal to the angle
+𝑙𝑥𝑦. Using the values of the angles 𝑙𝑥𝑦 calculated for local systems,
+it is possible to construct rays that pass through their vertex point,
+with their origins locating in the centroids.
+To set the equations of rays (straight lines) passing through two
+points, we use rectangular Galactic coordinates of specific centroids
+and points lying on unit circles built around these centroids in the
+𝑋𝑂𝑌 plane. If their radii 𝑟 are taken equal to 1 kpc, then the coordi-
+nates of the second point can be found as follows: 𝑋 = 𝑋𝑐 +𝑟 cos 𝑙𝑥𝑦,
+𝑌 = 𝑌𝑐 −𝑟 sin 𝑙𝑥𝑦. Finding the coordinates of the intersection point of
+two arbitrary rays (straight lines) allows one to find the distance from
+the Sun to this point in the 𝑋𝑂𝑌 coordinate system. Pairwise inter-
+sections of rays form a certain region of intersection in the Galactic
+plane. Fig. 8 show the region of intersection of the rays. Its rather
+large sizes and non-uniformity are visible (there are many peaks or
+nodes). The largest of these ray intersection nodes is located approx-
+imately at a distance of 9.3 kpc from the Sun. At the same time, it
+is not located on the 𝑂𝑋 axis of the rectangular Galactic coordinate
+system, but it is 0.5 kpc away from it along the 𝑂𝑌 axis in the posi-
+tive direction. Similarly, we can select other nodes that can be seen in
+Fig. 8. The presence of many nodes can be explained by the present
+nonlinear dependence 𝑙𝑥𝑦(𝑅, 𝜃), which was considered earlier.
+The coordinates of the vertices 𝐺 𝑗
+V(𝑋 𝑗
+V,𝑌 𝑗
+V) , at which the func-
+tions 𝑙𝑥𝑦(𝑅, 𝜃) will have the best convergence in certain ranges of
+Galactocentric distances 𝑅, were estimated using the least squares
+method. To this end, a system of equations was compiled for those
+rays whose origins got inside the chosen range of Galactocentric
+distances Δ𝑅. An additional condition for including the ray equation
+into the system of equations was the presence of at least 25 thousand
+stars in the stellar system. The solution of the system was the desired
+coordinates of the point of intersection of all rays — the vertex. So,
+for the range 5 < 𝑅 < 10 kpc, we got point 𝐺1
+V with coordinates:
+Figure 8. Rays directed to the vertex in the Galactic plane and the area of
+their intersection.
+Table 1. Results of estimation of Galactic rectangular coordinates of vertices
+𝐺0
+V = 𝐺V
+𝐺1
+V
+𝐺2
+V
+Δ𝑅, kpc
+0–16
+5–10
+10–15
+𝑋𝑉 , kpc
+8.96
+9.35
+8.87
+𝑌𝑉 , kpc
+-0.08
+0.24
+-0.4
+𝑅𝑉 , kpc
+8.96
+9.35
+8.88
+𝜖 (𝑋𝑉 ), kpc
+0.04
+0.03
+0.08
+𝜖 (𝑌𝑉 ), kpc
+0.02
+0.02
+0.03
+𝑋1
+V = 9.35 kpc, 𝑌1
+V = 0.24 kpc. This point is very close to the
+center of the most massive node, which is visible in Fig. 8. For the
+range of Gatatocentric distances 10 < 𝑅 < 15 kpc, we got another
+point - 𝐺2
+𝑉 , which already has different coordinates: 𝑋2
+V = 8.87 kpc,
+𝑌2
+V = −0.4 kpc. Also, we performed an estimate of the coordinates
+of the point 𝐺0
+V = 𝐺V, obtained using the entire available range of
+𝑅 and called by us the general vertex. The coordinates of this point
+are: 𝑋0
+V = 8.96 kpc, 𝑌0
+V = −0.08 kpc.
+In table 1, we provide for all points𝐺 𝑗
+V the coordinates and errors of
+their determination, as well as the calculated heliocentric distances.
+Now we can use the coordinates of the point 𝐺 𝑗
+V(𝑋,𝑌) instead of
+the 𝑋-th coordinate of the Galactic center, equal to 𝑅⊙. This allows
+us to define new Galactocentric cylindrical coordinate systems and
+orient properly the local coordinate system in order to construct new
+dependencies 𝑙𝑥𝑦(𝑅, 𝜃) and 𝑙𝑥𝑦(𝑋,𝑌). These results are shown in
+Figs. 9 and 10.
+Comparing Figs. 6 and 9, one can note an improvement in the con-
+MNRAS 000, 1–8 (2023)
+
+8
+6
+4
+2
+kpc
+0
+K
+-2
+-4
+9-
+-8
+-8
+9-
+-4
+-2
+0
+2
+4
+6
+8
+10
+12
+14
+16
+X, kpc2
+1
+kpc
+0
+Y
+-1
+-2
+4
+5
+6
+7
+8
+9
+10
+11
+12
+X, kpcThe vertex coordinates
+7
+Figure 9. Vertex deviation depending on Galactocentric cylindrical coordi-
+nates. The determined vertex coordinates have been used to set the orientation
+of local Galactic rectangular coordinate systems.
+vergence of the dependence 𝑙𝑥𝑦(𝑅, 𝜃) in those ranges of 𝑅 where the
+corresponding 𝐺 𝑗
+V was used. A similar conclusion can be reached by
+comparing Figs. 7 and 10. It should be noted that the local structures,
+which are clearly visible on the maps, remained the same, which in-
+directly confirms their purely kinematic, and not geometric, nature.
+The deviation of the angle 𝑙𝑥𝑦 in the entire region under study from
+zero can be considered as a measure of the non-axisymmetry of the
+Galaxy.
+5 SUMMER AND CONCLUSIONS
+The use of 𝐺𝑎𝑖𝑎 DR3 data makes it possible to determine the vertex
+coordinates not only in the near–Solar neighborhood, but also in that
+Figure 10. Vertex deviation depending on rectangular Galactic coordi-
+nates.The determined vertex coordinates have been used to set the orientation
+of local Galactic rectangular coordinate systems.
+MNRAS 000, 1–8 (2023)
+
+Ixy° (using Gy)
+90
+270
+75
+60
+240
+45
+30
+210
+15
+0
+0
+180%
+-15
+-30
+150
+-45
+-60
+120
+-75
+-90
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpclky° (using Gly)
+90
+270
+75
+60
+240
+45
+30
+210
+15
+L
+0
+0
+180%
+-15
+-30
+150
+-45
+-60
+120
+-75
+-90
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpcly° (using G2)
+90
+270
+75
+60
+240
+45
+30
+210
+15
+0
+0
+180%
+-15
+-30
+150
+-45
+-60
+120
+-75
+-90
+90
+2
+4
+6
+8
+10
+12
+14
+16
+R, kpclxy° (using Gv)
+30
+8
+25
+6
+20
+15
+4
+10
+2
+5
+0
+Y
+-5
+-2
+-10
+.4
+-15
+-20
+-6
+-25
+-8
+-30
+-8
+9-
+-4
+-2
+0
+2
+4
+6
+8
+X, kpcxy° (using Glv)
+30
+8
+25
+20
+15
+10
+2
+5
+0
+Y
+-5
+-2
+-10
+-4
+-15
+-20
+-6
+-25
+-8
+-30
+-8
+9-
+-4
+-2
+0
+2
+4
+6
+8
+X, kpclxy° (using G2)
+30
+25
+6
+20
+15
+4
+10
+2
+5
+0
+Y
+-5
+-2
+-10
+.4
+-15
+-20
+-6
+-25
+-8
+-30
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+X, kpc8
+Dmytrenko et al.
+part of the Galaxy for which astrometric parameters are available in
+𝐺𝑎𝑖𝑎 DR3. The approach proposed in the works by Fedorov et al.
+(2021, 2023) for the analysis of kinematic parameters, when a ficti-
+tious observer, being at an arbitrary point in the Galaxy, determines
+them in the framework of the O–M model, made it possible to obtain
+a number of new results related to a significant part of the Galaxy.
+In this work, we have applied this approach to determine the coor-
+dinates of the vertices of various stellar systems contained in spher-
+ical regions with a radius of 1 kpc, whose centers are located in the
+Galactic plane. Since the analysis revealed that the 𝑀+ deformation
+velocity tensors calculated in local coordinate systems are almost
+flat, we used only four components 𝑚+
+11, 𝑚+
+12, 𝑚+
+21, 𝑚+
+22. It turned
+out that the deviations of the vertices of these stellar systems obey a
+certain law and are presented mainly in the form of the dependence
+𝑙(𝑅) and weakly depend on 𝑡ℎ𝑒𝑡𝑎. This result could not have been
+obtained without knowledge of the spatial coordinates and velocities
+of the stars contained in 𝐺𝑎𝑖𝑎 DR3 and allowing one to set the local
+Galactic coordinate system at an arbitrary point in the Galactic plane.
+Usually, in works on determining the deviations of the vertex, it
+is explicitly or implicitly assumed that the distance from the Sun to
+the Galactic center and to the vertex are the same. This is indeed true
+for axisymmetric systems. For the case when 𝑚+
+11 and 𝑚+
+22 are not
+equal to zero, we show that the vertices of different stellar systems are
+located at different distances that do not coincide with the accepted
+distance of the Sun 𝑅⊙=8.28 kpc to the center of the Galaxy. This
+indicates that for the investigated part of the Galaxy there is no single
+center of rotation, as in the case of axisymmetric systems. Although
+we cannot specify the exact values of the distances from the Sun to the
+vertices, it is still possible to indicate suitable values of 𝑅V, at which
+the function 𝑙(𝑅), in a specific range of Galactocentric distances, has
+a minimum stratification.
+Our results indicate that not only the spherical coordinates of the
+Galactic center and the vertices of stellar systems do not coincide,
+but also their distances to the Sun do not coincide. These results can
+be useful in many kinematic and dynamic problems.
+6 ACKNOWLEDGEMENTS
+This work has made use of data from the European Space
+Agency (ESA) mission Gaia (https://www.cosmos.esa.int/
+gaia), processed by the Gaia Data Processing and Analysis
+Consortium (DPAC,https://www.cosmos.esa.int/web/gaia/
+dpac/consortium). Funding for the DPAC has been provided by
+national institutions, in particular the institutions participating in the
+Gaia Multilateral Agreement.
+We are immensely grateful to the Armed Forces of Ukraine for
+the fact that in wartime we still have the opportunity to work and do
+science.
+We sincerely thank the anonymous reviewer for a careful reading
+and insightful comments and suggestions on this paper.
+DATA AVAILABILITY
+The used catalogue data is available in a standardised format for
+readers via the CDS (https://cds.u-strasbg.fr). The software code
+used in this paper can be made available upon request by emailing
+the corresponding author.
+REFERENCES
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+
diff --git a/ldAyT4oBgHgl3EQfYfc8/content/tmp_files/load_file.txt b/ldAyT4oBgHgl3EQfYfc8/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/ldAyT4oBgHgl3EQfYfc8/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf,len=433
+page_content='MNRAS 000, 1–8 (2023)  Preprint 3 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='0  The vertex coordinates of the Galaxy’s stellar systems according to the  Gaia DR3 catalogue  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Dmytrenko,1★ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Fedorov,1† V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Akhmetov,1,2‡ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Velichko1,3 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Denyshchenko1  1Institute of astronomy of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='Karazin Kharkiv national university, Svobody sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 4, 61022, Kharkiv, Ukraine  2INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, Pino Torinese, Turin, I-10025, Italy  3Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  We present the results of determining the coordinates of the vertices of various stellar systems, the centroids of which are located  in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' To do this, the positions, parallaxes, proper motions, and radial velocities of red giants and subgiants  contained in the 𝐺𝑎𝑖𝑎 DR3 catalogue have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' When determining the components of the deformation velocity tensors in  local coordinate systems, we found the coordinates of the vertices of the stellar systems under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' It turned out that there is a  complex dependence of vertex deviations 𝑙𝑥𝑦 in Galactocentric cylindrical (𝑅, 𝜃) and Galactic rectangular (𝑋,𝑌) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Based on the approach proposed in this paper, heliocentric distances to vertices have been determined for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The  results obtained show that in addition to the fact that the spherical coordinates of the Galactic center and the vertices of stellar  systems do not coincide, their heliocentric distances do not coincide as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This indicates that there are structures in the Galaxy  that noticeably affect its axisymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Key words: methods: data analysis–proper motions–stars: kinematics and dynamics–Galaxy: kinematics and dynamics–solar  neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  1 INTRODUCTION  The third release of the 𝐺𝑎𝑖𝑎 mission, 𝐺𝑎𝑖𝑎 DR3 data (Gaia collab-  oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2016, 2022)), made new data available for studying the  stellar kinematics not only in the Solar neighborhood, but also in a  significant part of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The availability of high-precision  data of millions of stars in the releases of the 𝐺𝑎𝑖𝑎 mission makes it  possible to obtain new information about the kinematics of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Particularly valuable for kinematic studies is the availability of  data on radial velocities and parallaxes, which, together with the stel-  lar proper motions, make it possible to analyze the three-dimensional  velocity field V(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This entails the emergence of new possibilities  for determining some global kinematic parameters, as shown in the  works by Fedorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2021, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In this work, we present the  results of determining the kinematic centers of rotation of various  stellar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' And although there are certain difficulties in inter-  preting the results obtained, caused, for example, by the discrepancy  between the distances to sources determined from the 𝐺𝑎𝑖𝑎 paral-  laxes and using the Bayesian method (Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2021)),  usage of different estimates of the distances 𝑅⊙ to the Galactic cen-  ter, or relatively small number of stars with known radial velocities  (∼33 million), the relevance of such works is beyond doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In this paper, we determine the vertex coordinates of stellar sam-  ples whose centroids are located in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' One way to  determine the coordinates of the vertex is to analyze the strain rate  ★ E-mail: astronom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='karazin007@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='com (AMD)  † E-mail: pnfedorov@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='com (PNF)  ‡ E-mail: akhmetovvs@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='com (VSA)  tensor of the stellar system (Bobylev (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Bobylev & Bajkova  (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In section 2, we describe the ba-  sic steps for organizing the selection of stellar systems in a rectangular  coordinate system from giants and subgiants, which are contained in  the 𝐺𝑎𝑖𝑎 DR3 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In section 3 we present the formulas that  are used in this paper to calculate the angle 𝑙𝑥𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Section 4 contains  the progress of solving the problem and analysis of the results of  determining the coordinates of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  2 THE SAMPLE  It is well known, that the Galactic rectangular coordinate system with  the origin at the barycenter of the Solar System is defined by a right-  handed triple of mutually orthogonal unit vectors (i, j, k) directed  as follows: the X axis from the observer towards the galactic center  𝐿 = 0◦, 𝐵 = 0◦, the axis Y in the direction of Galactic rotation  𝐿 = 90◦, 𝐵 = 0◦, the Z axis is parallel to the direction to the  North Pole of the Galaxy 𝐵 = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' As noted in (Fedorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2021,  2023)), a similar coordinate system can be introduced at any arbitrary  point on the Galactic plane, provided that the spatial coordinates and  components of the spatial velocity are known for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The  transition from the Galactic Cartesian coordinate system with the  origin at the barycenter of the Solar System (𝑋𝑌𝑍) to such a local  Cartesian system (𝑥𝑦𝑧) with the origin at the chosen point (𝑥𝑦𝑧) is  equivalent to moving a fictitious observer from the barycenter of the  Solar System to the point specified by the coordinates of the chosen  origin of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In the local Cartesian coordinate system, as  © 2023 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='00203v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='GA]  31 Dec 2022  2  Dmytrenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Selection of red giants and subgiants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  well as in the Galactic Cartesian system, the x-axis (𝑙 = 𝐿 = 0◦,  𝑏 = 𝐵 = 0◦) is always directed from a particular centroid to the  center of the Galaxy, the 𝑂𝑦-axis (𝑙 = 𝐿 = 90◦ , 𝑏 = 𝐵 = 0◦) in  the direction of galactic rotation and perpendicular to 𝑂𝑥, while the  𝑂𝑧-axis (𝑏 = 𝐵 = 90◦) is always perpendicular to the plane of the  Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The orientation of the x and y axes of the local coordinate  systems was specified using the value R⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='28 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (GRAVITY  Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In a particular case, for an observer  who is in the Sun, the local coordinate system will coincide with the  rectangular Galactic coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In this work, 33 million stars from 𝐺𝑎𝑖𝑎 DR3 were selected for  which the radial velocities are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' From this sample, as in our  previous work (Fedorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2023)), were excluded those stars for  which the following conditions are not satisfied (Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (2018)):  ����  ����  𝑅𝑈𝑊𝐸 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='4,  𝜋/𝜎𝜋 > 5,  (𝜇𝛼/𝜎𝜇𝛼)2 + (𝜇𝛿/𝜎𝜇𝛿)2 > 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  By cutting off the main sequence on the 𝑀𝐺 − (𝐵𝑃 − 𝑅𝑃)  Hertzsprung-Russell diagram with two linear functions, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 1, from the approximately 30 million stars (remaining after  applying the Lindegren criteria), giants and subgiants were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Finally, a sample of approximately 15 million giants and subgiants  was used further in kinematic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The points from which, fic-  titious observations were made by a fictitious observer, were given  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Firstly, we single out spherical regions with a radius of  1 kpc, whose centers are located at the nodes of a rectangular grid  coinciding with the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Coincidence with the Galactic  plane is provided by setting the condition 𝑍 = 0 for the coordinates  of any node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Thus, the position of each node is uniquely specified by  a pair of coordinates 𝑋 and 𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The distance between adjacent nodes  along both coordinates was set equal to 250 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Each sphere circum-  scribed around a given node includes stars located at distances not  exceeding 1 kpc from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Thus, the nodes of the rectangular grid are  centroids whose velocities are equal to the average velocity of the  stars located within the corresponding spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  3 DETERMINING THE COORDINATES OF THE VERTEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In this paper, deformation velocity tensors were computed for all  stellar systems whose centers are located at the nodes of a rectangular  grid, as indicated above (see also Fedorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2023)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' If in the local  Cartesian coordinate system the radius-vector of stars is denoted as  r = (𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 𝑧) = 𝑞𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' and their velocities are V = (𝑉𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='𝑉𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='𝑉𝑧) = 𝑉𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  then from the expansion of the velocity field V(r) in the vicinity of  the node (centroid),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' we get:  𝑉𝑖(𝑑r) = 𝑉𝑖(0) + 1  2  � 𝜕𝑉𝑖  𝜕𝑞𝑘  − 𝜕𝑉𝑘  𝜕𝑞𝑖  �  0  𝑑𝑞𝑘 + 1  2  � 𝜕𝑉𝑖  𝜕𝑞𝑘  + 𝜕𝑉𝑘  𝜕𝑞𝑖  �  0  𝑑𝑞𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (1)  where  1  2  � 𝜕𝑉𝑖  𝜕𝑞𝑘  − 𝜕𝑉𝑘  𝜕𝑞𝑖  �  0  = 𝜔𝑖𝑘 = −𝜔𝑘𝑖 = 𝑀−  are components of antisymmetric and  1  2  � 𝜕𝑉𝑖  𝜕𝑞𝑘  + 𝜕𝑉𝑘  𝜕𝑞𝑖  �  0  = 𝑚+  𝑖𝑘 = 𝑚+  𝑘𝑖 = 𝑀+  symmetric tensors respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 𝑖, 𝑘 = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  By making the appropriate substitutions, this equation can also be  written in Galactic spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In this form, it is known  in the literature as the Ogorodnikov–Milne (O–M) kinematic model  (Ogorodnikov (1932, 1965)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The second rank symmetric tensor 𝑀+  is called the deformation velocity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The matrix of this tensor  has the form:  𝑀+ = ��  �  𝑚+  11  𝑚+  12  𝑚+  13  𝑚+  21  𝑚+  22  𝑚+  23  𝑚+  31  𝑚+  32  𝑚+  33  ��  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (2)  The 𝑀+ tensor completely determines the rate of deformation  motion in the stellar system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In the rectangular  Galactic coordinate system, it has 9 components, 6 of which are in-  dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' As is known, the kinematic interpretation of the diagonal  components of the tensor 𝑚+  11, 𝑚+  22, 𝑚+  33 is that these quantities are  the velocities of relative elongation (contraction/expansion) along  the axes of the coordinate system, and the non-diagonal components  𝑚+  12 = 𝑚+  21, 𝑚+  13, = 𝑚+  31, 𝑚+  23 = 𝑚+  32, characterize the velocities of  angular deformation in the planes (𝑥𝑂𝑦), (𝑦𝑂𝑧) and (𝑥𝑂𝑧), respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The velocity of angular deformation is understood as a change  in the right angle in these planes as a result of deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  It is convenient to present the results of calculations in the cylin-  drical Galactocentric coordinate system 𝑅, 𝜃, 𝑍, since its unit vectors  are parallel to the unit vectors of local rectangular coordinate systems  in each centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' For example, a node centered on the Sun will have  the following coordinates: 𝑅 = 𝑅⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='28 kpc, 𝜃 = 180◦, 𝑍 = 0 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 2 shows the dependencies of the diagonal components of  the deformation velocity tensors 𝑚+  11, 𝑚+  22, 𝑚+  33 in the Galactocen-  tric cylindrical coordinates 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' These dependencies are additionally  color-coded for different values of the coordinate 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 3 shows  the dependencies of the non-diagonal components of the velocity  deformation tensors 𝑚+  12, 𝑚+  13, 𝑚+  23 on 𝑅 and 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' It is known from  continuum mechanics that no matter how a particle of a continuous  medium moves, all its deformation can be reduced to the simplest  expansion (contraction) along three mutually perpendicular direc-  tions, which are the main axes 𝑥′𝑦′𝑧′ of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In the system of  its main axes 𝑥′𝑦′𝑧′, the velocity deformation tensor 𝑀+ will have  the following matrix:  𝑀+ = ��  �  𝜆1  0  0  0  𝜆2  0  0  0  𝜆3  ��  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (3)  where the diagonal components 𝜆1, 𝜆2, 𝜆3 are the principal values of  MNRAS 000, 1–8 (2023)  2  0  MG, mag  2  4  6  8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='5  3  BP-RP, magThe vertex coordinates  3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Diagonal components of the 𝑀+ tensor as a function of Galacto-  centric cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  the tensor, which are determined by solving the characteristic cubic  equation:  𝑀+ = ��  �  𝑚+  11 − 𝜆1  𝑚+  12  𝑚+  13  𝑚+  21  𝑚+  22 − 𝜆2  𝑚+  23  𝑚+  31  𝑚+  32  𝑚+  33 − 𝜆3  ��  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (4)  If 𝜆1, 𝜆2, 𝜆3 are positive, then the tensor surface is an ellipsoid, if  𝜆1, 𝜆2, 𝜆3 have different signs, the tensor surface is a hyperboloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 4 shows the dependencies of the principal values of the tensor  in the Galactocentric cylindrical coordinates 𝑅 and 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' As can be  seen from the figures, 𝜆1 and 𝜆2 have opposite signs and different  behavior, while the value of 𝜆3 is almost independent of 𝑅 and 𝜃  and is close to zero on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Only at the Galactocentric distance  greater than 11 kpc do we see a slight systematic deviation of 𝜆3 from  zero, which we neglected in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The numerical values of the  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Non-diagonal components of the 𝑀+ tensor as a function of Galac-  tocentric cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  parameters 𝜆1, 𝜆2, 𝜆3 indicate that the velocity deformation tensor  in the principal axes is almost independent of the 𝑍 coordinate and,  therefore, is very close to a flat (two-dimensional) tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  The kinematic interpretation of the parameters 𝜆1, 𝜆2, 𝜆3 in the  system of principal axes 𝑥′𝑦′𝑧′ is identical to that of 𝑚+  11, 𝑚+  22,  𝑚+  33 in the system of axes 𝑥𝑦𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' These quantities are the velocities  of relative elongations (contractions/expansions) along the principal  axes of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' As can be seen from the figures, the transition  from the Galactic coordinate system to the system of principal axes  practically does not change the 𝑚+  33 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Indeed, it turned out  that 𝑚+  33 almost coincides with 𝜆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' However, it is clearly seen that  the components 𝑚+  11 and 𝑚+  22 were not completely transformed into  𝜆2 and 𝜆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This is due to the fact that the 𝑚+  12 component, being  non-zero, contributes to 𝜆2 and 𝜆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Considering the components 𝜆2  and 𝜆1 in the range of 4-12 kpc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' one can see that the dependence of  MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 1–8 (2023)  m  11  30  270  25  20  240  15  10  210  5  0  180%  5  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' UI  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpcm  30  270  25  20  240  15  10  210  5  0  180%  5  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpcm  33  30  270  25  20  240  15  kpc*  10  210  5  0  180%  5  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' U  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpcm  30  270  25  20  240  15  kpc*  10  210  5  0  180%  5  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' UI  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpcm  23  30  270  25  20  240  15  kpc-  10  210  5  0  180%  5  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpcm  23  30  270  25  20  240  15  kpc-  10  210  5  0  180%  5  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpc4  Dmytrenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Eigenvalues of the 𝑀+ tensor as a function of Galactocentric  cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  𝑚+  11 on 𝑅 and 𝜃, similar to a sinusoidal one, has been transformed  into an almost analogous dependence of the component 𝜆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' And the  dependence 𝑚+  22 on 𝑅 and 𝜃 has been transformed into a similar  dependence 𝜆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' These facts indicate that the main axes 𝑂𝑥′ and 𝑂𝑦′  of the deformation velocity tensors are located almost in the Galactic  plane, and the 𝑂𝑧′ axis practically coincides with the 𝑂𝑧 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This  is also confirmed by the behavior of the 𝑧-dependent parameters  𝑚+  33, 𝑚+  13 and 𝑚+  23, which were presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The values of these  parameters in the entire range of 𝑅 and 𝜃 are close to zero and  indicate that the motion of stars is predominantly parallel to the  Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Therefore, in the further analysis, we consider the  tensor 𝑀+ as flat, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' having only four components 𝑚+  11, 𝑚+  12, 𝑚+  21,  𝑚+  22, three of which are independent (𝑚+  12 = 𝑚+  21):  𝑀+ =  �𝑚+  11  𝑚+  12  𝑚+  21  𝑚+  22  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (5)  The components of this tensor are related to the gradients of the veloc-  ity components in the local rectangular Galactic (𝑉𝑥, 𝑉𝑦) and in the  Galactocentric cylindrical (𝑉𝑅, 𝑉𝜃) coordinate systems along their  coordinate axes by the following relations (Chandrasekhar (1945);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Ogorodnikov (1965)):  𝐴 = 𝑚+  12 = 1  2  � 𝜕𝑉𝑥  𝜕𝑦 + 𝜕𝑉𝑦  𝜕𝑥  �  = 1  2  �  1  𝑅  𝜕𝑉𝑅  𝜕𝜃 − 𝑉𝜃  𝑅 + 𝜕𝑉𝜃  𝜕𝑅  �  𝐵 = 𝜔3 = 1  2  � 𝜕𝑉𝑦  𝜕𝑥 − 𝜕𝑉𝑥  𝜕𝑦  �  = 1  2  � 𝜕𝑉𝜃  𝜕𝑅 − 1  𝑅  𝜕𝑉𝑅  𝜕𝜃 + 𝑉𝜃  𝑅  �  𝐶 = 𝑚+  11−𝑚+  22  2  = 1  2  � 𝜕𝑉𝑥  𝜕𝑥 − 𝜕𝑉𝑦  𝜕𝑦  �  = 1  2  � 𝜕𝑉𝑅  𝜕𝑅 − 1  𝑅  𝜕𝑉𝜃  𝜕𝜃 − 𝑉𝑅  𝑅  �  𝐾 = 𝑚+  11+𝑚+  22  2  = 1  2  � 𝜕𝑉𝑥  𝜕𝑥 + 𝜕𝑉𝑦  𝜕𝑦  �  = 1  2  � 𝜕𝑉𝑅  𝜕𝑅 + 1  𝑅  𝜕𝑉𝜃  𝜕𝜃 + 𝑉𝑅  𝑅  �  (6)  where 𝐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 𝐵,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 𝐶 and 𝐾 are parameters similar to the generalized  Oort constants that can be found for each region of stars under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In continuum mechanics, it is shown that for a two-dimensional  tensor 𝑀+, the angles 𝛽1 and 𝛽2, which form the main axes of the  tensor 𝑀+ with the axes 𝑂𝑥 and 𝑂𝑦 of the coordinate system used,  are found from the expression:  𝑡𝑔(2𝛽) = 𝑡𝑔(2𝛽1) = 𝑡𝑔(2𝛽2) =  2𝑚+  12  𝑚+  11 − 𝑚+  22  ,  (7)  where 𝛽1 is the angle between 𝑂𝑥 and 𝑂𝑥′, and 𝛽2 is between 𝑂𝑦  and 𝑂𝑦′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  If the components of the 𝑀+ tensor are expressed in terms of the  parameters 𝐴, 𝐵, 𝐶, 𝐾 then the matrix 5 will have the following form:  𝑀+ =  �𝐾 + 𝐶  𝐴  𝐴  𝐾 − 𝐶  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (8)  and formula 7 will be accordingly transformed to the form:  𝑡𝑔(2𝛽) =  2𝐴  (𝐾 + 𝐶) − (𝐾 − 𝐶) = 𝐴  𝐶 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (9)  In the case of a purely Oort rotation, there is only the rotational  component 𝑉𝜃, while the components 𝑉𝑧 and 𝑉𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In this case,  𝑉𝜃 does not depend on 𝜃, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  𝜕𝑉𝑅  𝜕𝑅 = 0,  𝑉𝑅 + 𝜕𝑉𝜃  𝜕𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (10)  As a result, the terms by which the parameters𝐶 and 𝐾 are determined  in a cylindrical Galactocentric coordinate system, are equal to zero:  𝑉𝑅 = 0,  𝜕𝑉𝜃  𝜕𝜃 = 0,  (11)  and hence 𝐶 and 𝐾 are also equal to zero, and the tensor 𝑀+ will  have zero diagonal components:  𝑀+ =  �0  𝐴  𝐴  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (12)  Thus, for a purely Oort rotation, it follows from formula 9 that the  main axes of the deformation velocity tensor will be directed at an  angle 𝛽 = 45◦ to the axes of the local rectangular Galactic coordinate  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In this case, the angle 𝛽, for any non-zero values of 𝐴, will  be equal to 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' If 𝐶 or 𝐾 is non-zero, this means that the rotation is  not Oort (axisymmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In this case, the angle 𝛽 ≠ 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content="  MNRAS 000, 1–8 (2023)  30  270  25  20  240  15  10  210  5  0  180%  5  3  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R, kpc30  270  25  20  240  15  10  210  kpc'  5  0  180%  5  2  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R, kpc30  270  25  20  240  15  10  210  kpc'  5  0  180%  5  10  150  15  20  120  25  30  90  2  4  6  8  10  12  14  16  R, kpcThe vertex coordinates  5  Figure 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The values of the angle between the axes of local Galactic coordinate  systems and the main axes of the tensor 𝑀+ depending on Galactocentric  cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Vertex deviations depending on Galactocentric cylindrical coordi-  nates (𝑅, 𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  4 SOLUTION OF THE PROBLEM AND ANALYSIS OF THE  OBTAINED RESULTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  The term vertex is commonly referred to as a point on the sky,  relative to which the stellar system under consideration rotates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' To  determine the coordinates of the vertex, we used the deformation  velocity tensors derived by expanding the velocity field of various  subsamples (stellar systems) described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 5 shows the angles 𝛽 between the principal axes 𝑥′ of the  deformation velocity tensors of the stellar systems under study and  the 𝑥 axes of the local rectangular Galactic coordinate systems in  which these tensors have been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 5 shows the changes in  the angle 𝛽 depending on the cylindrical Galactocentric coordinates  𝑅 and 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The angles have been calculated from the formula 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' As  can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 5, the values of the angles 𝛽 depending on 𝑅  and 𝜃 differ markedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' As shown above, with axisymmetric (Oort)  rotation, the angle 𝑙𝑥𝑦 = (𝛽 − 45◦), called the deviation of the vertex  longitudes from the direction to the Galactic center, is exactly equal  to zero and does not depend on the coordinate angle 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 6, we present the dependencies of the vertex deviations  𝑙𝑥𝑦 on the cylindrical coordinates 𝑅, 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' And Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 7 shows the same  values, but in the form of a map, where the rectangular Galactic  coordinates 𝑋 and 𝑌 are plotted along the axes, and the 𝑙𝑥𝑦 values  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Vertex deviations depending on rectangular Galactic coordinates  (𝑋, 𝑌 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  are displayed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The advantage of such a graphical presentation  of the results is that one can immediately see the behavior of 𝑙𝑥𝑦 in  the entire range of rectangular Galactic coordinates 𝑋 and 𝑌, where  the kinematic analysis has been carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 6 it is clearly seen that the angle 𝑙𝑥𝑦 is not equal to zero,  and it changes with 𝑅 and 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' One can clearly see the “stratification  and intertwining” of the dependencies 𝑙𝑥𝑦(𝑅) corresponding to var-  ious fixed values of the angle 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This behavior of the dependencies  𝑙𝑥𝑦(𝑅, 𝜃) is probably due to the difference in the deformation veloci-  ties in different parts of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This assumption is confirmed by  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 7, which demonstrates the differences in the orientations of the  tensor surfaces due to the difference in the deformation velocities in  different parts of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' However, the similarity of the depen-  dencies 𝑙𝑥𝑦(𝑅) in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 6, at different values of the angle 𝜃, as well  as a noticeable predominance of red color in the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 7,  and blue at the bottom, suggests that "stratification and intertwining"  may not be due to kinematic causes alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  One of these reasons may be the incorrect value of the accepted  Galactocentric distance of the Sun 𝑅⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='28 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In this case, the  use of the accepted value of 𝑅⊙ to determine the rotation angles of  the 𝑥 axes of local coordinate systems in the direction of the Galactic  center will cause an inaccuracy in determining these angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' As a  result, the behavior of 𝑙𝑥𝑦(𝑅, 𝜃) will be determined not only by  kinematic differences, but also by the orientation inaccuracy of the  axes of local coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Checking this assumption, we found that the maximum conver-  gence of the functions 𝑙𝑥𝑦(𝑅, 𝜃) is realized at a certain value 𝑅V,  which differs noticeably from the accepted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In fact, when using  the vertex coordinates to set the orientation of local coordinate sys-  tems, the dependencies 𝑙𝑥𝑦(𝑅, 𝜃) become closest and practically turn  into one, “least stratified” function 𝑙𝑥𝑦(𝑅), which weakly depends  on 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  It also turned out that the use of one vertex does not provide  the best convergence of the functions 𝑙𝑥𝑦(𝑅, 𝜃) in the entire range  of distances 𝑅 used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Thus, achieving the best convergence of the  functions 𝑙𝑥𝑦(𝑅, 𝜃) in the range of 5-–10 kpc results in to a noticeable  deterioration in convergence within the range of 10–15 kpc (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 9  below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This result means that the vertices of different star systems  are at different distances from the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  MNRAS 000, 1–8 (2023)  βo  90  270  75  60  240  45  30  210  15  p  0  180%  15  30  150  45  60  120  75  90  90  2  4  6  8  10  12  14  16  R, kpc90  270  75  60  240  45  30  210  15  0  0  180%  15  30  150  45  60  120  75  90  90  2  4  6  8  10  12  14  16  R, kpc30  25  20  15  10  2  5  0  Y  5  2  10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='4  15  20  6  25  8  30  8  6  4  2  0  2  4  6  8  X, kpc6  Dmytrenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Amendt & Cuddeford (1991);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Kuijken & Gilmore (1991);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Smith et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2012) showed that in a stationary, axisymmetric disk galaxy, the  axes of the stellar velocity ellipsoid of any local stellar system ideally  coincide with the galactic coordinate axes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=', Binney & Tremaine  (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The main axes of the deformation ve-  locity tensor for the Oort (axisymmetric) rotation, as shown above,  rotate relative to the local axes by an angle of 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' It was shown  by Dehnen (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Minchev & Famaey (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Vorobyov & Theis  (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (2013) that structures which are not axisymmetric  can have a noticeable effect on the observed orientation of the stellar  velocity ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' They will have a similar effect on the orientation  of the principal axes of the deformation velocity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Therefore, in the general case, when a non-axisymmetric rotation  is realized, the direction to the Galactic center (a point on the celestial  sphere with coordinates 𝛼𝐺𝐶 = 266◦,40499, 𝛿𝐺𝐶 = −28◦,93617,  accepted by the Hipparcos consortium Perryman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' (1997)) and  the direction to the vertex (the point of the celestial sphere relative  to which the stellar system rotates) do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Our result shows  that not only the spherical coordinates of the Galactic center and the  vertices of stellar systems, but also their distances from the Sun do  not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In addition, it turned out that the values of the 𝑅V distance can  be estimated using the approach we proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Assuming that the  accepted value 𝑅⊙=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='28 kpc is correct, the 𝑥 axes of local coordinate  systems will always be directed to the same point – the Galactic center  with coordinates 𝐿 = 0◦, 𝐵 = 0◦, 𝑅⊙=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='28 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In this case, the main  axis 𝑥′ of the deformation velocity tensor, which is calculated in the  local coordinate system 𝑋𝑂𝑌, being rotated by 45 degrees, will be  directed to the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' In other words, the longitude of the vertex in  the local coordinate system will be numerically equal to the angle  𝑙𝑥𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Using the values of the angles 𝑙𝑥𝑦 calculated for local systems,  it is possible to construct rays that pass through their vertex point,  with their origins locating in the centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  To set the equations of rays (straight lines) passing through two  points, we use rectangular Galactic coordinates of specific centroids  and points lying on unit circles built around these centroids in the  𝑋𝑂𝑌 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' If their radii 𝑟 are taken equal to 1 kpc, then the coordi-  nates of the second point can be found as follows: 𝑋 = 𝑋𝑐 +𝑟 cos 𝑙𝑥𝑦,  𝑌 = 𝑌𝑐 −𝑟 sin 𝑙𝑥𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Finding the coordinates of the intersection point of  two arbitrary rays (straight lines) allows one to find the distance from  the Sun to this point in the 𝑋𝑂𝑌 coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Pairwise inter-  sections of rays form a certain region of intersection in the Galactic  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 8 show the region of intersection of the rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Its rather  large sizes and non-uniformity are visible (there are many peaks or  nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The largest of these ray intersection nodes is located approx-  imately at a distance of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='3 kpc from the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' At the same time, it  is not located on the 𝑂𝑋 axis of the rectangular Galactic coordinate  system, but it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='5 kpc away from it along the 𝑂𝑌 axis in the posi-  tive direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Similarly, we can select other nodes that can be seen in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The presence of many nodes can be explained by the present  nonlinear dependence 𝑙𝑥𝑦(𝑅, 𝜃), which was considered earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  The coordinates of the vertices 𝐺 𝑗  V(𝑋 𝑗  V,𝑌 𝑗  V) , at which the func-  tions 𝑙𝑥𝑦(𝑅, 𝜃) will have the best convergence in certain ranges of  Galactocentric distances 𝑅, were estimated using the least squares  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' To this end, a system of equations was compiled for those  rays whose origins got inside the chosen range of Galactocentric  distances Δ𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' An additional condition for including the ray equation  into the system of equations was the presence of at least 25 thousand  stars in the stellar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The solution of the system was the desired  coordinates of the point of intersection of all rays — the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' So,  for the range 5 < 𝑅 < 10 kpc, we got point 𝐺1  V with coordinates:  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Rays directed to the vertex in the Galactic plane and the area of  their intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Results of estimation of Galactic rectangular coordinates of vertices  𝐺0  V = 𝐺V  𝐺1  V  𝐺2  V  Δ𝑅, kpc  0–16  5–10  10–15  𝑋𝑉 , kpc  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='96  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='35  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='87  𝑌𝑉 , kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='4  𝑅𝑉 , kpc  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='96  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='35  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='88  𝜖 (𝑋𝑉 ), kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='08  𝜖 (𝑌𝑉 ), kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='03  𝑋1  V = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='35 kpc, 𝑌1  V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='24 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This point is very close to the  center of the most massive node, which is visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' For the  range of Gatatocentric distances 10 < 𝑅 < 15 kpc, we got another  point - 𝐺2  𝑉 , which already has different coordinates: 𝑋2  V = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='87 kpc,  𝑌2  V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='4 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Also, we performed an estimate of the coordinates  of the point 𝐺0  V = 𝐺V, obtained using the entire available range of  𝑅 and called by us the general vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The coordinates of this point  are: 𝑋0  V = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='96 kpc, 𝑌0  V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='08 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In table 1, we provide for all points𝐺 𝑗  V the coordinates and errors of  their determination, as well as the calculated heliocentric distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Now we can use the coordinates of the point 𝐺 𝑗  V(𝑋,𝑌) instead of  the 𝑋-th coordinate of the Galactic center, equal to 𝑅⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This allows  us to define new Galactocentric cylindrical coordinate systems and  orient properly the local coordinate system in order to construct new  dependencies 𝑙𝑥𝑦(𝑅, 𝜃) and 𝑙𝑥𝑦(𝑋,𝑌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' These results are shown in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 6 and 9, one can note an improvement in the con-  MNRAS 000, 1–8 (2023)  8  6  4  2  kpc  0  K  2  4  9-  8  8  9-  4  2  0  2  4  6  8  10  12  14  16  X, kpc2  1  kpc  0  Y  1  2  4  5  6  7  8  9  10  11  12  X, kpcThe vertex coordinates  7  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Vertex deviation depending on Galactocentric cylindrical coordi-  nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The determined vertex coordinates have been used to set the orientation  of local Galactic rectangular coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  vergence of the dependence 𝑙𝑥𝑦(𝑅, 𝜃) in those ranges of 𝑅 where the  corresponding 𝐺 𝑗  V was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' A similar conclusion can be reached by  comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 7 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' It should be noted that the local structures,  which are clearly visible on the maps, remained the same, which in-  directly confirms their purely kinematic, and not geometric, nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  The deviation of the angle 𝑙𝑥𝑦 in the entire region under study from  zero can be considered as a measure of the non-axisymmetry of the  Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  5 SUMMER AND CONCLUSIONS  The use of 𝐺𝑎𝑖𝑎 DR3 data makes it possible to determine the vertex  coordinates not only in the near–Solar neighborhood, but also in that  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Vertex deviation depending on rectangular Galactic coordi-  nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='The determined vertex coordinates have been used to set the orientation  of local Galactic rectangular coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 1–8 (2023)  Ixy° (using Gy)  90  270  75  60  240  45  30  210  15  0  0  180%  15  30  150  45  60  120  75  90  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpclky° (using Gly)  90  270  75  60  240  45  30  210  15  L  0  0  180%  15  30  150  45  60  120  75  90  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpcly° (using G2)  90  270  75  60  240  45  30  210  15  0  0  180%  15  30  150  45  60  120  75  90  90  2  4  6  8  10  12  14  16  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' kpclxy° (using Gv)  30  8  25  6  20  15  4  10  2  5  0  Y  5  2  10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='4  15  20  6  25  8  30  8  9-  4  2  0  2  4  6  8  X, kpcxy° (using Glv)  30  8  25  20  15  10  2  5  0  Y  5  2  10  4  15  20  6  25  8  30  8  9-  4  2  0  2  4  6  8  X, kpclxy° (using G2)  30  25  6  20  15  4  10  2  5  0  Y  5  2  10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='4  15  20  6  25  8  30  8  6  4  2  0  2  4  6  8  X, kpc8  Dmytrenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  part of the Galaxy for which astrometric parameters are available in  𝐺𝑎𝑖𝑎 DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The approach proposed in the works by Fedorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  (2021, 2023) for the analysis of kinematic parameters, when a ficti-  tious observer, being at an arbitrary point in the Galaxy, determines  them in the framework of the O–M model, made it possible to obtain  a number of new results related to a significant part of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  In this work, we have applied this approach to determine the coor-  dinates of the vertices of various stellar systems contained in spher-  ical regions with a radius of 1 kpc, whose centers are located in the  Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Since the analysis revealed that the 𝑀+ deformation  velocity tensors calculated in local coordinate systems are almost  flat, we used only four components 𝑚+  11, 𝑚+  12, 𝑚+  21, 𝑚+  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' It turned  out that the deviations of the vertices of these stellar systems obey a  certain law and are presented mainly in the form of the dependence  𝑙(𝑅) and weakly depend on 𝑡ℎ𝑒𝑡𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This result could not have been  obtained without knowledge of the spatial coordinates and velocities  of the stars contained in 𝐺𝑎𝑖𝑎 DR3 and allowing one to set the local  Galactic coordinate system at an arbitrary point in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Usually, in works on determining the deviations of the vertex, it  is explicitly or implicitly assumed that the distance from the Sun to  the Galactic center and to the vertex are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This is indeed true  for axisymmetric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' For the case when 𝑚+  11 and 𝑚+  22 are not  equal to zero, we show that the vertices of different stellar systems are  located at different distances that do not coincide with the accepted  distance of the Sun 𝑅⊙=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='28 kpc to the center of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' This  indicates that for the investigated part of the Galaxy there is no single  center of rotation, as in the case of axisymmetric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Although  we cannot specify the exact values of the distances from the Sun to the  vertices, it is still possible to indicate suitable values of 𝑅V, at which  the function 𝑙(𝑅), in a specific range of Galactocentric distances, has  a minimum stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  Our results indicate that not only the spherical coordinates of the  Galactic center and the vertices of stellar systems do not coincide,  but also their distances to the Sun do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' These results can  be useful in many kinematic and dynamic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  6 ACKNOWLEDGEMENTS  This work has made use of data from the European Space  Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='int/  gaia), processed by the Gaia Data Processing and Analysis  Consortium (DPAC,https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='int/web/gaia/  dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' Funding for the DPAC has been provided by  national institutions, in particular the institutions participating in the  Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  We are immensely grateful to the Armed Forces of Ukraine for  the fact that in wartime we still have the opportunity to work and do  science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  We sincerely thank the anonymous reviewer for a careful reading  and insightful comments and suggestions on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  DATA AVAILABILITY  The used catalogue data is available in a standardised format for  readers via the CDS (https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='u-strasbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' The software code  used in this paper can be made available upon request by emailing  the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
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+page_content=' E, 2013, ApJ, 764:123  Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=', Whiteoak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=', Evans, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=', 2012, ApJ, 746:181  Vorobyov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=', Theis, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=', 2008, MNRAS, 383, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content=' 817–830  This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
+page_content='  MNRAS 000, 1–8 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfYfc8/content/2301.00203v1.pdf'}
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+Genetic-tunneling driven energy optimizer for magnetic system
+Qichen Xu,1, 2 Zhuanglin Shen,3 Manuel Pereiro,4 Pawel Herman,5, 2 Olle Eriksson,4 and Anna Delin1, 2
+1Department of Applied Physics, School of Engineering Sciences, KTH Royal
+Institute of Technology, AlbaNova University Center, SE-10691 Stockholm, Sweden
+2SeRC (Swedish e-Science Research Center), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden
+3CAS Key Laboratory of Quantitative Engineering Biology, Shenzhen Institute of Synthetic Biology,
+Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China
+4Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden
+5Division of Computational Science and Technology, School of Electrical Engineering and Computer
+Science, KTH Royal Institute of Technology, AlbaNova University Center, SE-10691 Stockholm, Sweden
+(Dated: January 3, 2023)
+Novel topological spin textures, such as magnetic skyrmions, benefit from their inherent stability,
+acting as the ground state in several magnetic systems. In the current study of atomic monolayer
+magnetic materials, reasonable initial guesses are still needed to search for those magnetic patterns.
+This situation underlines the need to develop a more effective way to identify the ground states.
+To solve this problem, in this work, we propose a genetic-tunneling-driven variance-controlled opti-
+mization approach, which combines a local energy minimizer back-end and a metaheuristic global
+searching front-end. This algorithm is an effective optimization solution for searching for magnetic
+ground states at extremely low temperatures and is also robust for finding low-energy degenerated
+states at finite temperatures.
+We demonstrate here the success of this method in searching for
+magnetic ground states of 2D monolayer systems with both artificial and calculated interactions
+from density functional theory. It is also worth noting that the inherent concurrent property of this
+algorithm can significantly decrease the execution time. In conclusion, our proposed method builds
+a useful tool for low-dimensional magnetic system energy optimization.
+Optimization algorithms hold a fundamental connec-
+tion inside the interdisciplinary boundaries of physics
+and computer science which enhance the understand-
+ing of novel physical phenomena, e.g., topologically non-
+trivial defects and textures[1–3]. It can also accelerate
+new functional material findings, including atomic mono-
+layer magnetic materials, which is currently an active re-
+search field in the magnetism community[4].
+When it
+comes to 2D magnetic materials, typically, there are sets
+of algorithms for the energy optimization process of the
+magnetic system at finite temperature, e.g., the gradient
+descent family, Monte Carlo approaches, and spin dy-
+namic methods with damping.[5–7] These conventional
+algorithms are being plagued by the possibility of get-
+ting trapped into the local energy minimum rather than
+the global minimum. Thus, there is a need to use meta-
+heuristic methods to provide a better route to the global
+minimum of the potential energy surface.
+In particu-
+lar, the Markov chain Monte Carlo based heat-bath opti-
+mizations, a group of non-gradient sampling algorithms,
+are proven to be effective and robust in searching for
+low-energy states at finite temperatures.[8, 9] Unfortu-
+nately, in current implementations[10, 11], there is still
+some prior knowledge needed for getting acceptable re-
+sults, e.g., appropriate initial guesses and manual con-
+vergence analysis. This situation brings a fundamental
+challenge, i.e., finding a more effective and automatic way
+to apply a heuristic optimal searching for the magnetic
+ground state without any initial guess or prior knowledge.
+In order to overcome this challenge, a hybrid model
+can be a straightforward solution.
+There are several
+successful hybrid approaches under the idea of combin-
+ing meta-heuristic algorithms and typical optimizing ap-
+proaches, e.g., hybrid Monte Carlo[12], neural annealing
+optimization[13] or neural evolutionary method.[14, 15]
+But unfortunately, those approaches are mainly designed
+for the Ising model and may face problems when handling
+realistic material with long-range interactions.
+In this work, inspired by the idea of tunneling on
+potential energy surface[16] and hybrid meta-heuristic
+solutions[17, 18], a metaheuristic energy minimization
+approach is proposed and tested for magnetic systems
+with non-trivial topological charge.
+The method is
+based on several variance-threshold local optimizers, e.g.,
+spin dynamic optimizer and heat-bath Monte Carlo op-
+timizer.
+This involves a real space genetic tunneling
+front-end for searching of optimal initial guess and a se-
+lected local optimizer with adaptive variance-based con-
+vergence criteria as the back-end. We analyze the per-
+formance of this algorithm that has the potential to es-
+cape from local traps in energy minimization. We inves-
+tigate the efficiency by simulations on a 2D monolayer
+with model exchange parameters that give rise to Bloch-
+type skyrmions. We also investigate an experimentally
+well-studied system:
+Pd/Fe/Ir(111), which contains a
+N´eel-type skyrmionic phase. The validation process was
+performed on the aforementioned systems with different
+applied fields at finite temperatures, and we compared
+the proposed method with conventional approaches.
+arXiv:2301.00207v1  [physics.comp-ph]  31 Dec 2022
+
+2
+FIG. 1. (a) Conceptual illustration of the variance-threshold controlled localized optimization process to find low energy spin
+configurations. The contour map in the middle shows an example of a potential energy surface. The darker zones in the
+coutour map represents lower energy areas. The arrows connect the local optimization process and the configuration point
+in the potential energy surface. At the top of the figure, five colored blocks, i.e., Early, Rough, Medium, Fine, and Precise,
+denote different converge levels of the search algorithm. The ”Early” level means the lowest convergence, and the ”Precise” level
+represents the highest convergence. (b) Conceptual illustration of how the genetic operators are applied to the spin system. The
+whole process involves three subprocesses, i.e., spin configuration segmentation, crossover, and perturbation-based mutation
+(for details, see the Method section). In the spin configuration segmentation part, the real-space spin textures are viewed as
+information carriers similar to those in chromosomes in biological systems. The spin textures are divided into several segments
+that can be used in the same way as gene segments in crossover and mutation operations. As shown in the top left of Figure (b),
+the number indicates spin segments that have come from four different configurations of a parent configuration (see Method).
+In the mutation subplot (located in the top right of Figure (b)), the perturbation windows indicate the mutation operation
+which works on part of the configuration (for details about perturbation, see methods section). (c) Conceptual illustration
+of how to use the genetic tunnel to dig a tunnel through the energy barriers and heuristically search for a spin configuration
+with lower energy, with the aim of finding the global minimum. The curve represents the potential energy surface and the
+colored star represents the acceptable solution set at temperature T. The height of the shadow colored regions equals KbT.
+The stars in the plot represent a single-spin configuration. The black points represent configurations of the energy landscape
+that are not identified by the algorithm. (d) The flowchart of the whole procedure. The dark yellow box represents input data
+that need to be prepared before execution, and the light yellow box represents generated spin configurations. The box with a
+dashed boundary represents an optional choice. The dark blue rounded rectangles represent operations. The white diamond
+box represents conditional statements. The notations H (C0
+r), H (C0
+i ), ∆E, Tseed, C, C0, CLOPT and COPT represent the
+energy of a randomly generated spin configuration, the energy of any spin configuration in the initial generation, the threshold
+of energy difference, the predefined threshold for select initial spin configurations, the current spin configuration set, the initial
+generation set with a predefined quantity of spin configurations, the local optimized spin configuration, and the final optimized
+spin configurations, respectively (for details, see the Methods section).
+
+(a)
+(d)
+Crystal
+Magnetic
+Random
+information
+interactions
+Seeds
+Early
+Medium
+Rough
+Fine
+Precise
+Energy
+ Random configuration
+ Simulated Annealing
+generator
+(SA) generator
+Metropolis
+SLLG
+optimizer
+optimizer
+ 
+Local optimization steps
+Initial spin configuration filter
+No
+(b)
+HP(C9)
+Artificial spin
+configurations
+Yes
+(optional)
+Quantity satisfied
+Yes
+Initial generation of spin
+configurations c0
+Crossover
+Mutation
+Genetic selection operators
+(base on Hamiltonian )
+Genetic tunneling operators
+Local optimized spin segment
+Metropolis
+SLLG
+optimizer
+ optimizer
+(c)
+Local optimizied spin
+AXX（
+Genetic operators
+ configurations CLOPT
+Yes
+Energy
+Up to max iterations?
+No
+No
+Converged?
+KbT
+Yes
+Spin configurations3
+I.
+RESULT
+A.
+Spin system parametrization
+Ground state searching problems of magnetic material
+at zero kelvin can be reformulated as finding the global
+minimum of the potential energy surface (PES), which is
+here constructed by a classical Heisenberg spin Hamilto-
+nian in the form:
+H =
+�
+i̸=j
+JijSi · Sj +
+�
+i̸=j
+Dij · (Si × Sj)
++
+�
+i
+Bext · Si +
+�
+i
+KU
+uni (Si · ez)2
+(1)
+where Si and Sj are spin moments, Jij, Dij, KU
+ani ez and
+Bext are Heisenberg exchange interactions, Dzyaloshin-
+skii–Moriya interactions, uniaxial anisotropy, easy axis
+vector, and the applied field, respectively.
+Typically,
+these four Hamiltonian terms are enough to construct
+a potential energy surface for the ground state that one
+wishes to identify as specified by the magnetic config-
+uration (see e.g.
+Ref.[6]).
+Note that extra terms can
+also be involved in, e.g., biquadratic exchange coupling or
+spin-lattice coupling [6]. In this work, both model inter-
+actions and realistic materials specific magnetic parame-
+ters (which are calculated by using ab-initio density func-
+tional theory - DFT) are included to demonstrate the ef-
+fectiveness and efficiency of a proposed energy optimizer
+for various complex potential energy surfaces[6, 19].
+B.
+Genetic tunneling procedure
+Finding global optima in a complex potential energy
+surface of a spin system with long-range interaction nu-
+merically is commonly a non-deterministic polynomial-
+time hard (NP-hard) problem, and it is difficult to find
+appropriate solutions[13]. To solve this problem, we pro-
+pose a genetic tunneling algorithm to provide global en-
+ergy optimization. Full detail of the method is described
+in the Methods section, but we outline the most salient
+features here. The procedure is shown schematically in
+Figure1, where in particular, Figure1 (d) illustrates a flow
+chart of the method used. The figure shows the connec-
+tion between the local optimization module and an evo-
+lutionary global searching method, that allows to evolve
+the spin configuration where segments of this configura-
+tion are regarded as being similar to information carriers
+of genes in biological material. By using numerical pro-
+tocols that mimic gene flow along generations, we explore
+avenues to reach global minimum with minimal numerical
+effort while avoiding being trapped in metastable config-
+urations.
+The optimizer workflow used here can be classed into
+two parts: Firstly, it starts with the input, which con-
+tains physical information of a given system, e.g., crystal
+lattice, atomic position, and magnetic interactions. The
+initial generation of spin configurations is generated hav-
+ing random orientations of the atomic spins. Directly af-
+ter getting the initial generation, one local optimization
+module is involved in relaxing all magnetic orientations
+into the closest local minimum. The local optimization
+is controlled by variance threshold as shown in Figure 1.
+(a) and discussed in the Method section. The other part
+is metaheuristic searching. Once one has a spin config-
+uration of a local minimum for initial generation (see
+Methods section), the procedure will come to segmenta-
+tion of the spin configuration and tunneling mechanisms
+that bridge over energy maxima, as shown in Figure 1
+(b). This metaheuristic progress tries iterative searching
+for the global minima until the evolutionary process gets
+to convergence. See the Methods section for more detail
+on the genetic tunneling method.
+C.
+Finding a low-temperature ground state
+In this section, we investigate the efficiency of the ge-
+netic tunneling optimizer at low temperatures with dif-
+ferent local optimizer backends. The simulation is per-
+formed for a spin Hamiltonian that is appropriate for
+a monolayer of Pd/Fe/Ir(111).
+All interactions used
+for constructing the spin Hamiltonian are calculated by
+DFT[19]. In order to ensure an energy landscape that is
+hard, so that precise global optima exist, we employed in
+these simulations a temperature of 0.1 mK .
+The performance of the genetic tunneling optimizer
+with different local backends is shown in Figure 2. Note
+that we have used both Markov Chain Monte Carlo
+(MCMC) and a spin dynamic local optimizer as back-
+ends. For analysis performance of genetic tunneling op-
+erators, each local optimizer is combined with three typ-
+ical genetic selection operators,i.e., Rank(R), Tourna-
+ment(T), and Roulette Wheel(RW), respectively.
+Ad-
+ditionally, for the purpose of studying the impact of
+having a pre-optimized initial configuration, a simulated
+annealing initial spin configuration generator was used
+to combine the MCMC backend with Rank selection.
+For comparison, a classical Markov Chain Monte Carlo
+based simulated annealing simulation with fine tempera-
+ture mesh and 2.5×106 steps is used as a baseline (results
+denoted SA, see the Methods section for more detail on
+simulated annealing).
+An obvious trend in Figure 2 (a) is that the genetic tun-
+neling optimizer with different kinds of backends can find
+en energy minimum of the system, but a significant dif-
+ference can be found between Markov Chain Monte Carlo
+group and the spin dynamic-based group. It is notewor-
+thy that with finite local optimization steps and global
+searching epochs, all spin dynamic optimizers may not
+achieve convergence and in general Monte Carlo back-
+ends work better, yielding consistently lower energy for
+the same number of iterations. Genetic tunneling with
+Markov chain Monte Carlo backend invariably reaches
+
+4
+FIG. 2. (a) Performance benchmark of the genetic tunneling optimizer at 0.1 mK . The simulation system is Pd/Fe/Ir(111)
+with an external magnetic field of 2.7 T out of the plane. Here MCMC-R, MCMC-RW, and MCMC-T represent Markov chain
+Monte Carlo local optimizers with Rank (R), Roulette Wheel (RW), and Tournament (T) genetic selection, respectively (for
+details, see the Method section). The SLLG-R, SLLG-RW, and SLLG-T represent spin dynamic optimizers that solve the
+stochastic Landau–Lifshitz–Gilbert (SLLG) equation with an artificial damping value of 0.4 combined with Rank(R), Roulette
+Wheel(RW) and Tournament(T) genetic selection, respectively. Moreover, SA represents the conventional simulated annealing
+method and is used as reference energy. For better comparison, the result from SLLG R, SLLG RW, and SLLG T are shown
+from generation 12 of the evolutionary algorithm (see Methods section). (b) The time consumption in units of node-hours for
+each optimizer. All of the simulations are performed on an Intel Xeon Gold 6130 CPU node with 32 cores without concurrent
+processes. (c)-(k) Visualization of the final spin configurations from each optimizer. The color, which changes from red to blue,
+represents the spin moment’s direction from out of the plane (red) to in plane (blue).
+convergence with lower energy than any other method,
+including the reference energy obtained by simulated an-
+nealing (thus defining the reference energy or the base-
+line). The energy of the predicted system is also reflected
+in the real space spin configuration, which according to
+Figure 2 (h)-(k), has no stable hexagonal skyrmion lat-
+tice identified within the simulation time for any of the
+methods based on the spin dynamic backend.
+For results obtained by Markov chain Monte Carlo
+backend, it appears that all flavours of this method
+achieve similar results, as may be seen in Figure 2 (a) and
+Figure 2 (c)-(g). All of them identify a perfect hexagonal
+skyrmion lattice, with a unit cell size which is in good
+agreement with previous studies[20, 21].
+The applied
+quick simulated annealing initial configuration generator
+is seen to influence the first generation’s performance and
+to some degree the convergence speed, but it has no ef-
+fect on the final result. Thus, for the purpose of studying
+investigations of genetic tunneling mechanisms, instead
+of using the quick simulated annealing module, we only
+adopt the Markov Chain Monte Carlo based rank selec-
+tion module in the rest of this work. In Figure 2 (b) we
+show the time consumption of each method investigated
+here, and it is clear that methods with Markov chain
+Monte Carlo backend perform substantially better.
+D.
+Performance study on system sizes and applied
+fields
+In Figure 3(a) and (b), a set of different system sizes
+are applied in simulation with a spin Hamiltonian appro-
+priate for Pd/Fe/Ir(111) monolayer at T= 0.1 mK with a
+2.7 Tesla applied field. All simulations are performed on
+1 computing node paralleled with 32 CPU cores. The re-
+sult shows that the genetic tunneling optimizer can find
+the ground state of the system with lower energy than
+the conventional simulated annealing method, without
+
+(a)
+(b)
+-3.824-
+MCMC R
+SLLG R
+MCMC RW
+SLLG RW
+MCMC T
+SLLG T
+(mRy/atom)
+(node-hour)
+MCMC(SA) R
+SLLG(SA) R
+-3.828-
+SA
+100
+3.832-
+Energy
+Time
+MCMC R
+SLLG R
+MCMC RW
+SLLG RW
+3.836-
+MCMC T
+SLLG T
+E
+MCMC(SA) R
+SLLG(SA) R
+10-2
+-3.840-
+0
+10
+20
+30
+40
+50
+60
+0
+10
+20
+30
+40
+50
+60
+Iterations
+Iterations
+(c)
+(d)
+(e)
+(f)
+(g)
+SA
+MCMC R
+MCMC T
+MCMC(SA) R
+MCMC RW
+(i)
+(h)
+()
+(k)
+SLLG R
+SLLG RW
+SLLG T
+SLLG(SA) R5
+FIG. 3. Robust validation of genetic tunneling algorithm at low temperature on Fe/Pd/Ir (111) system. The caption GTO
+represents genetic tunneling optimizer with rank selection and SA represents conventional simulated annealing (a) and (b).
+The predicted ground state energy and simulation execution time with variable system size. (c)First generation simulation
+execution time in second and predicted ground state energy performed from 32 CPU cores to 512 CPU cores for 100 × 100
+spin system. (d) First generation simulation execution time in second and predicted ground state energy performed from 32
+CPU cores to 1024 CPU cores for 200 × 200 spin system. Each point in (c) and (d) represents 5 simulations. The time error
+bar in (c) and (d) shows the highest, lowest, and average first-generation execution time. The energy error bar in (c) and (d)
+shows the highest and lowest predicted ground state energy. (e) Energy-field phase diagram at 0.1 mK, in which SS, SKL, and
+FM represent spin spiral, skyrmion lattice, and ferromagnetic state, separately. The cyan and purple color zone represent the
+SS-SKL and SKL-FM transition zone. The black arrow indicates the Y-axis scale. (f) Executive time with different applied
+fields.
+exception.
+The figure shows that the associated time
+consumption is notably lower than the conventional sim-
+ulated annealing method, especially for simulations with
+small system sizes.
+However, the genetic tunnel opti-
+mizer has a computational cost that increases faster with
+system size compared to the simulated annealing method
+(Figure 3(b)). However, because of the intrinsic property
+of the heuristic searching method, the here suggested ge-
+netic tunneling optimizer also has a chance to quickly
+reach convergence in some particular simulation,e.g., in
+the case of simulation system size increased to 220×220,
+the time consumption is lower than the simulation with
+
+(a)
+(b)
+GTO
+GTO
+6 
+SA
+SA
+-3.838
+3.838
+4
+Time (
+-3.838
+2
+3.839-
+0
+60x60
+0z0zz 002x00z 08x08 091x091 0x0 0z1x0z 001x001 08x08
+60x60
+80x80
+System size (atoms)
+System size (atoms)
+(c)
+(d)
+100
+1400-
+200
+200-
+-3.838
+Execution time
+Execution time
+-3.838
+system
+system
+1200-
+175-
+size
+Final system energy
+(mRy/atom)
+size
+Final system energy
+150-
+1000-
+125-
+800
+-3.839
+3.839
+Time
+100-
+600
+Energy (
+75 
+400
+50 -
+-3.840
+200
+-3.840
+25 
+0
+32
+64
+128
+256
+512
+32
+64
+128
+256
+512
+1024
+CPU cores
+CPU cores
+(e)
+(f)
+ss
+SKL
+FM
+1.6
+Transition zone
+100
+80
+1.4-
+GTO
+-- GTO
+Q
+100
+system
+ SA
+ size
+ SA
+1.2
+60
+char
+1.0
+-3.840
+(m
+0.8
+rgy
+Time
+0.6-
+-3.844
+20
+100
+0.4
+GTO
+ system
+'size
+SA
+3.848-
+0.2
+1.31.51.71.92.12.32.52.72.93.13.33.53.7
+1.31.5
+1.71.92.12.32.52.72.93.13.33.53.7
+Applied field (T)
+Applied field (T)6
+FIG. 4. Ground state search for artificial Bloch-type skyrmionic system and N´eel-type skyrmionic Pd/Fe/Ir(111) system with
+the simulation temperature at 8K. The red line represents the Hamiltonian of the best individual of each generation (Elite
+group), the yellow band represents the energy distribution of elite individuals in each generation, and the blue line represents
+the energy of spin configuration predicted baseline simulated annealing. To show the optimization process in detail, we set the
+convergence limit to an extremely low value. Based on this, all optimization will run up to the maximum iteration threshold
+of 50 in this work. The real space spin configuration visualization of the genetic tunneling optimizer and simulated annealing
+optimizer is shown in the middle of each figure. The color bar inside of each figure indicates the direction of the spin moment,
+and the color change from red to blue represents the spin moment’s direction from out of the plane to in plane. (a), (b), and
+(c) include the simulation on a system with parameterized exchange and with 40T,150T, and 400T applied fields, respectively.
+(d), (e), and (f) indicated the simulation of the Pd/Fe/Ir(111) system with 1T, 2.7T, and 4.4T. The ground state of the system
+in (a)-(b), (c)-(d), and (e)-(f) are spin spiral, skyrmion lattice, and ferromagnetic state, separately.
+180 × 180 spins.
+To investigate the efficiency of the genetic tunneling al-
+gorithm with concurrent computing, a set of simulations
+was performed on a variety of computing nodes. A com-
+parison of the results is shown in Figure 3(c) and (d). In
+Figure 3(c), a Pd/Fe/Ir(111) system with 100×100 spins
+are chosen to run on computer architectures ranging from
+32 CPU cores to 512 CPU cores. It appears that the low-
+est energy is almost equivalent, while the execution time
+of generations decreases with the increase of the number
+
+Bloch-type artificial skyrmion system
+Néel-type Pd/Fe/lr system
+(a)
+(b)
+-4.550-
+B=40T
+B=1T
+Best individual
+Elites
+SA
+3.777-
+-4.560-
+0.77
+3.779-
+-4.570-
+3.781-
+-4.580-
+Best individual
+Elites
+SA
+3.783
+-4.590+
+0
+10
+20
+30
+40
+50
+60
+0
+10
+20
+30
+40
+50
+60
+(c)
+(d)
+Energy (mRy/atom)
+B=150T
+Best individual
+B=2.7T
+Best individual
+Elites
+-3.780-
+Elites
+SA
+ SA
+4.722
+-3.782-
+GTC
+4.723-
+3.784-
+0.77
+-3.786
+4.724-
+3.788-
+4.725-
+10
+20
+30
+40
+50
+60
+0
+10
+20
+30
+40
+50
+60
+0
+(e)
+(f)
+B=400T
+B=4.4T
+Best individual
+Best individual
+Elites
+Elites
+-3.794-
+5.680-
+SA
+GTO
+SA
+-5.680
+3.798-
+-0.93
+5.681-
+0.77
+-3.802
+-5.681-
+-3.806
+5.682-
+10
+20
+30 
+40
+50
+60
+0
+10
+20
+30 
+40
+50 
+60
+Generations7
+of cores. Since the simulation system size is too small
+to show the power of concurrent computing with more
+than 128 cores, we double the system size and show the
+simulation result in Figure 3(d). From these results, it
+may be concluded that the genetic tunneling algorithm
+can appreciably benefit from concurrent computing.
+To evaluate the performance of the genetic tunneling
+algorithm under different applied fields, we calculate a
+phase diagram shown in Figure 3(e). Here we also show
+a comparison of the conventional simulated annealing
+method. The execution time of the simulation is shown
+in Figure 3(f). It can be inferred that, in this case, the
+genetic tunneling algorithm can always identify lower en-
+ergy than the simulated annealing method.
+However,
+for this system, the energy difference between the two
+methods is negligible. More importantly for this system,
+the genetic tunneling algorithm has clearly lower time
+consumption(see Figure 3(f)).
+For topological charge,
+with the result from the genetic tunneling algorithm in
+Figure 3(e), we conclude that this system will hold the
+quantity of topological charge in the range of applied
+fields of 1.6 to 3.1T and have two transition zones, in the
+spin spiral to skyrmion lattice zone and skyrmion lattice
+to ferromagnetic zone, the topological charge will ascent
+and descent, respectively. More details about the energy
+difference between configurations with various topologi-
+cal charges in the second transition zone can be found in
+Supplementary C.
+E.
+Searching ground state at 8K with different
+applied fields
+In this section, we present results from investigations
+on how the genetic tunneling algorithm works with dif-
+ferent kinds of skyrmionic systems at a temperature of
+8K with a variety of applied fields. Two simulated sys-
+tems are shown in Figure 4. The first column includes
+Figure 4 (a/c/e), which represents simulations on an ar-
+tificial frustrated spin System that can exhibit spin spi-
+ral state, Bloch-type skyrmion state, and ferromagnetic
+state at low, medium, and high applied field, respec-
+tively.
+The simulated system of the second column in
+Figure 4 (b/d/f) is for a Pd/Fe/Ir(111) monolayer which
+also has three states at different fields, but instead of
+Bloch skyrmion, it contains N´eel-type skyrmion.
+As shown in Figure 4 (a/c/e), both the genetic tun-
+neling algorithm and conventional simulated annealing
+method can find the ground state of the artificial system
+with different applied fields, but as regards the energy
+itself, the genetic tunneling algorithm can always find
+spin configurations with lower energy. This result also
+reflects the real space spin configuration visualization,
+especially in Figure 4 (c), evidently while magnetic mo-
+ments in both systems follow the Boltzmann distribution
+at 8K, the hexagonal skyrmion lattice found by genetic
+tunneling algorithm relatively more stable.
+From Figure 4 (b/d/e), it seems that for a more com-
+plex, real system, the genetic tunneling algorithm still
+works better than the conventional simulated annealing.
+When the applied field is 1 Tesla, the ground state pre-
+dicted by genetic tunneling optimizer is a twisty spin
+spiral state which is in remarkably good agreement with
+experimental data in Ref.[21]. However, the simulated
+annealing gives a hybrid phase that contains spin spirals
+and bubbles. When it comes to 2.7 and 4.4 Tesla, the
+genetic tunneling algorithm apparently gives a more rea-
+sonable hexagonal skyrmion lattice, and thermal fluctu-
+ated ferromagnetic state. To summarize, the final system
+energy that we get from the genetic tunneling optimizer
+is lower than what we obtain from the simulated anneal-
+ing in both an artificial and a real Pd/Fe/Ir(111) system.
+It is affirmed that this algorithm is robust in different
+skyrmionic systems with different applied fields.
+F.
+Ground state searching at finite temperatures
+In order to analyze the performance of the genetic tun-
+neling algorithm under the influence of thermal fluctua-
+tion, we performed simulations with a variety of tempera-
+tures on the Pd/Fe/Ir(111) system with an applied field
+of 2.7 Tesla which is in the middle of skyrmion lattice
+zone. The results are shown in Figure 5. When tuning
+the system temperatures, the green and yellow line plot
+in Figure 5 (a) indicate that with different temperatures,
+the predicted ground state energy is broadly similar, but
+our genetic tunneling algorithm will find slightly lower
+energy than conventional simulated annealing.
+When analyzing the topological charge of the predicted
+ground state, our genetic tunneling algorithm success-
+fully found that the trend of the topological charge is
+decaying with increasing temperature increased, which is
+consistent with the experimental result[21, 22]. More de-
+tails are shown in Figure 5 (b)-(i), which demonstrates
+that the conventional simulated annealing has underesti-
+mated the stability of the skyrmion lattice, especially in
+Figure 5 (b/c) the result from genetic tunneling optimizer
+is more acceptable for the hexagonal lattice shape.
+II.
+DISCUSSION AND CONCLUSION
+To the best of our knowledge, this is the first study
+establishing a real-space genetic-tunneling optimization
+protocol for complex spin systems with long-range in-
+teractions.
+The algorithm presented in this work has
+general applicability to magnetic systems, and is here
+shown to successfully find the ground state for mono-
+layer spin systems at finite temperature and at a variety
+of applied magnetic fields. The approach contains two
+essential parts: (1) A variance-threshold controlled local
+optimizer, which includes a Markov chain Monte Carlo
+optimizer and a spin-dynamic optimizer and (2) a real-
+space genetic tunneling metaheuristic searching module.
+The algorithm is designed to be able to escape from a
+
+8
+FIG. 5. Ground state searching on Pd/Fe/Ir (111) system at finite temperatures. GTO represents the genetic tunneling method
+with Markov Chain Monte Carlo backend and rank selection strategy. SA represents simulated annealing. (a) The ground state
+energy and topological charge of the Pd/Fe/Ir (111) system at finite temperatures. (b)-(e) shows the acceptable ground state
+that finds by GTO at finite temperatures. (f)-(i) indicates the ground state that finds by conventional SA. The green dash
+cycle in (b),(c),(d), and (f) are eye guidings for the hexagonal skyrmion lattice. The black arrow indicates the Y-axis scale.
+local minimum by genetic tunneling operators and find
+an advisable global minimum for a given system without
+any initial guess. However, either fed artificial spin con-
+figuration or involved simulated annealing to the initial
+generation, it will, as demonstrated here, speed up the
+convergence.
+The efficiency of genetic tunneling is investigated on
+both a simple artificial system with magnetic frustration
+and a Pd/Fe/Ir(111) monolayer that includes complex
+Heisenberg and Dzyaloshinskii–Moriya interactions, as
+calculated from DFT. The result indicates that the ge-
+netic tunneling algorithm has better performance than
+conventional simulated annealing at extremely low tem-
+peratures as well as for finite temperatures, when it
+comes to finding stable spin configurations as a function
+of external parameters like temperature and applied mag-
+netic field. Most noteworthy, we here considered the spi-
+ral structure, the skyrmion lattice, and the ferromagnetic
+state. It can also be concluded that the performance algo-
+rithm is not limited by the system size, geometry, nature
+of the magnetic interactions, temperature or the applied
+field strength.
+In practice, he here proposed method needs a fine-
+tuning depending on the system under consideration.
+This includes the current version of ’cut-off’, where the
+variance threshold may increase the risk of premature
+convergence. For each particular system, a fine-tuning of
+the hyperparameter process is also needed, which should
+include a design of genetic operator and threshold value
+testing in order to get better performance.
+Therefore,
+further research should focus on those aspects for im-
+proving this prototype protocol.
+In conclusion, we have explored a genetic tunnel-
+ing protocol, which is designed to predict the magnetic
+ground state of a magnetic system at finite tempera-
+tures.
+We demonstrate that our method is robust for
+two-dimensional systems, both for a simpler model sys-
+tems and for a more complex Pd/Fe/Ir(111) system. We
+envision that our findings will pave the way for evolution-
+ary computing in finding the ground state of magnetic
+systems, e.g., for magnets with non-trivial topology and
+spin glass systems. It is possible that the here suggested
+protocols will find applications in other areas of solid-
+state science, or even in fields outside natural science.
+
+(a)
+-3.4
+50
+GTO
+GTO
+SA
+45
+SA
+Q
+3.5
+40
+charge
+35
+3.6
+30
+3.
+25
+20
+-3.8
+15
+-3.9-
+10
+0
+10
+20
+30
+40
+50
+09
+Temperature (K)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g
+(h)
+(i)9
+III.
+METHOD
+A.
+Variance-controlled local energy optimizer at
+finite temperature
+In a local optimization of spin configurations, which
+involves finding local minima obtained from given initial
+guesses, several approaches can be used, e.g, the Markov
+chain Monte Carlo (MCMC) optimizer or a spin dynam-
+ics simulation method with Gilbert damping. These two
+methods have been proven to be robust and efficient in
+describing complex spin systems[6, 23]. Generally, the
+Metropolis MCMC optimizer performs energy minimiza-
+tion under finite temperature by using the transition
+probability Pt between two spin configurations in the
+Markov chain:
+Pt =
+�
+exp
+�
+− ∆E
+KBT
+�
+,
+if ∆E > 0
+1,
+otherwise
+(2)
+where ∆E, KB, and T are the energy difference between
+spin configurations, the Boltzmann constant, and the
+temperature of the system.
+With a given initial spin
+configuration the method will iteratively minimize the
+energy of the system.
+The other approach of optimization of spin configu-
+rations is the spin dynamic optimizer, which uses the
+stochastic Landau–Lifshitz–Gilbert (SLLG) equation to
+simulate the time evolution of atomic magnetic moments.
+This method can reach a spin configuration near a local
+energy minimum from a given initial state when Gilbert
+damping[23] is included in the simulations, since energy
+is the allowed to dissipate from the system. The atom-
+istic SLLG equation is:
+dmi
+dt
+= − γLmi ×
+�
+Bi + Bf
+i
+�
+− γL
+α
+mi
+mi ×
+�
+mi ×
+�
+Bi + Bf
+i
+��
+(3)
+where Bi and Bf
+i are the effective magnetic field and a
+stochastic magnetic field that is related to thermal fluc-
+tuation of a heat bath with temperature T. In this equa-
+tion, the first term represents the precessional motion
+of atomic magnetic moments while the second term de-
+scribes the damping motion.
+In the expression above,
+γL is the renormalized gyromagnetic ratio which is cal-
+culated from:
+γL =
+γ
+(1 + α2)
+(4)
+where γ and α are the gyromagnetic ratio respectively
+the isotropic Gilbert damping constant.
+In the present work, we use both the Metropolis
+MCMC and SLLG local optimizations, with a given ini-
+tial spin configuration, using the Uppsala Atomistic Spin
+Dynamics (UppASD) package[10]. Both optimizers can
+successfully minimize the energy under finite tempera-
+ture from a given initial guess of a spin configuration.
+Additionally, for automatically stopping the minimiza-
+tion process at the desirable convergence level, as shown
+in Figure 1 (a), we introduce a variance threshold that
+for the spin Hamiltonian used here is defined as:
+Var(H ) = 1
+n
+n
+�
+i=1
+(Hi − ⟨Hi⟩)2
+(5)
+where ⟨Hi⟩ means the expectation value of spin Hamilto-
+nian. With the predefined variance threshold, a control-
+lable local optimization can be performed for a desirable
+optimum near the initial state.
+B.
+Metaheuristic genetic-tunneling methods
+The metaheuristic algorithm can interactively guide
+and modify the operations of subordinate heuristics, to
+efficiently produce preferable solutions within a high-
+dimensional search space[24, 25]. Representative meta-
+heuristic algorithms include the Simulated Annealing
+(SA), the Particle Swarm Optimization (PSO)[26] and
+Genetic Algorithms (GA)[27]. It has been shown that a
+hybrid algorithm, which combines the algorithms men-
+tioned above with a local optimizer, e.g., the gradient
+descent, can be an efficient way to solve the global opti-
+mization problem within a complex configuration space.
+In this study, as shown schematically in Figure 1 (c),
+we introduced a genetic tunneling strategy that mini-
+mizes the energy of the spin system globally. The whole
+procedure is demonstrated as a flow chart in Figure 1
+(d).
+Basically, the whole algorithm starts with a ran-
+dom initial generation C0 that contains relevant spin
+configurations. Typically, a limited number of spin con-
+figurations, NP , is considered, each labeled C0
+i where
+i = 1, 2, · · · , Np. Hence the connection between C0 and
+C0
+i is
+C0 =
+�
+C0
+i | i = 1, 2, · · · , Np
+�
+.
+(6)
+In practice, the work here has considered one configura-
+tion, C0
+i , as an object represented by 100 × 100 atomic
+spins, but this number can naturally change, depend-
+ing on the material in question that is studied. In this
+work, NP has been chosen to be 64.
+The strategy is
+then to view these 64 configurations as a parent gener-
+ation, and by inspiration from evolutionary science, one
+wishes to find methods to find adapted better generations
+Ck, where k labels the generation in question. Provided
+enough generations have been generated and scrutinized
+for performance, the hope is to have an optimal repre-
+sentative of the magnetic configuration, where the total
+energy represents the main search criteria of interest in
+this work.
+
+10
+Note that an optional simulated annealing process can
+be invoked to seed the initial generation with optimal
+configuration. With the initial generation, C0, the main
+metaheuristic searching loop begins with the selection
+of good candidates for the initial generation. Currently,
+three classical genetic selection operators are available in
+this work, i.e., roulette wheel selection, tournament se-
+lection, and ranking selection. As an example, in roulette
+wheel selection, the probability of choosing one candidate
+spin configuration C0
+i is equal to:
+P(C0
+i ) =
+Hi
+�Np
+i=1 Hi
+(7)
+where Hi is the Hamiltonian of a given spin configu-
+ration C0
+i .
+The selection progress will stop before the
+number of chosen spin configurations exceeds the preset
+parent group limitation, Np. In this work, we select from
+these Np configurations a smaller group (typically 4) that
+defines the parent group used to generate one single off-
+spring of the next generation. The probability of finding
+a single parent configuration is calculated from Eqn.7,
+and we choose in this way the four configurations that
+define a parent group.
+From these four configurations
+one offspring is generated by considering spin segments
+of each parent, and then to apply genetic tunneling oper-
+ators, which include crossover and mutation that enable
+to dig of a tunnel on the potential energy surface to an-
+other minimum with a deeper energy barrier.
+In this work, we use linear and squared crossover
+and mutation operators. As shown in Figure 1(b), the
+squared operators will split each given spin configuration
+in the group of (four) parents into four parts and put
+them into a spin segmentation group, that will contain
+a total of sixteen segments. If the operator represents
+crossover, four spin segments will be randomly selected
+from the spin segment group, to generate an offspring
+spin configuration by combination. If one considers dis-
+turbances to a given configuration, in the spirit of muta-
+tions of biological science, one can apply a perturbation
+before the combination takes place, as is shown in Fig-
+ure 1(b). Here, a perturbation operation is applied with
+some random shuffling or adding Gaussian noise to the
+spin moments within each spin segment. After such a
+possible mutation, the algorithm used here continues to
+combine as an offspring configuration of 64 members, and
+the whole procedure can be repeated. The linear opera-
+tors follow the same workflow as squared operators but
+have more freedom to choose the number of linear seg-
+ments to split from one spin configuration. In the sup-
+plementary section, we provide more details on the other
+two selection methods used in this work.
+C.
+Details of the procedure
+As shown in Figure 1(d), the flowchart of the proce-
+dure starts from the input of crystal information, ran-
+dom seed, and magnetic interactions. Those inputs can
+build up the simulation system, in which we use only
+nearest-neighbor interactions for the model system with
+selected (not calculated) exchange parameters. For this
+system, we choose parameters that are designed to gen-
+erate Bloch-type skyrmions. In addition, we used more
+than thirty neighbor interactions in the Pd/Fe/Ir(111)
+system, where all parameters were calculated from first-
+principles theory. We used a large truncation in number
+of neighbors, in order to avoid truncation errors.
+After preparing all input data, we adopted a random
+spin configuration generator, to produce random spin
+configurations.
+Then we optimize this random config-
+uration with the Metropolis or SLLG optimizer, in order
+to obtain locally optimized configurations. A larger num-
+ber of such configurations were realized and we selected
+64 by use of a condition that two configurations should
+not be too close in energy. To be precise we used the
+criterion that
+H
+�
+C0
+r
+�
+− H
+�
+C0
+i
+�
+) > ∆E,
+(8)
+where ∆E is a threshold energy difference, that guaran-
+tees that configurations C0
+r and C0
+i are not too similar.
+In this work, we use values ranging from 10−4 to 10−6
+mRy/atom for ∆E. Once the quantity of randomly gen-
+erated spin configurations C0 up the preset value, which
+is set to 64 in this work, as well as the offspring candi-
+date quantity, the procedure will end up random search-
+ing process and adapt genetic tunneling methods and try
+to search global minimum. The metaheuristic searching
+process will stop when the iterations are up to the pre-
+defined threshold, which is 50 in this work, or when the
+energy of each local optimized spin configuration in the
+generation CLOPT get converged.
+After the searching
+process stops, the procedure will extract final spin con-
+figurations COPT that represents the best solution (see
+flowchart in Figure 1(d)).
+D.
+Simulated annealing
+In this work, we introduced a quickly simulated an-
+nealing initial spin configuration generator as an optional
+module in the procedure workflow and a simulated an-
+nealing ground state searcher in the benchmark that is
+shown in Figure 2.
+The quickly simulated annealing generator has sparse
+temperature mesh with a size of four. In detail, the tem-
+perature mesh starts from the simulated temperature and
+adds 20K, 50K, and 200K to get the temperature mesh.
+In each temperature, it will execute 2000 times Metropo-
+lis progress. Suggest value for the application of a quick
+simulated annealing generator is not discussed in this pa-
+per but may be interesting for further study.
+For the simulated annealing that was used as the base-
+line, we compared two temperature strategies, both of
+which start from 900K, which is close to the curie tem-
+perature of the Fe monolayer. The first one is a common
+strategy typically used in UppASD simulations where we
+
+11
+set temperature mesh with fixed gaps from 200K to 1K,
+and the other one uses a temperature decrease ratio from
+0.95 to 0.9. In this work, we find the difference between
+those two strategies with the same simulated annealing
+steps is minimal, but the second one will consume more
+than ten times more than the first strategy. We adopt
+the first strategy to set the temperature stages for better
+inheritance from previous work and sustainability.
+IV.
+DATA AVAILABILITY
+All data needed for reproducing the results can be
+found in the GitHub repository https://github.com/
+MXJK851/GTO-2D
+V.
+CODE AVAILABILITY
+All code of GTO-2D is available at https://github.
+com/MXJK851/GTO-2D under the GPL-3.0 license.
+ACKNOWLEDGMENTS
+The
+authors
+acknowledge
+financial
+support
+from
+the Knut and Alice Wallenberg Foundation through
+Grant No.
+2018.0060.
+Q.X. acknowledges the China
+Scholarship Council (201906920083), O.E. acknowledges
+eSSENCE, the Swedish Research Council (VR) and the
+ERC (project FASTCORR-Grant No. 854843). A.D. ac-
+knowledges financial support from the Swedish Research
+Council (VR) through Grants No. 2019-05304 and 2016-
+05980.
+The computations/data handling were enabled
+by resources provided by the Swedish National Infras-
+tructure for Computing (SNIC), partially funded by the
+Swedish Research Council through grant agreement No.
+2016-07213. The authors would also like to extend our
+gratitude to Fredrik Dehlin, Liang Chen, Ivan Miranda,
+Filipp N. Rybakov, and Liuzhen Yang for their valuable
+comments and discussions during the writing of our pa-
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+
+12
+VI.
+SUPPLEMENTARY
+A.
+Implementation details of selection operators
+The pseudocode corresponding to the tournament se-
+lection and rank selection algorithms used in this work
+are provided in algorithm 1 and algorithm 2.
+B.
+Relative energies of different phases in the
+Pd/Fe/Ir (111) system at 0.1 mK in the transition
+zone from the skyrmion lattice phase to the
+ferromagnetic phase
+In Table I, we list the calculated average relative en-
+ergies per atom for four different phases in the Pd/Fe/Ir
+(111) system: the ferromagnetic phase (FM), the isolated
+skyrmion phase (Sk) and the 6 × 5 and 6 × 6 skyrmion
+lattice phase (SkL). The applied magnetic field is varied
+from 3.1 to 3.4 T. Zero energy denotes the found ground
+state in each case. This table proves that the algorithm
+we propose in this paper indeed correctly discovered the
+transition state in Figure 3(e).
+Algorithm1 Tournament selection for local optimized
+spin configuration
+Choose the tournament size K
+Choose the parent group set size O
+Set counter i = 0
+for i = 1 to O do
+Choose K spin configurations from the parent genera-
+tion at random
+Calculate the energies of these K spin configurations
+Choose the best spin configuration with the lowest en-
+ergy from the tournament
+end for
+Output selected spin configuration
+Algorithm2 Rank selection for local optimized spin
+configuration
+Choose the parent group set size O
+Set counter i = 0
+for i = 1 to O do
+Calculate the energies of all spin configurations from the
+parent generation
+Choose and remove the best spin configuration with the
+lowest energy from the parent generation
+end for
+Output selected spin configuration
+
+13
+TABLE I. Relative energies of different phases in the Pd/Fe/Ir (111) system (mRy/atom) as a function of magnetic field B.
+B (T)
+FM
+Isolated Sk
+6x5 SkL
+6x6 SkL
+3.10
+4.08450000E-04
+3.92660000E-04 0.00000000E+00 1.21990000E-04
+3.11
+3.63090000E-04
+3.48780000E-04 0.00000000E+00 1.35040000E-04
+3.12
+3.17720000E-04
+3.04880000E-04 0.00000000E+00 1.48080000E-04
+3.13
+2.72360000E-04
+2.60990000E-04 0.00000000E+00 1.61130000E-04
+3.14
+2.26990000E-04
+2.17100000E-04 0.00000000E+00 1.74170000E-04
+3.15
+1.81630000E-04
+1.73200000E-04 0.00000000E+00 1.87210000E-04
+3.16
+1.36260000E-04
+1.29310000E-04 0.00000000E+00 2.00250000E-04
+3.17
+9.09000000E-05
+8.54200000E-05 0.00000000E+00 2.13290000E-04
+3.18
+4.55400000E-05
+4.15300000E-05 0.00000000E+00 2.26340000E-04
+3.19
+2.53000000E-06 0.00000000E+00 2.37000000E-06 2.41750000E-04
+3.20
+1.05000000E-06 0.00000000E+00 4.62500000E-05 2.98680000E-04
+3.21
+0.0000000E+00
+4.10000000E-07
+9.05600000E-05 3.56030000E-04
+3.22
+0.00000000E+00 1.89000000E-06
+1.35920000E-04 4.14440000E-04
+3.23
+0.00000000E+00 3.36000000E-06
+1.81290000E-04 4.72850000E-04
+3.24
+0.00000000E+00 4.84000000E-06
+2.26660000E-04 5.31260000E-04
+3.25
+0.00000000E+00 6.31000000E-06
+2.72020000E-04 5.89660000E-04
+3.26
+0.00000000E+00 7.79000000E-06
+3.17390000E-04 6.48080000E-04
+3.27
+0.00000000E+00 9.25000000E-06
+3.62750000E-04 7.06480000E-04
+3.28
+0.00000000E+00 1.07300000E-05
+4.08120000E-04 7.64890000E-04
+3.29
+0.00000000E+00 1.22000000E-05
+4.53480000E-04 8.23300000E-04
+3.30
+0.00000000E+00 1.36700000E-05
+4.98850000E-04 8.81710000E-04
+
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+page_content='Genetic-tunneling driven energy optimizer for magnetic system  Qichen Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 2 Zhuanglin Shen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='3 Manuel Pereiro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4 Pawel Herman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 2 Olle Eriksson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4 and Anna Delin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 2  1Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' School of Engineering Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' KTH Royal  Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' AlbaNova University Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' SE-10691 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Sweden  2SeRC (Swedish e-Science Research Center),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' KTH Royal Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' SE-10044 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Sweden  3CAS Key Laboratory of Quantitative Engineering Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Shenzhen Institute of Synthetic Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Shenzhen Institute of Advanced Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' China  4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Uppsala University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Box 516,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' SE-75120 Uppsala,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Sweden  5Division of Computational Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' School of Electrical Engineering and Computer  Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' KTH Royal Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' AlbaNova University Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' SE-10691 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Sweden  (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 2023)  Novel topological spin textures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' such as magnetic skyrmions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' benefit from their inherent stability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  acting as the ground state in several magnetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In the current study of atomic monolayer  magnetic materials, reasonable initial guesses are still needed to search for those magnetic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  This situation underlines the need to develop a more effective way to identify the ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  To solve this problem, in this work, we propose a genetic-tunneling-driven variance-controlled opti-  mization approach, which combines a local energy minimizer back-end and a metaheuristic global  searching front-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' This algorithm is an effective optimization solution for searching for magnetic  ground states at extremely low temperatures and is also robust for finding low-energy degenerated  states at finite temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  We demonstrate here the success of this method in searching for  magnetic ground states of 2D monolayer systems with both artificial and calculated interactions  from density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' It is also worth noting that the inherent concurrent property of this  algorithm can significantly decrease the execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In conclusion, our proposed method builds  a useful tool for low-dimensional magnetic system energy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Optimization algorithms hold a fundamental connec-  tion inside the interdisciplinary boundaries of physics  and computer science which enhance the understand-  ing of novel physical phenomena, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', topologically non-  trivial defects and textures[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' It can also accelerate  new functional material findings, including atomic mono-  layer magnetic materials, which is currently an active re-  search field in the magnetism community[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  When it  comes to 2D magnetic materials, typically, there are sets  of algorithms for the energy optimization process of the  magnetic system at finite temperature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', the gradient  descent family, Monte Carlo approaches, and spin dy-  namic methods with damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' [5–7] These conventional  algorithms are being plagued by the possibility of get-  ting trapped into the local energy minimum rather than  the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Thus, there is a need to use meta-  heuristic methods to provide a better route to the global  minimum of the potential energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In particu-  lar, the Markov chain Monte Carlo based heat-bath opti-  mizations, a group of non-gradient sampling algorithms,  are proven to be effective and robust in searching for  low-energy states at finite temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' [8, 9] Unfortu-  nately, in current implementations[10, 11], there is still  some prior knowledge needed for getting acceptable re-  sults, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', appropriate initial guesses and manual con-  vergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' This situation brings a fundamental  challenge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', finding a more effective and automatic way  to apply a heuristic optimal searching for the magnetic  ground state without any initial guess or prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In order to overcome this challenge, a hybrid model  can be a straightforward solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  There are several  successful hybrid approaches under the idea of combin-  ing meta-heuristic algorithms and typical optimizing ap-  proaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', hybrid Monte Carlo[12], neural annealing  optimization[13] or neural evolutionary method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' [14, 15]  But unfortunately, those approaches are mainly designed  for the Ising model and may face problems when handling  realistic material with long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In this work, inspired by the idea of tunneling on  potential energy surface[16] and hybrid meta-heuristic  solutions[17, 18], a metaheuristic energy minimization  approach is proposed and tested for magnetic systems  with non-trivial topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The method is  based on several variance-threshold local optimizers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=',  spin dynamic optimizer and heat-bath Monte Carlo op-  timizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  This involves a real space genetic tunneling  front-end for searching of optimal initial guess and a se-  lected local optimizer with adaptive variance-based con-  vergence criteria as the back-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' We analyze the per-  formance of this algorithm that has the potential to es-  cape from local traps in energy minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' We inves-  tigate the efficiency by simulations on a 2D monolayer  with model exchange parameters that give rise to Bloch-  type skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' We also investigate an experimentally  well-studied system:  Pd/Fe/Ir(111), which contains a  N´eel-type skyrmionic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The validation process was  performed on the aforementioned systems with different  applied fields at finite temperatures, and we compared  the proposed method with conventional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='00207v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='comp-ph]  31 Dec 2022  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (a) Conceptual illustration of the variance-threshold controlled localized optimization process to find low energy spin  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The contour map in the middle shows an example of a potential energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The darker zones in the  coutour map represents lower energy areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The arrows connect the local optimization process and the configuration point  in the potential energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' At the top of the figure, five colored blocks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', Early, Rough, Medium, Fine, and Precise,  denote different converge levels of the search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The ”Early” level means the lowest convergence, and the ”Precise” level  represents the highest convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (b) Conceptual illustration of how the genetic operators are applied to the spin system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The  whole process involves three subprocesses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', spin configuration segmentation, crossover, and perturbation-based mutation  (for details, see the Method section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In the spin configuration segmentation part, the real-space spin textures are viewed as  information carriers similar to those in chromosomes in biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The spin textures are divided into several segments  that can be used in the same way as gene segments in crossover and mutation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' As shown in the top left of Figure (b),  the number indicates spin segments that have come from four different configurations of a parent configuration (see Method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In the mutation subplot (located in the top right of Figure (b)), the perturbation windows indicate the mutation operation  which works on part of the configuration (for details about perturbation, see methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (c) Conceptual illustration  of how to use the genetic tunnel to dig a tunnel through the energy barriers and heuristically search for a spin configuration  with lower energy, with the aim of finding the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The curve represents the potential energy surface and the  colored star represents the acceptable solution set at temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The height of the shadow colored regions equals KbT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The stars in the plot represent a single-spin configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The black points represent configurations of the energy landscape  that are not identified by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (d) The flowchart of the whole procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The dark yellow box represents input data  that need to be prepared before execution, and the light yellow box represents generated spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The box with a  dashed boundary represents an optional choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The dark blue rounded rectangles represent operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The white diamond  box represents conditional statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The notations H (C0  r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' H (C0  i ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' ∆E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Tseed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' CLOPT and COPT represent the  energy of a randomly generated spin configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' the energy of any spin configuration in the initial generation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' the threshold  of energy difference,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' the predefined threshold for select initial spin configurations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' the current spin configuration set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' the initial  generation set with a predefined quantity of spin configurations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' the local optimized spin configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' and the final optimized  spin configurations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' respectively (for details,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' see the Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='(d)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Crystal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Magnetic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Random  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='interactions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Seeds  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Early  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Medium  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Rough  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Fine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Precise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Energy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Random configuration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Simulated Annealing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='generator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='(SA) generator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Metropolis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='SLLG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='optimizer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='optimizer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Local optimization steps  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Initial spin configuration filter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='HP(C9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Artificial spin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='configurations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='(optional)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Quantity satisfied  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Initial generation of spin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='configurations c0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Crossover  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Mutation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Genetic selection operators  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='(base on Hamiltonian )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Genetic tunneling operators  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Local optimized spin segment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Metropolis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='SLLG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='optimizer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='optimizer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='(c)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Local optimizied spin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='AXX（  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Genetic operators  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='configurations CLOPT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Energy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Up to max iterations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  No  No  Converged?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  KbT  Yes  Spin configurations3  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  RESULT  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Spin system parametrization  Ground state searching problems of magnetic material  at zero kelvin can be reformulated as finding the global  minimum of the potential energy surface (PES),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' which is  here constructed by a classical Heisenberg spin Hamilto-  nian in the form:  H =  �  i̸=j  JijSi · Sj +  �  i̸=j  Dij · (Si × Sj)  +  �  i  Bext · Si +  �  i  KU  uni (Si · ez)2  (1)  where Si and Sj are spin moments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Jij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Dij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' KU  ani ez and  Bext are Heisenberg exchange interactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Dzyaloshin-  skii–Moriya interactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' uniaxial anisotropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' easy axis  vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' and the applied field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Typically,  these four Hamiltonian terms are enough to construct  a potential energy surface for the ground state that one  wishes to identify as specified by the magnetic config-  uration (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Note that extra terms can  also be involved in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', biquadratic exchange coupling or  spin-lattice coupling [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In this work, both model inter-  actions and realistic materials specific magnetic parame-  ters (which are calculated by using ab-initio density func-  tional theory - DFT) are included to demonstrate the ef-  fectiveness and efficiency of a proposed energy optimizer  for various complex potential energy surfaces[6, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Genetic tunneling procedure  Finding global optima in a complex potential energy  surface of a spin system with long-range interaction nu-  merically is commonly a non-deterministic polynomial-  time hard (NP-hard) problem, and it is difficult to find  appropriate solutions[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' To solve this problem, we pro-  pose a genetic tunneling algorithm to provide global en-  ergy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Full detail of the method is described  in the Methods section, but we outline the most salient  features here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The procedure is shown schematically in  Figure1, where in particular, Figure1 (d) illustrates a flow  chart of the method used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The figure shows the connec-  tion between the local optimization module and an evo-  lutionary global searching method, that allows to evolve  the spin configuration where segments of this configura-  tion are regarded as being similar to information carriers  of genes in biological material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' By using numerical pro-  tocols that mimic gene flow along generations, we explore  avenues to reach global minimum with minimal numerical  effort while avoiding being trapped in metastable config-  urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The optimizer workflow used here can be classed into  two parts: Firstly, it starts with the input, which con-  tains physical information of a given system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', crystal  lattice, atomic position, and magnetic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The  initial generation of spin configurations is generated hav-  ing random orientations of the atomic spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Directly af-  ter getting the initial generation, one local optimization  module is involved in relaxing all magnetic orientations  into the closest local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The local optimization  is controlled by variance threshold as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  (a) and discussed in the Method section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The other part  is metaheuristic searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Once one has a spin config-  uration of a local minimum for initial generation (see  Methods section), the procedure will come to segmenta-  tion of the spin configuration and tunneling mechanisms  that bridge over energy maxima, as shown in Figure 1  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' This metaheuristic progress tries iterative searching  for the global minima until the evolutionary process gets  to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' See the Methods section for more detail  on the genetic tunneling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Finding a low-temperature ground state  In this section, we investigate the efficiency of the ge-  netic tunneling optimizer at low temperatures with dif-  ferent local optimizer backends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The simulation is per-  formed for a spin Hamiltonian that is appropriate for  a monolayer of Pd/Fe/Ir(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  All interactions used  for constructing the spin Hamiltonian are calculated by  DFT[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In order to ensure an energy landscape that is  hard, so that precise global optima exist, we employed in  these simulations a temperature of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='1 mK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The performance of the genetic tunneling optimizer  with different local backends is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Note  that we have used both Markov Chain Monte Carlo  (MCMC) and a spin dynamic local optimizer as back-  ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' For analysis performance of genetic tunneling op-  erators, each local optimizer is combined with three typ-  ical genetic selection operators,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', Rank(R), Tourna-  ment(T), and Roulette Wheel(RW), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Ad-  ditionally, for the purpose of studying the impact of  having a pre-optimized initial configuration, a simulated  annealing initial spin configuration generator was used  to combine the MCMC backend with Rank selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  For comparison, a classical Markov Chain Monte Carlo  based simulated annealing simulation with fine tempera-  ture mesh and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='5×106 steps is used as a baseline (results  denoted SA, see the Methods section for more detail on  simulated annealing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  An obvious trend in Figure 2 (a) is that the genetic tun-  neling optimizer with different kinds of backends can find  en energy minimum of the system, but a significant dif-  ference can be found between Markov Chain Monte Carlo  group and the spin dynamic-based group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' It is notewor-  thy that with finite local optimization steps and global  searching epochs, all spin dynamic optimizers may not  achieve convergence and in general Monte Carlo back-  ends work better, yielding consistently lower energy for  the same number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Genetic tunneling with  Markov chain Monte Carlo backend invariably reaches  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (a) Performance benchmark of the genetic tunneling optimizer at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='1 mK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The simulation system is Pd/Fe/Ir(111)  with an external magnetic field of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7 T out of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Here MCMC-R, MCMC-RW, and MCMC-T represent Markov chain  Monte Carlo local optimizers with Rank (R), Roulette Wheel (RW), and Tournament (T) genetic selection, respectively (for  details, see the Method section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The SLLG-R, SLLG-RW, and SLLG-T represent spin dynamic optimizers that solve the  stochastic Landau–Lifshitz–Gilbert (SLLG) equation with an artificial damping value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4 combined with Rank(R), Roulette  Wheel(RW) and Tournament(T) genetic selection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Moreover, SA represents the conventional simulated annealing  method and is used as reference energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' For better comparison, the result from SLLG R, SLLG RW, and SLLG T are shown  from generation 12 of the evolutionary algorithm (see Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (b) The time consumption in units of node-hours for  each optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' All of the simulations are performed on an Intel Xeon Gold 6130 CPU node with 32 cores without concurrent  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (c)-(k) Visualization of the final spin configurations from each optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The color, which changes from red to blue,  represents the spin moment’s direction from out of the plane (red) to in plane (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  convergence with lower energy than any other method,  including the reference energy obtained by simulated an-  nealing (thus defining the reference energy or the base-  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The energy of the predicted system is also reflected  in the real space spin configuration, which according to  Figure 2 (h)-(k), has no stable hexagonal skyrmion lat-  tice identified within the simulation time for any of the  methods based on the spin dynamic backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  For results obtained by Markov chain Monte Carlo  backend, it appears that all flavours of this method  achieve similar results, as may be seen in Figure 2 (a) and  Figure 2 (c)-(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' All of them identify a perfect hexagonal  skyrmion lattice, with a unit cell size which is in good  agreement with previous studies[20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The applied  quick simulated annealing initial configuration generator  is seen to influence the first generation’s performance and  to some degree the convergence speed, but it has no ef-  fect on the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Thus, for the purpose of studying  investigations of genetic tunneling mechanisms, instead  of using the quick simulated annealing module, we only  adopt the Markov Chain Monte Carlo based rank selec-  tion module in the rest of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In Figure 2 (b) we  show the time consumption of each method investigated  here, and it is clear that methods with Markov chain  Monte Carlo backend perform substantially better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Performance study on system sizes and applied  fields  In Figure 3(a) and (b), a set of different system sizes  are applied in simulation with a spin Hamiltonian appro-  priate for Pd/Fe/Ir(111) monolayer at T= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='1 mK with a  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7 Tesla applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' All simulations are performed on  1 computing node paralleled with 32 CPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The re-  sult shows that the genetic tunneling optimizer can find  the ground state of the system with lower energy than  the conventional simulated annealing method, without  (a)  (b)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='824-  MCMC R  SLLG R  MCMC RW  SLLG RW  MCMC T  SLLG T  (mRy/atom)  (node-hour)  MCMC(SA) R  SLLG(SA) R  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='828-  SA  100  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='832-  Energy  Time  MCMC R  SLLG R  MCMC RW  SLLG RW  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='836-  MCMC T  SLLG T  E  MCMC(SA) R  SLLG(SA) R  10-2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='840-  0  10  20  30  40  50  60  0  10  20  30  40  50  60  Iterations  Iterations  (c)  (d)  (e)  (f)  (g)  SA  MCMC R  MCMC T  MCMC(SA) R  MCMC RW  (i)  (h)  ()  (k)  SLLG R  SLLG RW  SLLG T  SLLG(SA) R5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Robust validation of genetic tunneling algorithm at low temperature on Fe/Pd/Ir (111) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The caption GTO  represents genetic tunneling optimizer with rank selection and SA represents conventional simulated annealing (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The predicted ground state energy and simulation execution time with variable system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (c)First generation simulation  execution time in second and predicted ground state energy performed from 32 CPU cores to 512 CPU cores for 100 × 100  spin system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (d) First generation simulation execution time in second and predicted ground state energy performed from 32  CPU cores to 1024 CPU cores for 200 × 200 spin system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Each point in (c) and (d) represents 5 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The time error  bar in (c) and (d) shows the highest, lowest, and average first-generation execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The energy error bar in (c) and (d)  shows the highest and lowest predicted ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (e) Energy-field phase diagram at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='1 mK, in which SS, SKL, and  FM represent spin spiral, skyrmion lattice, and ferromagnetic state, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The cyan and purple color zone represent the  SS-SKL and SKL-FM transition zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The black arrow indicates the Y-axis scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (f) Executive time with different applied  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The figure shows that the associated time  consumption is notably lower than the conventional sim-  ulated annealing method, especially for simulations with  small system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  However, the genetic tunnel opti-  mizer has a computational cost that increases faster with  system size compared to the simulated annealing method  (Figure 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' However, because of the intrinsic property  of the heuristic searching method, the here suggested ge-  netic tunneling optimizer also has a chance to quickly  reach convergence in some particular simulation,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', in  the case of simulation system size increased to 220×220,  the time consumption is lower than the simulation with  (a)  (b)  GTO  GTO  6  SA  SA  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='838  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='838  4  Time (  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='838  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='839-  0  60x60  0z0zz 002x00z 08x08 091x091 0x0 0z1x0z 001x001 08x08  60x60  80x80  System size (atoms)  System size (atoms)  (c)  (d)  100  1400-  200  200-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='838  Execution time  Execution time  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='838  system  system  1200-  175-  size  Final system energy  (mRy/atom)  size  Final system energy  150-  1000-  125-  800  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='839  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='839  Time  100-  600  Energy (  75  400  50 -  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='840  200  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='840  25  0  32  64  128  256  512  32  64  128  256  512  1024  CPU cores  CPU cores  (e)  (f)  ss  SKL  FM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='6  Transition zone  100  80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4-  GTO  -- GTO  Q  100  system  SA  size  SA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='2  60  char  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='840  (m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='8  rgy  Time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='6-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='844  20  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content="4  GTO  system  'size  SA  3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='848-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
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+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
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+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7  Applied field (T)  Applied field (T)6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Ground state search for artificial Bloch-type skyrmionic system and N´eel-type skyrmionic Pd/Fe/Ir(111) system with  the simulation temperature at 8K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The red line represents the Hamiltonian of the best individual of each generation (Elite  group), the yellow band represents the energy distribution of elite individuals in each generation, and the blue line represents  the energy of spin configuration predicted baseline simulated annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' To show the optimization process in detail, we set the  convergence limit to an extremely low value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Based on this, all optimization will run up to the maximum iteration threshold  of 50 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The real space spin configuration visualization of the genetic tunneling optimizer and simulated annealing  optimizer is shown in the middle of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The color bar inside of each figure indicates the direction of the spin moment,  and the color change from red to blue represents the spin moment’s direction from out of the plane to in plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (a), (b), and  (c) include the simulation on a system with parameterized exchange and with 40T,150T, and 400T applied fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  (d), (e), and (f) indicated the simulation of the Pd/Fe/Ir(111) system with 1T, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7T, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The ground state of the system  in (a)-(b), (c)-(d), and (e)-(f) are spin spiral, skyrmion lattice, and ferromagnetic state, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  180 × 180 spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  To investigate the efficiency of the genetic tunneling al-  gorithm with concurrent computing, a set of simulations  was performed on a variety of computing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' A com-  parison of the results is shown in Figure 3(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In  Figure 3(c), a Pd/Fe/Ir(111) system with 100×100 spins  are chosen to run on computer architectures ranging from  32 CPU cores to 512 CPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' It appears that the low-  est energy is almost equivalent, while the execution time  of generations decreases with the increase of the number  Bloch-type artificial skyrmion system  Néel-type Pd/Fe/lr system  (a)  (b)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='550-  B=40T  B=1T  Best individual  Elites  SA  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='777-  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='560-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='77  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='779-  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='570-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='781-  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='580-  Best individual  Elites  SA  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='783  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='590+  0  10  20  30  40  50  60  0  10  20  30  40  50  60  (c)  (d)  Energy (mRy/atom)  B=150T  Best individual  B=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7T  Best individual  Elites  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='780-  Elites  SA  SA  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='722  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='782-  GTC  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='723-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='784-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='77  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='786  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='724-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='788-  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='725-  10  20  30  40  50  60  0  10  20  30  40  50  60  0  (e)  (f)  B=400T  B=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4T  Best individual  Best individual  Elites  Elites  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='794-  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='680-  SA  GTO  SA  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='680  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='798-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='93  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='681-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='77  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='802  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='681-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='806  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='682-  10  20  30  40  50  60  0  10  20  30  40  50  60  Generations7  of cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Since the simulation system size is too small  to show the power of concurrent computing with more  than 128 cores, we double the system size and show the  simulation result in Figure 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' From these results, it  may be concluded that the genetic tunneling algorithm  can appreciably benefit from concurrent computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  To evaluate the performance of the genetic tunneling  algorithm under different applied fields, we calculate a  phase diagram shown in Figure 3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Here we also show  a comparison of the conventional simulated annealing  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The execution time of the simulation is shown  in Figure 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' It can be inferred that, in this case, the  genetic tunneling algorithm can always identify lower en-  ergy than the simulated annealing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  However,  for this system, the energy difference between the two  methods is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' More importantly for this system,  the genetic tunneling algorithm has clearly lower time  consumption(see Figure 3(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  For topological charge,  with the result from the genetic tunneling algorithm in  Figure 3(e), we conclude that this system will hold the  quantity of topological charge in the range of applied  fields of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='6 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='1T and have two transition zones, in the  spin spiral to skyrmion lattice zone and skyrmion lattice  to ferromagnetic zone, the topological charge will ascent  and descent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' More details about the energy  difference between configurations with various topologi-  cal charges in the second transition zone can be found in  Supplementary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Searching ground state at 8K with different  applied fields  In this section, we present results from investigations  on how the genetic tunneling algorithm works with dif-  ferent kinds of skyrmionic systems at a temperature of  8K with a variety of applied fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Two simulated sys-  tems are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The first column includes  Figure 4 (a/c/e), which represents simulations on an ar-  tificial frustrated spin System that can exhibit spin spi-  ral state, Bloch-type skyrmion state, and ferromagnetic  state at low, medium, and high applied field, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The simulated system of the second column in  Figure 4 (b/d/f) is for a Pd/Fe/Ir(111) monolayer which  also has three states at different fields, but instead of  Bloch skyrmion, it contains N´eel-type skyrmion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  As shown in Figure 4 (a/c/e), both the genetic tun-  neling algorithm and conventional simulated annealing  method can find the ground state of the artificial system  with different applied fields, but as regards the energy  itself, the genetic tunneling algorithm can always find  spin configurations with lower energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' This result also  reflects the real space spin configuration visualization,  especially in Figure 4 (c), evidently while magnetic mo-  ments in both systems follow the Boltzmann distribution  at 8K, the hexagonal skyrmion lattice found by genetic  tunneling algorithm relatively more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  From Figure 4 (b/d/e), it seems that for a more com-  plex, real system, the genetic tunneling algorithm still  works better than the conventional simulated annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  When the applied field is 1 Tesla, the ground state pre-  dicted by genetic tunneling optimizer is a twisty spin  spiral state which is in remarkably good agreement with  experimental data in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' However, the simulated  annealing gives a hybrid phase that contains spin spirals  and bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' When it comes to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4 Tesla, the  genetic tunneling algorithm apparently gives a more rea-  sonable hexagonal skyrmion lattice, and thermal fluctu-  ated ferromagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' To summarize, the final system  energy that we get from the genetic tunneling optimizer  is lower than what we obtain from the simulated anneal-  ing in both an artificial and a real Pd/Fe/Ir(111) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  It is affirmed that this algorithm is robust in different  skyrmionic systems with different applied fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Ground state searching at finite temperatures  In order to analyze the performance of the genetic tun-  neling algorithm under the influence of thermal fluctua-  tion, we performed simulations with a variety of tempera-  tures on the Pd/Fe/Ir(111) system with an applied field  of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7 Tesla which is in the middle of skyrmion lattice  zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' When tuning  the system temperatures, the green and yellow line plot  in Figure 5 (a) indicate that with different temperatures,  the predicted ground state energy is broadly similar, but  our genetic tunneling algorithm will find slightly lower  energy than conventional simulated annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  When analyzing the topological charge of the predicted  ground state, our genetic tunneling algorithm success-  fully found that the trend of the topological charge is  decaying with increasing temperature increased, which is  consistent with the experimental result[21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' More de-  tails are shown in Figure 5 (b)-(i), which demonstrates  that the conventional simulated annealing has underesti-  mated the stability of the skyrmion lattice, especially in  Figure 5 (b/c) the result from genetic tunneling optimizer  is more acceptable for the hexagonal lattice shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSION  To the best of our knowledge, this is the first study  establishing a real-space genetic-tunneling optimization  protocol for complex spin systems with long-range in-  teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The algorithm presented in this work has  general applicability to magnetic systems, and is here  shown to successfully find the ground state for mono-  layer spin systems at finite temperature and at a variety  of applied magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The approach contains two  essential parts: (1) A variance-threshold controlled local  optimizer, which includes a Markov chain Monte Carlo  optimizer and a spin-dynamic optimizer and (2) a real-  space genetic tunneling metaheuristic searching module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The algorithm is designed to be able to escape from a  8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Ground state searching on Pd/Fe/Ir (111) system at finite temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' GTO represents the genetic tunneling method  with Markov Chain Monte Carlo backend and rank selection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' SA represents simulated annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (a) The ground state  energy and topological charge of the Pd/Fe/Ir (111) system at finite temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (b)-(e) shows the acceptable ground state  that finds by GTO at finite temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' (f)-(i) indicates the ground state that finds by conventional SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The green dash  cycle in (b),(c),(d), and (f) are eye guidings for the hexagonal skyrmion lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The black arrow indicates the Y-axis scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  local minimum by genetic tunneling operators and find  an advisable global minimum for a given system without  any initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' However, either fed artificial spin con-  figuration or involved simulated annealing to the initial  generation, it will, as demonstrated here, speed up the  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The efficiency of genetic tunneling is investigated on  both a simple artificial system with magnetic frustration  and a Pd/Fe/Ir(111) monolayer that includes complex  Heisenberg and Dzyaloshinskii–Moriya interactions, as  calculated from DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The result indicates that the ge-  netic tunneling algorithm has better performance than  conventional simulated annealing at extremely low tem-  peratures as well as for finite temperatures, when it  comes to finding stable spin configurations as a function  of external parameters like temperature and applied mag-  netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Most noteworthy, we here considered the spi-  ral structure, the skyrmion lattice, and the ferromagnetic  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' It can also be concluded that the performance algo-  rithm is not limited by the system size, geometry, nature  of the magnetic interactions, temperature or the applied  field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In practice, he here proposed method needs a fine-  tuning depending on the system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  This includes the current version of ’cut-off’, where the  variance threshold may increase the risk of premature  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' For each particular system, a fine-tuning of  the hyperparameter process is also needed, which should  include a design of genetic operator and threshold value  testing in order to get better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Therefore,  further research should focus on those aspects for im-  proving this prototype protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In conclusion, we have explored a genetic tunnel-  ing protocol, which is designed to predict the magnetic  ground state of a magnetic system at finite tempera-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  We demonstrate that our method is robust for  two-dimensional systems, both for a simpler model sys-  tems and for a more complex Pd/Fe/Ir(111) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' We  envision that our findings will pave the way for evolution-  ary computing in finding the ground state of magnetic  systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', for magnets with non-trivial topology and  spin glass systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' It is possible that the here suggested  protocols will find applications in other areas of solid-  state science, or even in fields outside natural science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  (a)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4  50  GTO  GTO  SA  45  SA  Q  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='5  40  charge  35  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='6  30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  25  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='8  15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='9-  10  0  10  20  30  40  50  09  Temperature (K)  (b)  (c)  (d)  (e)  (f)  (g  (h)  (i)9  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  METHOD  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Variance-controlled local energy optimizer at  finite temperature  In a local optimization of spin configurations, which  involves finding local minima obtained from given initial  guesses, several approaches can be used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g, the Markov  chain Monte Carlo (MCMC) optimizer or a spin dynam-  ics simulation method with Gilbert damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' These two  methods have been proven to be robust and efficient in  describing complex spin systems[6, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Generally, the  Metropolis MCMC optimizer performs energy minimiza-  tion under finite temperature by using the transition  probability Pt between two spin configurations in the  Markov chain:  Pt =  �  exp  �  − ∆E  KBT  �  ,  if ∆E > 0  1,  otherwise  (2)  where ∆E, KB, and T are the energy difference between  spin configurations, the Boltzmann constant, and the  temperature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  With a given initial spin  configuration the method will iteratively minimize the  energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The other approach of optimization of spin configu-  rations is the spin dynamic optimizer, which uses the  stochastic Landau–Lifshitz–Gilbert (SLLG) equation to  simulate the time evolution of atomic magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  This method can reach a spin configuration near a local  energy minimum from a given initial state when Gilbert  damping[23] is included in the simulations, since energy  is the allowed to dissipate from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The atom-  istic SLLG equation is:  dmi  dt  = − γLmi ×  �  Bi + Bf  i  �  − γL  α  mi  mi ×  �  mi ×  �  Bi + Bf  i  ��  (3)  where Bi and Bf  i are the effective magnetic field and a  stochastic magnetic field that is related to thermal fluc-  tuation of a heat bath with temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In this equa-  tion, the first term represents the precessional motion  of atomic magnetic moments while the second term de-  scribes the damping motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In the expression above,  γL is the renormalized gyromagnetic ratio which is cal-  culated from:  γL =  γ  (1 + α2)  (4)  where γ and α are the gyromagnetic ratio respectively  the isotropic Gilbert damping constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In the present work, we use both the Metropolis  MCMC and SLLG local optimizations, with a given ini-  tial spin configuration, using the Uppsala Atomistic Spin  Dynamics (UppASD) package[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Both optimizers can  successfully minimize the energy under finite tempera-  ture from a given initial guess of a spin configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Additionally, for automatically stopping the minimiza-  tion process at the desirable convergence level, as shown  in Figure 1 (a), we introduce a variance threshold that  for the spin Hamiltonian used here is defined as:  Var(H ) = 1  n  n  �  i=1  (Hi − ⟨Hi⟩)2  (5)  where ⟨Hi⟩ means the expectation value of spin Hamilto-  nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' With the predefined variance threshold, a control-  lable local optimization can be performed for a desirable  optimum near the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Metaheuristic genetic-tunneling methods  The metaheuristic algorithm can interactively guide  and modify the operations of subordinate heuristics, to  efficiently produce preferable solutions within a high-  dimensional search space[24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Representative meta-  heuristic algorithms include the Simulated Annealing  (SA), the Particle Swarm Optimization (PSO)[26] and  Genetic Algorithms (GA)[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' It has been shown that a  hybrid algorithm, which combines the algorithms men-  tioned above with a local optimizer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', the gradient  descent, can be an efficient way to solve the global opti-  mization problem within a complex configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In this study, as shown schematically in Figure 1 (c),  we introduced a genetic tunneling strategy that mini-  mizes the energy of the spin system globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The whole  procedure is demonstrated as a flow chart in Figure 1  (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Basically, the whole algorithm starts with a ran-  dom initial generation C0 that contains relevant spin  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Typically, a limited number of spin con-  figurations, NP , is considered, each labeled C0  i where  i = 1, 2, · · · , Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Hence the connection between C0 and  C0  i is  C0 =  �  C0  i | i = 1, 2, · · · , Np  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  (6)  In practice, the work here has considered one configura-  tion, C0  i , as an object represented by 100 × 100 atomic  spins, but this number can naturally change, depend-  ing on the material in question that is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In this  work, NP has been chosen to be 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The strategy is  then to view these 64 configurations as a parent gener-  ation, and by inspiration from evolutionary science, one  wishes to find methods to find adapted better generations  Ck, where k labels the generation in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Provided  enough generations have been generated and scrutinized  for performance, the hope is to have an optimal repre-  sentative of the magnetic configuration, where the total  energy represents the main search criteria of interest in  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  10  Note that an optional simulated annealing process can  be invoked to seed the initial generation with optimal  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' With the initial generation, C0, the main  metaheuristic searching loop begins with the selection  of good candidates for the initial generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Currently,  three classical genetic selection operators are available in  this work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=', roulette wheel selection, tournament se-  lection, and ranking selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' As an example, in roulette  wheel selection, the probability of choosing one candidate  spin configuration C0  i is equal to:  P(C0  i ) =  Hi  �Np  i=1 Hi  (7)  where Hi is the Hamiltonian of a given spin configu-  ration C0  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The selection progress will stop before the  number of chosen spin configurations exceeds the preset  parent group limitation, Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In this work, we select from  these Np configurations a smaller group (typically 4) that  defines the parent group used to generate one single off-  spring of the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The probability of finding  a single parent configuration is calculated from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='7,  and we choose in this way the four configurations that  define a parent group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  From these four configurations  one offspring is generated by considering spin segments  of each parent, and then to apply genetic tunneling oper-  ators, which include crossover and mutation that enable  to dig of a tunnel on the potential energy surface to an-  other minimum with a deeper energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In this work, we use linear and squared crossover  and mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' As shown in Figure 1(b), the  squared operators will split each given spin configuration  in the group of (four) parents into four parts and put  them into a spin segmentation group, that will contain  a total of sixteen segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' If the operator represents  crossover, four spin segments will be randomly selected  from the spin segment group, to generate an offspring  spin configuration by combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' If one considers dis-  turbances to a given configuration, in the spirit of muta-  tions of biological science, one can apply a perturbation  before the combination takes place, as is shown in Fig-  ure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Here, a perturbation operation is applied with  some random shuffling or adding Gaussian noise to the  spin moments within each spin segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' After such a  possible mutation, the algorithm used here continues to  combine as an offspring configuration of 64 members, and  the whole procedure can be repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The linear opera-  tors follow the same workflow as squared operators but  have more freedom to choose the number of linear seg-  ments to split from one spin configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In the sup-  plementary section, we provide more details on the other  two selection methods used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Details of the procedure  As shown in Figure 1(d), the flowchart of the proce-  dure starts from the input of crystal information, ran-  dom seed, and magnetic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Those inputs can  build up the simulation system, in which we use only  nearest-neighbor interactions for the model system with  selected (not calculated) exchange parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' For this  system, we choose parameters that are designed to gen-  erate Bloch-type skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In addition, we used more  than thirty neighbor interactions in the Pd/Fe/Ir(111)  system, where all parameters were calculated from first-  principles theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' We used a large truncation in number  of neighbors, in order to avoid truncation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  After preparing all input data, we adopted a random  spin configuration generator, to produce random spin  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Then we optimize this random config-  uration with the Metropolis or SLLG optimizer, in order  to obtain locally optimized configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' A larger num-  ber of such configurations were realized and we selected  64 by use of a condition that two configurations should  not be too close in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' To be precise we used the  criterion that  H  �  C0  r  �  − H  �  C0  i  �  ) > ∆E,  (8)  where ∆E is a threshold energy difference, that guaran-  tees that configurations C0  r and C0  i are not too similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In this work, we use values ranging from 10−4 to 10−6  mRy/atom for ∆E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Once the quantity of randomly gen-  erated spin configurations C0 up the preset value, which  is set to 64 in this work, as well as the offspring candi-  date quantity, the procedure will end up random search-  ing process and adapt genetic tunneling methods and try  to search global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The metaheuristic searching  process will stop when the iterations are up to the pre-  defined threshold, which is 50 in this work, or when the  energy of each local optimized spin configuration in the  generation CLOPT get converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  After the searching  process stops, the procedure will extract final spin con-  figurations COPT that represents the best solution (see  flowchart in Figure 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Simulated annealing  In this work, we introduced a quickly simulated an-  nealing initial spin configuration generator as an optional  module in the procedure workflow and a simulated an-  nealing ground state searcher in the benchmark that is  shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The quickly simulated annealing generator has sparse  temperature mesh with a size of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In detail, the tem-  perature mesh starts from the simulated temperature and  adds 20K, 50K, and 200K to get the temperature mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  In each temperature, it will execute 2000 times Metropo-  lis progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Suggest value for the application of a quick  simulated annealing generator is not discussed in this pa-  per but may be interesting for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  For the simulated annealing that was used as the base-  line, we compared two temperature strategies, both of  which start from 900K, which is close to the curie tem-  perature of the Fe monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The first one is a common  strategy typically used in UppASD simulations where we  11  set temperature mesh with fixed gaps from 200K to 1K,  and the other one uses a temperature decrease ratio from  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='95 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' In this work, we find the difference between  those two strategies with the same simulated annealing  steps is minimal, but the second one will consume more  than ten times more than the first strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' We adopt  the first strategy to set the temperature stages for better  inheritance from previous work and sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  DATA AVAILABILITY  All data needed for reproducing the results can be  found in the GitHub repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='com/  MXJK851/GTO-2D  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  CODE AVAILABILITY  All code of GTO-2D is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  com/MXJK851/GTO-2D under the GPL-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The  authors  acknowledge  financial  support  from  the Knut and Alice Wallenberg Foundation through  Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='0060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' acknowledges the China  Scholarship Council (201906920083), O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' acknowledges  eSSENCE, the Swedish Research Council (VR) and the  ERC (project FASTCORR-Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 854843).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' ac-  knowledges financial support from the Swedish Research  Council (VR) through Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' 2019-05304 and 2016-  05980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  The computations/data handling were enabled  by resources provided by the Swedish National Infras-  tructure for Computing (SNIC), partially funded by the  Swedish Research Council through grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  2016-07213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The authors would also like to extend our  gratitude to Fredrik Dehlin, Liang Chen, Ivan Miranda,  Filipp N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Rybakov, and Liuzhen Yang for their valuable  comments and discussions during the writing of our pa-  per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
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+page_content='  SUPPLEMENTARY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Implementation details of selection operators  The pseudocode corresponding to the tournament se-  lection and rank selection algorithms used in this work  are provided in algorithm 1 and algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  Relative energies of different phases in the  Pd/Fe/Ir (111) system at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='1 mK in the transition  zone from the skyrmion lattice phase to the  ferromagnetic phase  In Table I, we list the calculated average relative en-  ergies per atom for four different phases in the Pd/Fe/Ir  (111) system: the ferromagnetic phase (FM), the isolated  skyrmion phase (Sk) and the 6 × 5 and 6 × 6 skyrmion  lattice phase (SkL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' The applied magnetic field is varied  from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Zero energy denotes the found ground  state in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' This table proves that the algorithm  we propose in this paper indeed correctly discovered the  transition state in Figure 3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Algorithm1 Tournament selection for local optimized  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='spin configuration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Choose the tournament size K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Choose the parent group set size O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Set counter i = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='for i = 1 to O do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Choose K spin configurations from the parent genera-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='tion at random  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Calculate the energies of these K spin configurations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Choose the best spin configuration with the lowest en-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='ergy from the tournament  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Output selected spin configuration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Algorithm2 Rank selection for local optimized spin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='configuration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Choose the parent group set size O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Set counter i = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='for i = 1 to O do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Calculate the energies of all spin configurations from the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='parent generation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Choose and remove the best spin configuration with the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='lowest energy from the parent generation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='Output selected spin configuration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content=' Relative energies of different phases in the Pd/Fe/Ir (111) system (mRy/atom) as a function of magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='  B (T)  FM  Isolated Sk  6x5 SkL  6x6 SkL  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='08450000E-04  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='92660000E-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
+page_content='00000000E+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf'}
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+Patient-specific Finite Element Modeling of 
+Aneurysmal dilatation after chronic type B 
+aortic dissection 
+Shaojie Zhanga, Joan D. Laubriea, S. Jamaleddin Mousavia, Stéphane Avrila 
+and Sabrina Ben Ahmedb 
+a Mines Saint-Etienne, Univ Lyon, Univ Jean Monnet, INSERM, U1059 Sainbiose, Centre 
+CIS, F-42023 Saint-Etienne, France.  
+b Univ Jean Monnet, INSERM, U1059 Sainbiose and University Hospital of Saint-Etienne, F-
+42000 Saint-Etienne,  
+ 
+ 
+Abstract 
+Progressive aneurysmal dilatation is a well-recognized com-
+plication in patients with chronic type B aortic dissection (cTBAD), which may 
+lead to a delayed rupture and create a life-threatening condition. However, our 
+understanding of such aortic expansion in cTBAD remains weak. In the pre-
+sent paper, we propose to use numerical simulations to study the role of growth 
+and remodeling (G&R) in aneurysmal dilatation after cTBAD. We set up a 3D 
+finite-element model of G&R for aortic dissection within an open-source code. 
+Constitutive equations, momentum balance equations, and equations related 
+to the mechanobiology of the artery were formulated based on the homoge-
+nized constrained mixture theory. The model was first applied to idealized aor-
+tic geometries with cylindrical and toric shapes to demonstrate its feasibility 
+and efficiency. The model was then applied to a patient-specific aortic segment 
+to show its potential in more relevant and complex patient-specific clinical ap-
+plications. It was found that the G&R tends to naturally trigger the aneurys-
+mal dilatation after dissection, in order to restore its tensional equilibrium. 
+Our results indicated that the value of the gain parameter, related to collagen 
+G&R, plays an important role in the stability of aortic expansion after cTBAD. 
+A small gain parameter will induce an excessive aneurysmal degeneration 
+whilst a large gain parameter helps to recover a stabilized state of the artery 
+after dissection. Finally, it was found that other mechanobiology-related pa-
+rameters, such as the circumferential length of the dissection, as well as the 
+pressure in the false lumen, may also be determinant for the stability of aneu-
+rysmal dilatation after cTBAD. Both a wide tear and an elevated false lumen 
+pressure favor an unstable development of aortic expansion after cTBAD. As 
+future work, the present model will be validated through predictions of aneu-
+rysmal dilatation in patient-specific clinical cases, in comparison with datasets 
+followed over a significant period of time. 
+ 
+ 
+ 
+
+2  
+I. 
+INTRODUCTION 
+ 
+Chronic type B aortic dissection (cTBAD) is defined when a tear originates in 
+the descending aorta and remains 3 months after its onset [1]. Patients with uncom-
+plicated cTBAD are preferentially treated medically with periodic clinical and im-
+aging surveillance, regarding the acceptable survival rate generally observed in a 
+short-term follow-up [2], [3]. However, the long-term outcome of such conservative 
+management remains questionable mainly due to the progressive aneurysmal dila-
+tation [4]. Invasive surgical interventions, such as endovascular repair or open sur-
+gery are then needed [5]. Up to now, little is known about the aneurysmal dilatation 
+after cTBAD, either it is stable with a moderated progression rate or there is an 
+excessive aneurysmal degeneration. It is yet of crucial importance for surgeons to 
+be able to assess the risk of aortic expansions in patients with early-stage cTBAD 
+to choose the optimal treatment approach. Patients identified at high risk for aortic 
+enlargement may therefore benefit from early surgical interventions and reduce 
+mortality from delayed aneurysm ruptures. 
+Published studies on this topic remain scarce. It has been widely accepted that 
+the presence of an excessive aortic diameter, typically greater than 40 mm, and a 
+patent false lumen are two high-risk factors for late aneurysm development after 
+cTBAD [6]-[8]. Besides, older age and elevated mean blood pressure were also 
+found to promote aneurysmal degeneration in cTBAD [6]. Tsai et al. reported that 
+the size, the number, as well as location of tears have significant impacts on the 
+pressure in the false lumen, and therefore influencing the false lumen expansion [9]. 
+Recently, Trimarchi et al. revealed that there are many other factors that may affect 
+aneurysmal dilatation after cTBAD, including demographic, clinical, pharmaco-
+logic, and radiologic variables, such as connective tissues disorders, gender, the 
+presence of thrombus in the false lumen, etc [10]. However, all the above researches 
+were based on observational studies or clinical trials with data collected over a long 
+follow-up period.  
+Considering the recent advances in computational mechanics of arteries [11], 
+[12] and more specifically the growth and remodeling (G&R) models [13]-[17], 
+numerical models can be an interesting alternative option for studying these influ-
+encing factors. However, to the author’s best knowledge, G&R after cTBAD has 
+never been modeled so far. There is still an important potential for G&R models to 
+understand vascular adaptation in chronic type B aortic dissection, where the patient 
+can undergo a long-term process of G&R after breaking the initial mechanical equi-
+librium due to tear opening. 
+In this specific context, we developed a 3D finite-element model of vascular ad-
+aptation to study the aneurysmal dilatation after cTBAD, within an open-source 
+code written in python and C++ [18], [19]. The G&R model of the arterial wall is 
+based on the homogenized constrained mixture theory (CMT) and the aortic dissec-
+tion is modeled through an original two-continuum arterial wall concept. We also 
+performed a sensitivity analysis to evaluate the influence of several selected mech-
+anobiological parameters on the aneurysmal dilatation after cTBAD. 
+
+3 
+Details of the model are given in this book chapter, by first introducing the math-
+ematical framework of the CMT method for G&R with respect to cTBAD, then 
+describing the two-continuum aortic dissection model, and finally showing poten-
+tials of the model, from a simple validation test case to academic applications with 
+idealized geometries, until a more relevant patient-specific application. 
+II. 
+Material and Methods 
+A. Constitutive and balance equations 
+The CMT was first proposed by Humphrey and Rajagopal as a hybrid method to 
+describe mechano-regulated G&R of arteries [20]. It was then largely used for mod-
+eling aneurysm formation [13]-[17]. In this work, we employ the homogenized 
+CMT [17] to model arterial G&R after cTBAD. Basic equations formulated under 
+the homogenized CMT framework are briefly introduced in this section. Readers 
+can refer to reference publications for more detailed mathematical formulations and 
+their interpretations [15], [17], [20]. 
+First, we assume that the arterial wall can be modeled as an homogenized mixture 
+made up by a matrix containing a network of elastic fibers, passive reinforcements 
+represented by 4 collagen fiber families (respectively oriented in circumferential, 
+axial and diagonal (+/- 45°) direction) and active reinforcements accounting for the 
+contractility of smooth muscle cells (SMCs) in the circumferential direction. Let 
+Ω𝑅 ⊂ ℝ3 and Ω𝑡 ⊂ ℝ3 denote, respectively, the initial traction-free reference con-
+figuration at time 𝑡 = 0 and current deformed configuration at time 𝑡 > 0 of the ar-
+terial wall. According to homogenized CMT, we assume that all constituents in the 
+arterial wall deform together with a same deformation gradient 𝐅: 
+𝐅 = 𝜕𝒙
+𝜕𝑿 
+(1) 
+where 𝑿 represents a material point in Ω𝑅 and 𝒙 represents the associated spatial 
+point in Ω𝑡. Moreover, based on the theory of Rodriguez and Hoger [21], this de-
+formation gradient tensor 𝐅 can be split into an elastic part and an inelastic part for 
+each constituent 𝑖 ∈ [𝑒, 𝑐𝑗, 𝑚], such as 
+𝐅 = 𝐅𝑒𝑙
+𝑖 𝐅𝑔𝑟
+𝑖  
+(2) 
+where 𝑒, 𝑐𝑗, 𝑚 represents respectively the elastic matrix, the 𝑗𝑡ℎ collagen fiber fam-
+ily and smooth muscle cells. More precisely, 𝐅𝑒𝑙
+𝑖  represents the elastic deformation 
+tensor related to stresses that balance external mechanical loads over the arterial 
+wall, while 𝐅𝑔𝑟
+𝑖  represents the inelastic deformation tensor related to G&R, i.e. re-
+lated to the continuous mass turnover of each constituent. Besides, we assume that 
+G&R is a fully stress-mediated process. Other non-mechanical effects related to the 
+mass turnover, such as immune-mediated chemical remodeling, damage, or me-
+chanical fatigue, are neglected in this work. Therefore, the temporal homogenized 
+mass deposition or degradation rate of each constituent can be expressed as 
+
+4  
+𝜌̇𝑅
+𝑖 = 𝜌𝑅
+𝑖 𝑘𝜎
+𝑖 𝜎𝑖 − 𝜎ℎ
+𝑖
+𝜎ℎ
+𝑖
+ 
+(3) 
+where 𝜌𝑅
+𝑖  is the reference mass density of constituent 𝑖, related to the reference con-
+figuration of the arterial wall. The right-hand side term of Equation 3 describes the 
+mass turnover due to the stress difference between the current stress 𝜎𝑖 and the ho-
+meostatic stress 𝜎ℎ
+𝑖, where 𝑘𝜎
+𝑖  is a regularization parameter (named gain parameter) 
+with respect to each constituent.  
+The homogenized CMT consists in the decomposition of the inelastic defor-
+mation gradient 𝐅𝑔𝑟
+𝑖  through two sub-gradient tensors 
+𝐅𝑔𝑟
+𝑖 = 𝐅𝑔
+𝑖𝐅𝑟
+𝑖 
+(4) 
+where 𝐅𝑔
+𝑖 is the growth-related tensor describing volume changes to due mass turn-
+over, and 𝐅𝑟
+𝑖 is the remodeling-related tensor describing how the prestretch of each 
+constituent is updated through continuous extant mass degradation and new mass 
+production. As suggested by Braeu et al. [15], we assume that the growth defor-
+mation is the same for all constituents in the arterial wall, such as 
+𝐅𝑔
+𝑖 = 𝐅𝑔 = 𝐈 + 𝜌𝑅
+𝜌𝑅0
+𝒂0
+⊥ ⊗ 𝒂0
+⊥ − 𝒂0
+⊥ ⊗ 𝒂0
+⊥ 
+(5) 
+where 𝜌𝑅 is the current reference mass density, 𝜌𝑅0 is the initial reference mass den-
+sity (at time 𝑡 = 0), 𝐈 is the second order identity and 𝒂0
+⊥ the growth direction [17]. 
+The remodeling process of elastin can be generally neglected (𝐅𝑟
+𝑒 = 𝐈) considering 
+its slow mass degradation rate (typically several decades for elastin half-life time). 
+We assume that the remodeling process of collagen fibers and SMCs occurs at a 
+constant volume and along the fiber direction, which can be expressed as [22] 
+𝐅𝑟
+𝑖 = 𝜆𝑟
+𝑖 𝒂0
+𝑖 ⊗ 𝒂0
+𝑖 + 1
+√𝜆𝑟𝑖 (𝐈 − 𝒂0
+𝑖 ⊗ 𝒂0
+𝑖 ) 
+(6) 
+where 𝜆𝑟
+𝑖  is the respective remodeling stretch of fiber 𝑖 ∈ [𝑐𝑗, 𝑚] along its fiber di-
+rection 𝒂0
+𝑖  with its time evolution 𝜆̇𝑟
+𝑖  given by [13] 
+𝜆̇𝑟
+𝑖 = (𝜌̇𝑅
+𝑖
+𝜌𝑅
+𝑖 + 1
+𝑇𝑖)
+𝜆𝑖
+(𝜆𝑒𝑙
+𝑖 )
+2 (𝜕𝜎𝑖
+𝜕𝜆𝑒𝑙
+𝑖 )
+−1
+× (𝜎𝑖 − 𝜎ℎ
+𝑖) 
+(7) 
+where 𝑇𝑖 is the average mass turnover time during which old mass increment is 
+degraded and replaced by a new mass increment. 𝜆𝑒𝑙
+𝑖  is the elastic stretch of fiber 𝑖 
+defined as 𝜆𝑒𝑙
+𝑖 = √(𝐅𝑒𝑙
+𝑖 )
+𝑡𝐅𝑒𝑙
+𝑖 ∶ 𝒂0
+𝑖 ⊗ 𝒂0
+𝑖  and 𝜆𝑖 is the total stretch of fiber 𝑖 defined 
+as 𝜆𝑖 = √(𝐅)𝑡𝐅 ∶ 𝒂0
+𝑖 ⊗ 𝒂0
+𝑖  
+Finally, considering that the homeostatic configuration of the arterial wall is 
+achieved at time 𝑡 = 𝑡0 and defining the initial traction-free geometry of the arterial 
+wall at time 𝑡 = 0 as the same geometry as its homeostatic configuration, the initial 
+
+5 
+elastic prestretch of each constituent 𝐆ℎ
+𝑖  corresponding to the homeostatic configu-
+ration at time 𝑡0 can simply satisfy 
+𝐅𝑟
+𝑖(𝑡0) = (𝐆ℎ
+𝑖 )
+−1 
+(8) 
+due to the fact that 𝐈 = 𝐅(𝑡0) = 𝐅𝑒
+𝑖(𝑡0)𝐅𝑔
+𝑖(𝑡0)𝐅𝑟
+𝑖(𝑡0) = 𝐆ℎ
+𝑖 𝐅𝑔
+𝑖(𝑡0)𝐅𝑟
+𝑖(𝑡0) =
+𝐆ℎ
+𝑖 𝐅𝑟
+𝑖(𝑡0). The detailed expressions of 𝐆ℎ
+𝑖  are hereby given for each constituent 𝑖 ∈
+[𝑒, 𝑐𝑗, 𝑚] , with respect to a cylindrical coordinate system 
+𝐆ℎ
+𝑒 = 𝐝𝐢𝐚𝐠 ( 1
+𝜆𝜃
+𝑒 𝜆𝑧𝑒 ,   𝜆𝜃
+𝑒 ,   𝜆𝑧
+𝑒) 
+(9) 
+𝑮ℎ
+𝑐𝑖 = 𝜆ℎ
+𝑐𝑖𝒂0
+𝑐𝑖 ⊗ 𝒂0
+𝑐𝑖 +
+1
+√𝜆ℎ
+𝑐𝑖
+(𝑰 − 𝒂0
+𝑐𝑖 ⊗ 𝒂0
+𝑐𝑖) 
+(10) 
+𝑮ℎ
+𝑚 = 𝜆ℎ
+𝑚𝒂0
+𝑚 ⊗ 𝒂0
+𝑚 +
+1
+√𝜆ℎ
+𝑚 (𝑰 − 𝒂0
+𝑚 ⊗ 𝒂0
+𝑚) 
+(11) 
+where 𝜆𝜃
+𝑒  and 𝜆𝑧
+𝑒 are the initial deposition stretches of elastin, respectively in the 
+circumferential and axial direction, uniform over the whole arterial wall. 𝜆ℎ
+𝑐𝑖 and 𝜆ℎ
+𝑚 
+are respectively the initial deposition stretches of collagen fibers (same deposition 
+stretch for all four directions) and SMCs. 
+Based on CMT, the strain energy density function of the arterial wall can be 
+given by 
+𝛹 = 𝜌𝑅
+𝑒𝑊𝑒 + ∑ 𝜌𝑅
+𝑐𝑗𝑊𝑐𝑗
+4
+𝑗=1
++ 𝜌𝑅
+𝑚𝑊𝑚 
+(12) 
+where 𝜌𝑅
+𝑒, 𝜌𝑅
+𝑐𝑗and 𝜌𝑅
+𝑚 are respectively the reference mass densities of the elastic ma-
+trix, of the 𝑗𝑡ℎ collagen fiber family and of SMCs, and 𝑊𝑒, 𝑊𝑐𝑗 and 𝑊𝑚 are the 
+specific strain energy density functions with respect to each constituent. Moreover, 
+the strain energy density function 𝑊𝑖 of each constituent 𝑖 ∈ [𝑒, 𝑐𝑗, 𝑚], can be ex-
+pressed as a function of its elastic deformation tensor 𝐅𝑒𝑙
+𝑖 , or equivalently, its elastic 
+right Cauchy-Green tensor 𝐂𝑒𝑙
+𝑖 , defined as 𝐂𝑒𝑙
+𝑖 = (𝐅𝑒𝑙
+𝑖 )
+𝑡𝐅𝑒𝑙
+𝑖 . In the present work, the 
+elastic matrix is considered as a quasi-incompressible Neo-Hookean hyperelastic 
+material with its specific strain energy density function 𝑊𝑒 given by 
+𝑊𝑒 = 𝜇𝑒
+2 (tr(𝐂̅𝑒𝑙
+𝑒 ) − 3) + 𝜅(|𝐅𝑒𝑙
+𝑒 | − 1)2 
+(13) 
+where 𝜇𝑒 is a material parameter characterizing the shear stiffness of elastin and 𝜅 
+is an arbitrary but sufficiently high penalty parameter ensuring quasi incompressi-
+bility. 𝐂̅𝑒𝑙
+𝑒  is the isochoric elastic right Cauchy-Green tensor of elastin, defined as 
+𝐂̅𝑒𝑙
+𝑒 = (𝐅̅𝑒𝑙
+𝑒 )𝑡𝐅̅𝑒𝑙
+𝑒  and 𝐅̅𝑒𝑙
+𝑒 = 𝐅𝑒𝑙
+𝑒 |𝐅𝑒𝑙
+𝑒 |1/3
+⁄
+. The specific strain energy density function 
+
+6  
+of collagen fiber families is modeled by an anisotropic Fung-type exponential func-
+tion,  
+𝑊𝑐𝑗 = 𝑘1
+𝑐𝑗
+2𝑘2
+𝑐𝑗 (𝑒𝑘2
+𝑐𝑗(𝐼4𝑒𝑙
+𝑐𝑗−1)
+2
+− 1) 
+(14) 
+We also use the same anisotropic Fung-type exponential function to model the 
+passive behavior of SMCs [15], while an additional term is added for the active tone 
+contribution such as, 
+𝑊𝑚 = 𝑘1
+𝑚
+2𝑘2
+𝑚 (𝒆𝑘2𝑚(𝐼4𝑒𝑙
+𝑚−1)2
+− 1) + 𝜎𝑚𝑎𝑥
+𝜌𝑅0
+(𝜆𝑎𝑐𝑡 + 1
+3
+(𝜆𝑚𝑎𝑥
+𝑚
+− 𝜆𝑎𝑐𝑡)3
+(𝜆𝑚𝑎𝑥
+𝑚
+− 𝜆0
+𝑚)2 ) 
+(15) 
+where 𝑘1
+𝑐𝑗and 𝑘1
+𝑚 are stress-like material parameters, and 𝑘2
+𝑐𝑗 and 𝑘2
+𝑚are dimen-
+sionless material parameters. 𝐼4𝑒𝑙
+𝑐𝑗 and 𝐼4𝑒𝑙
+𝑚 are pseudo-invariants, which are addi-
+tional invariants defined in case of anisotropic materials such as 𝜆𝑒𝑙
+𝑖 = 𝐂𝑒𝑙
+𝑖 ∶ 𝒂0
+𝑖 ⊗
+𝒂0
+𝑖  with 𝑖 ∈ [𝑐𝑗, 𝑚] [23]. 𝜆𝑎𝑐𝑡 is the active stretch in the circumferential direction, 
+𝜎𝑚𝑎𝑥 is the maximum active Cauchy stress, 𝜌𝑅0 is the reference total mass density 
+of the arterial wall at time 𝑡 = 0, and 𝜆𝑚𝑎𝑥
+𝑚
+ and 𝜆0
+𝑚 are the active stretches respec-
+tively at maximum and zero active stress for SMCs. 
+The second Piola-Kirchhoff stress tensor 𝐒 and the fourth order elasticity tensor 
+of the arterial wall ℂ are then deduced by performing the first and second deriva-
+tives of the strain energy function 𝛹 with respect to the total Green-Lagrange strain 
+𝐄 
+𝐒 = 𝜕𝛹
+𝜕𝐄 = 𝜑𝑒𝐒𝑒 + ∑ 𝜑𝑐𝑗𝐒𝑐𝑗
+𝑖
++ 𝜑𝑚𝐒𝑚 
+(16) 
+ℂ = 𝜕2𝐒
+𝜕𝐄𝜕𝐄 = 𝜑𝑒ℂ𝑒 + ∑ 𝜑𝑐𝑗ℂ𝑐𝑗
+𝑖
++ 𝜑𝑚ℂ𝑚 
+(17) 
+where 𝜑𝑖, 𝐒𝑖 and ℂ𝑖 are the mass fraction, second Piola-Kirchhoff stress and forth 
+order elasticity tensor with respect to each constituent 𝑖 ∈ [𝑒, 𝑐𝑗, 𝑚] in the arterial 
+wall, defined as 
+𝜑𝑖 = 𝜌𝑅
+𝑖
+𝜌𝑅
+ 
+(18) 
+𝐒𝑖 = 𝜌𝑅
+𝜕𝑊𝑖
+𝜕𝐄  
+(19) 
+ℂ𝑖 = 𝜌𝑅
+𝜕2𝑊𝑖
+𝜕𝐄𝜕𝐄 
+(20) 
+with 𝜌𝑅 = 𝜌𝑅
+𝑒 + 𝜌𝑅
+𝑐𝑗 + 𝜌𝑅
+𝑚 the reference total mass density of the arterial wall. Fi-
+nally, assuming that the G&R occurs at a slow time scale and can be considered as 
+a quasi-static process, the dynamics effects such as inertia or viscoelasticity of the 
+
+7 
+arterial wall can be neglected. Therefore, the momentum balance equations of the 
+arterial wall can be simply written as 
+∇ ∙ 𝝈 + 𝜌𝒃 = 𝟎 
+(21) 
+𝜌 is the spatial density of the arterial wall, related to its reference density 𝜌𝑅, as 𝜌 =
+𝜌𝑅 |𝐅|
+⁄
+, 𝒃 is the body force vector given in the spatial configuration, 𝝈 is the Cau-
+chy stress derived from the previous second Piola-Kirchoff stress as  
+𝝈 = 𝟏
+|𝐅| 𝐅𝐒𝐅𝑡 
+(22) 
+The boundary conditions applied on the arterial wall can be Dirichlet boundary 
+conditions, assigning the predefined displacement field over the mesh nodes or 
+Robin boundary conditions, which are applied over the mesh surface, such as 
+𝝈 ∙ 𝒏 = 𝑃𝑇𝐿𝒏 + 𝒑𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 + 𝑭𝑠𝑝𝑟𝑖𝑛𝑔 
+(23) 
+where 𝑃𝑇𝐿 denotes the true luminal pressure of the artery due to blood flow, applied 
+on the inner surface of the media layer. 𝒏 is the outward pointing unit vector normal 
+to the arterial inner media surface. 𝒑𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 is the pressure in the false lumen after 
+cTBAD. 𝑭𝑠𝑝𝑟𝑖𝑛𝑔 is an additional spring-based elastic force, related to the two-con-
+tinnumm arterial wall concept proposed in this work. Details of the two-continuum 
+arterial wall concept, as well as, 𝑭𝑠𝑝𝑟𝑖𝑛𝑔, 𝒑𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛, will be given in the next sec-
+tion. 
+ 
+B. Dissection model 
+In this section, we will firstly present the two-continuum arterial wall concept, 
+dedicated to the modeling of G&R in the case of cTBAD. It is worth noting that the 
+mechanism of the initial tear formation or the subsequent tear progression is cur-
+rently not of primary interest in this paper. Under this specific context, we propose 
+to model the initial healthy arterial wall without dissection with two continuum bod-
+ies. As shown in Figure 1a, the arterial wall is made up of two layers, respectively 
+the inner media layer and the outer adventitia layer. The two layers are perfectly 
+connected by high stiffness elastic springs. More precisely, each spring connects 
+two adjacent mesh surfaces 𝑆𝑎 and 𝑆𝑚, respectively at the inner surface of the ad-
+ventitia and the outer surface of media. The force applied on each surface is com-
+puted on the relative displacement of the two connecting surfaces, as given in Equa-
+tion (24) 
+𝑭𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 = {−𝑘𝑠𝑝𝑟𝑖𝑛𝑔(𝑑𝑆𝑎 − 𝑑𝑆𝑚)𝒏𝑆𝑎,
+𝑎𝑑𝑣𝑒𝑛𝑡𝑖𝑡𝑖𝑎
+𝑘𝑠𝑝𝑟𝑖𝑛𝑔(𝑑𝑆𝑎 − 𝑑𝑆𝑚)𝒏𝑆𝑚,
+𝑚𝑒𝑑𝑖𝑎
+ 
+(24) 
+where 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 is the stiffness of the interfacial spring. 𝑑𝑆𝑎 and 𝑑𝑆𝑚 the nodal-aver-
+aged displacement of the mesh surface located respectively in the inner adventitia 
+
+8  
+and outer media surfaces. 𝒏𝑆𝑎 and 𝒏𝑆𝑚 are the outward pointing unit vectors nor-
+mal to the respective mesh. It is worth noting that displacement of one layer is trans-
+mitted to the other layer with low distortions due to the high stiffness of the springs. 
+Finally, we assume that this initial configuration of the arterial wall is at its home-
+ostatic state with a reference luminal pressure 𝑃𝑇𝐿. To validate this new concept, a 
+simple validation test case has been proposed in this work, in comparison with the 
+conventional arterial wall model with a single continuum body [13], [15], [16], [17]. 
+ 
+       
+Figure 1 - (a) Illustration of the two-continuum arterial wall model, with the adventitia layer 
+and the media layer connected by high stiffness elastic springs, representing a healthy arterial 
+wall under its homeostatic state. (b) Initial configuration of the arterial wall after cTBAD. The 
+false lumen is created by breaking interfacial elastic springs in a selected region between the 
+media and adventitia layers.  
+ 
+Based on this new concept of a two-continuum arterial wall, the initial configu-
+ration of cTBAD with false lumen can be obtained by simply vanishing interfacial 
+springs. As shown in Figure 1b, the false lumen (FL), as well as the free surfaces 
+(i.e. inner side of adventitia layer and outer side of media layer), is created by break-
+ing springs in a selected region where tears are assumed to be present. The force 
+induced by the pressure in the false lumen is then applied on each mesh surface, 
+𝑆𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛, of the newly created free surfaces 
+𝒑𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 = 𝑃𝐹𝐿𝒏𝑆𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 
+(25) 
+where 𝑃𝐹𝐿 is the constant pressure in the false lumen and 𝒏𝑆𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 is the outward 
+pointing unit vector normal to the free mesh surfaces in the false lumen. Finally, it 
+is worth noting that the presence of such pressurized false lumen will break the 
+mechanical equilibrium of the arterial initial homeostatic state, and therefore trig-
+ging the G&R of the arterial wall over a long period of time in case of cTBAD, until 
+the achievement of a new preferred mechanical state or eventually an excessive an-
+eurysmal degeneration. 
+
+adventitia
+adventitia
+false lumen (FL)
+media
+media
+VM
+W
+WW
+W
+true lumen (TL
+WW
+W
+(a)
+(b)9 
+C. Finite element implementation 
+The proposed model was implemented within an open-source finite element 
+code, written in Python and C++ [18], [19]. Three different steps, are defined in the 
+model: 
+ 
+1st step: Computation of the healthy arterial wall at homeostasis. The ini-
+tial arterial wall is loaded with a constant luminal pressure, 𝑃𝑇𝐿, on its en-
+tire inner surface of media.  
+ 
+2nd step: Opening of the arterial wall. Interfacial springs are removed in 
+a selected region between the adventitia layer and media layer, creating the 
+initial dissection tear of cTBAD. The same luminal pressure in the true 
+lumen, 𝑃𝑇𝐿, is maintained as in the previous step, applied on the entire in-
+ner surface of the media. In the meantime, the false lumen will be loaded 
+with a constant pressure, 𝑃𝐹𝐿, applied on the newly created surfaces related 
+to the tear-open region.  
+ 
+3rd step: Adaptation of the arterial wall with G&R after cTBAD, i.e., after 
+the creation of a pressurized false lumen. 
+Note that there is only one time increment in the first two steps while the third 
+step is composed of several time increments to obtain relevant results of a long-term 
+arterial wall adaptation after cTBAD. For each time increment, the same set of mo-
+mentum balance and constitutive equations is solved with the Newton-Raphson 
+method. Finally, at the end of each time increment, we obtain the displacement field 
+and the associated stress-strain information on each mesh node of the arterial wall. 
+III. 
+Numerical applications 
+ 
+In order to show the potentials of the present dissection model, four different 
+simulations were performed in this paper:  
+ 
+Validation test case: It consists of a simple test case to validate the 
+spring-connected two-continuum arterial wall concept proposed in this 
+work. The validation was achieved through the comparison with refer-
+ence results in the literature [13], [15], [16], [17] where the conventional 
+single continuum arterial wall model was employed to simulate the an-
+eurysm formation in response to external stimuli. 
+ 
+Application to a cylindrical artery: The second application is to study 
+the G&R after cTBAD in the case of an idealized cylindrical artery.  
+ 
+Application to a toric artery: In this third application, the dissection 
+model is applied to an idealized toric artery. 
+ 
+Application to a patient-specific artery: In this last application, we 
+demonstrate the feasibility of our dissection model for further more 
+complex and relevant clinical patient-specific applications, by model-
+ing the G&R on a dissected patient-specific human descending aortic 
+segment. 
+
+10  
+A. Validation test case 
+An idealized two-layered cylindrical artery is considered. The geometry was the 
+same as that has been used in the work of Braeu et al. [15], with an inner arterial 
+radius of 10 mm and a constant arterial thickness of 1.41 mm. Besides, we assume 
+that each layer of the arterial wall, i.e. adventitia layer and media layer, has the same 
+thickness of 0.705 mm. Moreover, it should be mentioned that a constant gap of 
+0.01 mm is defined between the two layers, allowing the presence of interfacial 
+connecting springs with respect to the two-continuum arterial wall concept. The 
+mesh was hexahedral and composed of 6 × 40 × 25 elements (thickness × circum-
+ferential × length). Finally, the whole geometry is assumed to be at a homeostatic 
+state, related to a reference luminal pressure of 13.3 kPa.  
+Following Braeu et al. [15], we apply a sudden degradation of the elastic matrix 
+such as  
+𝐷̇ 𝑒 = − 𝜌𝑅
+𝑒
+𝑇𝑒 − 𝐷𝑑𝑎𝑚
+𝑡𝑑𝑎𝑚
+𝜌𝑅0
+𝑒 𝑒
+−0.5(
+𝑧
+𝐿𝑑𝑎𝑚)
+2
+−
+𝑡
+𝑡𝑑𝑎𝑚 
+(26) 
+where 𝐿𝑑𝑎𝑚 and 𝑡𝑑𝑎𝑚 are respectively the spatial and the temporal damage spread 
+parameter, 𝐷𝑑𝑎𝑚 is the maximum damage, 𝜌𝑅0
+𝑒  is the initial reference mass density 
+of elastin, 𝑧 is the axial position of the cylinder. Noting that due to the symmetry of 
+the problem, only half of the cylinder is modeled in this simulation with 𝑧 varying 
+from 0 mm to 90 mm. The first term at the right-hand side (RHS) of Equation 26 
+describes the natural elastin degradation by aging effect. The second term of the 
+RHS is related to sudden external stimuli, causing a maximum elastin degradation 
+at the center of the cylinder, i.e. 𝑧 = 0 mm. Material properties used in this valida-
+tion test case are taken from the 3D model of Braeu et al. [15] and are summarized 
+in Table 1 together with other simulation parameters. The simulation results ob-
+tained in this simulation are compared to the reference results in the literature [13], 
+by using three different values of collagen gain parameter 𝑘𝜎
+𝑐𝑖. 
+Table 1. Material properties used in the validation test case. 
+Symbol 
+Value 
+Unit 
+𝜇𝑒 
+72 
+J ∙ kg−1 
+𝜅 
+720 
+J ∙ kg−1 
+𝑘1
+𝑐𝑗 
+568 
+J ∙ kg−1 
+𝑘2
+𝑐𝑗 
+11.2 
+− 
+𝑘1
+𝑚 
+7.6 
+J ∙ kg−1 
+𝑘2
+𝑚 
+11.4 
+− 
+𝜌𝑅0
+𝑒  
+241.5 
+kg ∙ m−3 
+
+11 
+𝜌𝑅0
+𝑐1 = 𝜌𝑅0
+𝑐2  
+65.1 
+kg ∙ m−3 
+𝜌𝑅0
+𝑐3 = 𝜌𝑅0
+𝑐4  
+260.4 
+kg ∙ m−3 
+𝜌𝑅0
+𝑚  
+157.5 
+kg ∙ m−3 
+𝜆𝑧𝑒 
+1.25 
+− 
+𝜆𝜃
+𝑒 
+1.34 
+− 
+𝜆ℎ
+𝑐𝑗 
+1.062 
+− 
+𝜆ℎ
+𝑚 
+1.1 
+− 
+𝑇𝑒 
+101. 16 
+𝑦ears 
+𝑇𝑐𝑗 
+101 
+days 
+𝑇𝑚 
+101 
+days 
+𝐿𝑑𝑎𝑚 
+8 
+mm 
+𝑡𝑑𝑎𝑚 
+40 
+days 
+𝐷𝑚𝑎𝑥 
+0.5 
+− 
+𝜎𝑎𝑐𝑡𝑚𝑎𝑥 
+54 
+kPa 
+𝜆0 
+0.8 
+− 
+𝜆𝑚 
+1.4 
+− 
+𝑘𝑠𝑝𝑟𝑖𝑛𝑔 
+1000  
+kPa ∙ mm−1 
+ 
+B. Application to a cylindrical artery 
+After the validation of the two-continuum arterial wall model, we first apply the 
+G&R after cTBAD in the case of an idealized two-layered cylindrical artery as 
+shown in Figure 2. The same geometry as in the previous validation case was used, 
+except that the length of the artery is reduced from 90 mm to 50 mm. The mesh was 
+hexahedral and composed of 6 × 60 × 20 elements (thickness × circumferential × 
+length). Similarly, we assume that this geometry was related to a homeostatic state 
+under an inner true lumen pressure of 100 mmHg. The initial tear of the dissection 
+is created by breaking springs in regions where 𝑥 ≤ 10 and 𝑧 ≤ 50, to model a rep-
+resentative initial tear of cTBAD. A pure sliding boundary condition is assigned on 
+the cross-section at two extremities. The outer surface of the adventitia is free. The 
+reference pressure in the false lumen is assumed to be the same as in the true lumen. 
+The same material properties as reported in Table 1 has been used, considering ad-
+ditionally the layered distribution of different material constituents as suggested by 
+Mousavi et al. [16] for human ascending thoracic aorta, i.e., the media has 97% of 
+
+12  
+the total elastin, 15% of the total axial and diagonal collagen fibers, and 100% of 
+the total SMCs, while the adventitia has 3% of the total elastin, 85% of the total 
+axial and diagonal collagen and 100 % of the total circumferential collagen.  
+ 
+ 
+Figure 2 – Schematic representation of the idealized cylindrical artery on which the presented 
+dissection model is applied for modeling G&R after cTBAD. 
+ 
+C. Application to a toric artery 
+In order to further verify the applicability of the present dissection model, we 
+employed here an idealized toric geometry, as shown in Figure 3, to simulate the 
+arterial G&R after cTBAD.  
+ 
+ 
+Figure 3 - Schematic representation of the idealized toric artery on which the present dissection 
+model was applied for modeling G&R after cTBAD. 
+ 
+The geometry was a fourth of a torus with an arch radius of 65 mm, similar to 
+the ones already used in the literature [16], [24]. The arterial section is defined with 
+an inner radius of 18 mm and an outer radius of 20.38 mm. The thickness of the 
+artery is 2.38 mm, including a constant gap of 0.01 mm between two equal-thick-
+ness adventitia and media layers. The mesh was hexahedral and composed of 6 × 
+60 × 20 elements (thickness × circumferential × length). Once again, we assume 
+that this geometry was related to a homeostatic state, under a constant inner true 
+lumen pressure of 80 mmHg. Similarly, a pure sliding boundary condition is as-
+signed to cross-sections at the two extremities while the outer surface of the adven-
+titia is let free. Mechanical parameters, as well as simulation parameters, are re-
+ported in Table 2, based on Laubrie et al. [25]. The initial tear of the dissection is 
+
+(0,0,0)
+10mm
+11.42mm
+50mm65mm
+18mml
+20.38mm
+(0,0,0)
+Y13 
+created by breaking springs in the regions defined by √𝑥2 + 𝑦2 ≥ 70 and 
+𝑎𝑟𝑐𝑡𝑎𝑛(𝑥 𝑦
+⁄ ) ≤ 60°, to model a representative initial tear of cTBAD. 
+Table 2. Material properties and simulation parameters used for the toric artery simulation. 
+Symbol 
+Value 
+Unit 
+𝜇𝑒 
+80 
+J ∙ kg−1 
+𝜅 
+800 
+J ∙ kg−1 
+𝑘1
+𝑐𝑗 
+292.0 
+J ∙ kg−1 
+𝑘2
+𝑐𝑗 
+5.6 
+− 
+𝑘1
+𝑚 
+13.8 
+J ∙ kg−1 
+𝑘2
+𝑚 
+6.0 
+− 
+𝜌𝑅0
+𝑒  (media) 
+169.0 
+kg ∙ m−3 
+𝜌𝑅0
+𝑐1 = 𝜌𝑅0
+𝑐2  (media) 
+14.6 
+kg ∙ m−3 
+𝜌𝑅0
+𝑐3 = 𝜌𝑅0
+𝑐4  (media) 
+58.4 
+kg ∙ m−3 
+𝜌𝑅0
+𝑚  (media) 
+735.0 
+kg ∙ m−3 
+𝜌𝑅0
+𝑒  (adventitia) 
+565.0 
+kg ∙ m−3 
+𝜌𝑅0
+𝑐1 = 𝜌𝑅0
+𝑐2  (adventitia) 
+48.5 
+kg ∙ m−3 
+𝜌𝑅0
+𝑐3 = 𝜌𝑅0
+𝑐4  (adventitia) 
+194.0 
+kg ∙ m−3 
+𝜌𝑅0
+𝑚  (adventitia) 
+0.0 
+kg ∙ m−3 
+𝜆ℎ
+𝑐𝑗 
+11 
+− 
+𝜆ℎ
+𝑚 
+1.1 
+− 
+𝑇𝑒 
+101. 16 
+𝑦ears 
+𝑇𝑐𝑗 
+101 
+days 
+𝑇𝑚 
+101 
+days 
+𝜎𝑎𝑐𝑡𝑚𝑎𝑥 
+54 
+kPa 
+𝜆0 
+0.8 
+− 
+𝜆𝑚 
+1.4 
+− 
+𝑘𝑠𝑝𝑟𝑖𝑛𝑔 
+1000 
+kPa ∙ mm−1 
+
+14  
+D. Application to a patient-specific artery 
+In this last test case, our dissection model was applied to a patient-specific human 
+descending thoracic aortic segment, as shown in Figure 4a. It was taken from a pa-
+tient-specific aortic arch geometry, reconstructed from a patient's CT scan [16], as 
+shown in Figure 4b. The exact location of the modeled aortic segment is shown in 
+Figure 4b with the blue mesh. The thickness of the adventitia and media was as-
+sumed to be equal. Besides, the two layers were separated with a constant gap of 
+0.01 mm. The mesh was hexahedral and composed of 6 × 48 × 42 elements (thick-
+ness × circumferential × length).  Similar to previous test cases, we assumed that 
+this initial geometry was related to a homeostatic state, with an inner true lumen 
+pressure of 80 mmHg. The reference pressure in the false lumen was assumed to be 
+the same as in the true lumen. However, this value may change as a sensitivity study 
+was performed on the false lumen pressure in this patient-specific case test, which 
+is detailed in the results section. Moreover, it should be mentioned that the pure 
+sliding boundary condition was applied on the cross-section at the two extremities 
+and the outer surface of the adventitia was let free. The same material properties 
+and simulation parameters as summarized in Table 2. Finally, it is worth noting that 
+the non-uniform prestretches were used in this patient-specific artery, which was 
+computed based on an iterative method previously developed by Laubrie et al. [25].  
+ 
+ 
+Figure 4 - (a) Schematic representation of the patient-specific human descending thoracic aor-
+tic segment on which the presented dissection model was applied for modeling G&R after 
+cTBAD. (b) Illustration of the modeled aortic segment location, i.e., regions covered by the 
+blue mesh, with respect to the whole patient-specific aortic arch.  
+ 
+In order to better describe the initial dissection tear of cTBAD, we introduce a 
+numerical parameter, 𝛼, for each circumferential cross-section of the arterial seg-
+ment, as shown in Figure 5. Note that the unit vector normal to this cross-section is 
+computed from the arterial centerline points. The unit vertical direction of the cross-
+section is approximately defined as the averaged projection vector of this section on 
+the xy plane. After the definition of directions, we define 𝛼 for each interfacial con-
+necting spring  
+α = 𝑙
+𝐷 
+(27) 
+
+(0,0,0)
+(a)
+(b)15 
+where 𝑙 is the length of averaged spring positions, projected on the cross-section's 
+vertical direction. The tear opening criterion is thus defined as 𝛼 ≥ 𝛼𝑚𝑖𝑛. Therefore, 
+with 𝛼𝑚𝑖𝑛 = 0, all interfacial springs between the adventitia and media layers will 
+be broken, creating a full separation of the two layers. While if 𝛼𝑚𝑖𝑛 is equal to 1, 
+no interfacial springs will be broken and thus no presence of the tear. In this patient-
+specific simulation, the effect of the tear opening length to the G&R after cTBAD 
+was considered, by varying the values of 𝛼𝑚𝑖𝑛 from 0.8 (a narrow tear) to 0.5 (a 
+wide tear).  
+ 
+Figure 5 - Description of the initial tear opening criteria defined on each circumferential cross-
+section of the aortic segment. 
+ 
+E. Computational details 
+  
+All simulations were performed on a Macbook Pro with Intel Core i5 and 8 Go 
+of memory. The computation time for each simulation takes around 2 hours. The 
+low computation resources prove the computational efficiency of our dissection 
+model. 
+IV. 
+Results 
+A. Validation test case 
+Results of the validation test case are shown in Figure 6, illustrating the aneurys-
+mal expansion of the arterial wall due to elastin loss. The evolution of the maximum 
+inner radius of the aneurysm is shown, in comparison with the reference results [13], 
+over a period of 10 years. The results indicated that the current two-continuum ar-
+terial wall model is in good agreement with the conventional single continuum ar-
+terial wall model. The aneurysmal expansion tends to recover its stability with a 
+large gain parameter while a small gain parameter promotes an uncontrolled expan-
+sion of the aneurysm. With this validation test case, we justified the use of such a 
+two-continuum arterial wall concept for G&R problems.  
+
+α=1
+1
+D
+D
+0 =16  
+ 
+Figure 6 - Evolution of the maximum arterial inner radius over 10 years in response to an initial 
+sudden elastin loss for both the two-continuum arterial wall model (solid lines) and the refer-
+ence single-continuum arterial wall model (dash lines), with three different values of gain pa-
+rameter related to collagen G&R [13].  
+ 
+B. Application to a cylindrical artery 
+We first show the reference simulation results in the case of a cylindrical artery, 
+as illustrated in Figure 7, with respect to a reference value of gain parameter 𝑘𝜎
+𝑐𝑗 =
+𝑘𝜎
+𝑚 = 0.05. It can be seen that the dissected part of the artery, especially the outer 
+adventitia layer, continues to dilate due to the effect of G&R after the initial tear 
+opening. This aneurysmal dilatation tends to be unstable, with an increasing growth 
+rate over time. Besides, the maximum stress, which is located at the vicinity of the 
+tear edge, also increases rapidly over time.  
+Previous studies on G&R, which modeled aneurysm progression but disregarded 
+effects of the dissection, reported that gain parameters have a determinant effect on 
+the stability of aneurysmal dilatation[15], [16]. To investigate the effect of this gain 
+parameter in the specific context of cTBAD, we considered three different values 
+of gain parameters, ranging between 0.05 and 0.15, with results illustrated in Fig-
+ure 8, showing the temporal evolution of the maximum outer diameter of the dis-
+sected cylindrical artery. It can be seen that the same tendency as previously re-
+ported in the literature has been observed for G&R after cTBAD. A small gain 
+parameter tends to induce an unstable aneurysmal degeneration while a large gain 
+parameter tends to favor the stability of aneurysmal dilatation. 
+
+mm
+16.5
+kci=0.05
+15.2
+kci = 0.09
+kci = 0.13
+13.9
+12.6
+Maximum
+11.3
+10.0
+0
+2
+4
+6
+8
+10
+Time[years]17 
+ 
+Figure 7 - Reference results of the cylindrical artery with respect to a reference value of gain 
+parameter 𝑘𝜎
+𝑐𝑗 = 𝑘𝜎𝑚 = 0.05, showing geometrical and equivalent von Mises stress evolutions 
+after cTBAD over 9 years. 
+ 
+ 
+ 
+Figure 8 - Influence of the gain parameter to aneurysmal dilatation after cTBAD, showing the 
+temporal evolution of the maximum outer diameter of the dissected cylindrical artery. 
+ 
+C. Application to a toric artery 
+Reference simulation results in the case of a toric artery, with a reference value 
+of gain parameter 𝑘𝜎
+𝑐𝑗 = 𝑘𝜎
+𝑚 = 0.05, are shown in Figure 9. Similarly, the dissected 
+part of the artery undergoes an unstable aneurysmal dilatation over 6 years after the 
+initial tear opening. The maximum stress also increases over time. Besides, it is 
+interesting to note that for this toric artery case, the stress seems to increase over the 
+whole dissected adventitia layer, and is not limited to the vicinity of the tear edge 
+as observed in the previous cylindrical artery case.  
+The effect of the gain parameter has also been investigated in this dissected toric 
+artery. The temporal evolution of the maximum outer diameter of the dissected toric 
+
+t=0years
+t=6years
+OyM [MPa]
+0.06
+1.12
+t=3years
+t = 9 yearsImm
+lartery
+41.3
+Maximum outerradius of dissected
+37.8
+= 0.05
+kci
+= 0.10
+kct
+= 0.15
+34.2
+30.7
+27.2
+23.6
+0
+2
+4
+6
+9
+11
+Time [years]18  
+artery is shown in Figure 10. It can be seen that the results obtained are also in 
+agreement with previous findings of the gain parameter: a large gain parameter fa-
+vors a stable growth of aneurysm while a small gain parameter promotes an exces-
+sive enlargement of aneurysm after cTBAD. 
+ 
+ 
+Figure 9 - Reference results of the toric artery with respect to a reference value of gain parameter 
+𝑘𝜎
+𝑐𝑗 = 𝑘𝜎𝑚 = 0.05, showing geometrical and equivalent von Mises stress evolutions after 
+cTBAD over 6 years. 
+ 
+ 
+Figure 10 - Influence of the gain parameter to aneurysmal dilatation after cTBAD, showing the 
+temporal evolution of the maximum outer diameter of the dissected toric artery. 
+ 
+D. Application to a patient-specific artery 
+The reference simulation results in the case of a patient-specific artery, more 
+precisely, a patient-specific human descending aortic segment, are shown in Figure 
+11, with respect to a reference value of gain parameter 𝑘𝜎
+𝑐𝑗 = 𝑘𝜎
+𝑚 = 0.05. The re-
+sults illustrate the geometrical and stress evolutions of the aortic segment, and also 
+
+t =0years
+t=3years
+OvM [MPa]
+0.01
+0.51
+t=6years[mm]
+Maximum outer radius of dissected artery [
+72.7
+kci
+= 0.05
+66.5
+= 0.10
+= 0.15
+60.3
+54.1
+47.9
+41.8
+0
+2
+4
+6
+9
+11
+Time [years]19 
+the circumferential cross-section at its dissected extremity. We found that the dis-
+sected aortic segment undergoes continuous aneurysmal dilatation overtime after 
+the initial tear opening. Besides, it can be seen that there is a significant increase of 
+stress over time, mostly on the dissected part of the outer adventitia layer. 
+ 
+ 
+Figure 11 - Reference results of the patient-specific artery with respect to a reference value of 
+gain parameter 𝑘𝜎
+𝑐𝑗 = 𝑘𝜎
+𝑚 = 0.05, showing geometrical and equivalent von Mises stress evolu-
+tions after cTBAD over 6 years, as well as the dilated circumferential cross-section at its dis-
+sected maximum extremity, respectively at (a) 0 year of G&R, (b) 3 year of G&R and 6 year 
+of G&R.  
+ 
+Similarly, the effect of the gain parameter on aneurysmal dilatation after cTBAD 
+was investigated.  Results obtained confirm the same tendency as observed in pre-
+vious test cases: a large gain parameter tends to stabilize the aneurysmal dilatation 
+of the dissected artery while a small gain parameter promotes an excessive aneurys-
+mal degeneration after cTBAD. 
+ 
+Figure 12 - Influence of the gain parameter to the aneurysmal dilatation after cTBAD, show-
+ing the temporal evolution of the maximum outer diameter of the dissected patient-specific 
+artery. 
+
+t= 0years
+t =3years
+t =6years
+OvM [MPa]
+0.01
+0.19
+(a)
+(b)outer radius of dissected aorta[mm]
+45.1
+kct = 0.05
+43.0
+kei
+= 0.10
+k = 0.15
+40.8
+38.7
+36.5
+Maximum 
+34.3
+0
+2
+3
+5
+7
+8
+Time[years]20  
+Apart from the gain parameter, it has also been reported in the literature that the 
+tear size may have a significant influence on aneurysmal dilatation after cTBAD 
+[9], [26]. Being aware that effects of the tear size is very complex, which depend 
+not only on the position of the tear but also on its irregular shape, involving gener-
+ally measures in three dimensions. In this paper, note that we try to study only the 
+cicumferential length of the tear, with four different values of 𝛼𝑚𝑖𝑛 ranging from 
+0.8 to 0.5, i.e., from narrow circumferential tear to wide circumferential tear. Re-
+sults obtained are shown in Figure 13, describing the temporal evolution of the max-
+imum outer diameter of the dissected artery. It can be seen that a wider tear pro-
+motes an uncontrolled aneurysmal dilatation after cTBAD, while a narrow tear 
+tends to favor a stable aneurysmal dilatation with a moderate growth rate. 
+ 
+ 
+Figure 13 - Influence of the initial circumferential opening length of dissecting tear to the an-
+eurysmal dilatation after cTBAD, showing the temporal evolution of the maximum outer di-
+ameter of the dissected patient-specific artery. 
+ 
+Finally, in this patient-specific simulation, we also investigated the effect of the 
+pressure as it has been identified as a high-risk factor in cTBAD, especially in the 
+false lumen where the pressure may impact directly on the stress distribution of the 
+most weakened outer adventitia layer [6], [27], [28]. In order to evaluate the effect 
+of the pressure in the false lumen, three different values of false lumen pressure are 
+considered, respectively higher than the true lumen pressure (𝑝𝐹𝐿 𝑝𝑇𝐿
+⁄
+> 1), equal 
+to the true lumen pressure (𝑝𝐹𝐿 𝑝𝑇𝐿
+⁄
+= 1), and lower than the true lumen pressure 
+(𝑝𝐹𝐿 𝑝𝑇𝐿
+⁄
+< 1). Note that pressure in the true lumen is assumed to be constant. Re-
+sults are shown in Figure 14, showing the temporal evolution of the maximum outer 
+diameter of the dissected artery. It can be seen that a higher pressure promotes an-
+eurysmal dilatation after cTBAD. More precisely, regarding the aneurysmal dilata-
+tion rate at 3 years, a 10% pressure increase in the false lumen induces an "enlarg-
+ing" growth of the dissected artery (≥ 3mm/year) while for false lumen pressure 
+equal or below the true lumen pressure, aneurysmal dilatation can be considered as 
+"stable" (< 3mm/year) [1]. 
+
+mm
+Taorta
+46.4
+amin =0.5
+dissected
+43.9
+amin =0.6
+αmin = 0.7
+αmin=0.8
+outer radius of 
+41.5
+39.0
+36.6
+Maximum
+34.2
+0
+2
+3
+5
+7
+8
+Time[years]21 
+ 
+Figure 14 - Influence of the pressure in the false lumen to the aneurysmal dilatation after 
+cTBAD, showing the temporal evolution of the maximum outer diameter of the dissected pa-
+tient-specific artery. 
+V. 
+Discussion 
+ 
+cTBAD is associated with poor long term outcomes, mainly as a result of exces-
+sive aneurysmal dilatations. By consequence, a considerable part of patients with 
+cTBAD will require ultimately surgical interventions such as endovascular repair 
+or open surgery [29], [30].  However, there is a serious lack of risk assessment tools 
+because our current understanding of the aneurysmal dilatation mechanism after 
+cTBAD remains weak.  
+In this chapter, we proposed a numerical approach to study the role of G&R in 
+aneurysmal dilatation in cTBAD. We found that the G&R process triggers naturally 
+the aneurysmal dilatation. Moreover, it was found that with a large gain parameter 
+related to collagen G&R, the aneurysmal dilatation tends to be stable while with a 
+small gain parameter, there would be an excessive aneurysmal degeneration. It is 
+interesting to note that the results obtained are in agreement with clinical evidence 
+reported by Juvonen et al. [6], where older patients present a higher risk of aneurys-
+mal rupture in case of cTBAD. In fact, this gain parameter describes the capacity of 
+arteries to restore its tensional equilibrium state in case of a disturbance of its mech-
+anobiological equilibrium [14] and it has been reported that age may affect this gain 
+parameter with older patients generally having an impaired stress-regulated repair 
+mechanism compared to young patients [31].  
+ Based on sensitivity analysis performed on patient-specific simulations, we 
+found that the circumferential tear length also has a significant influence on the 
+G&R process after cTBAD. A wide tear promotes an unstable development of an-
+eurysmal dilatation while a narrow tear reduces the risk of uncontrolled aneurysmal 
+dilatation. The reason could be twofold. First, a wide initial opening tear means 
+
+mm
+aorta
+45.2
+dissected
+43.0
+40.7
+radiusof
+38.5
+outer
+PFL/PTL= 1.1
+36.3
+PFL/PTL = 1.0
+laximum
+PFL/PTL= 0.9
+34.0
+0
+2
+3
+4
+6
+Time[years]22  
+naturally a larger initial dissected arterial diameter once the false lumen is pressur-
+ized compared to a narrow initial tear. Secondly, the consequence of a larger initial 
+dissected arterial diameter is that the deformation and stress will be much higher, 
+especially in the dissected outer adventitia layer, accelerating the G&R process of 
+the artery. Indeed, the results remained very limited, considering the irregular three-
+dimensional tear shape and other complex mechanobiological phenomena ne-
+glected. However, it provides a mechanical proof that the tear size is also an im-
+portant influencing factor that needs to be considered in the risk assessment of pa-
+tients with cTBAD.  
+Finally, our results indicated that the pressure in the false lumen has a determi-
+nant role in the aneurysmal progression rate of the dissected artery. We found that 
+a relative 10% increase of pressure in the false lumen, compared to that of the true 
+lumen, is sufficient to promote an "unstable" growth of the dissecting aneurysm. 
+This can be critical for patients, as surgical interventions are usually recommended 
+for such situations [1]. 
+Despite the above promising results obtained, the current model still has some 
+shortcomings that could be addressed in the future. First, in the present model, the 
+tear configuration remains fixed after its initial creation. However, with the contin-
+uous aneurysmal dilatation and accumulation of stress especially near the edge of 
+the tear, the initial tear may propagate due to high-stress concentration and thus alter 
+the G&R process. Therefore, integration of the tear propagation models, such as 
+that reported in the literature [32]-[35], could be an essential step to build a more 
+reliable dissection model for evaluating the G&R effect to aneurysmal dilatation 
+after cTBAD. 
+Secondly, the dissection model is relatively simplified. Intraluminal thrombus, 
+which was often reported to have an important role in dissecting aneurysm pathol-
+ogies and rupture [36], [37], was literally neglected in this work. Besides, the effects 
+of surrounding tissues have also been neglected although it has already been re-
+ported in the literature that the surrounding connective tissues or vertebral column 
+may impact regional adaptation of aortic walls by changing the local wall stress 
+distribution or even more directly the aneurysmal aortic shape [38], [39]. Moreover, 
+potential dynamic effects of the blood flow inside the arterial wall are currently not 
+taken into consideration, despite its non-negligible effect directly on the wall stress 
+distribution, as reported in the literature [26], [40].  
+Finally, the initial tear configuration was restricted to its circumferential length. 
+Location of the tear, number of the tears or other dimensions related to the tears are 
+currently neglected. For further patient-specific simulations, these parameters 
+should be carefully considered as they may directly impact the pressure in both the 
+false lumen and the true lumen [9].  
+ 
+VI. 
+Conclusions 
+ 
+
+23 
+In summary, we introduced an efficient 3D finite-element model, based on an 
+open-source in-house code, to model the aneurysmal dilatation due to G&R after 
+cTBAD. We showed the potential of this dissection model to simulate G&R process 
+after cTBAD, from simple test cases with idealized arterial geometries to a more 
+relevant case with a patient-specific geometry. The effects of different parameters 
+on aneurysmal dilatation were assessed through a comprehensive sensitivity analy-
+sis. It was found that the gain parameter related to collagen G&R as well as the 
+circumferential initial tear length, has an undeniable impact on the stability of the 
+dissecting aneurysm. Moreover, our results indicated that the stability of the dis-
+secting aneurysm is very sensitive to the intraluminal false lumen pressure. A rela-
+tive pressure increase of 10% in the false lumen may induce an excessive aneurys-
+mal degeneration in patients with cTBAD. 
+Future work is twofold. The first one is coupling the present dissection model 
+with tear propagation models, for applications to more reliable patient-specific sim-
+ulations. The second one is to account for a more accurate configuration of opening 
+tears while considering the potential dynamic effects of the blood flow inside the 
+dissected arteries. 
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+
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf,len=599
+page_content='Patient-specific Finite Element Modeling of  Aneurysmal dilatation after chronic type B  aortic dissection  Shaojie Zhanga, Joan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Laubriea, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Jamaleddin Mousavia, Stéphane Avrila  and Sabrina Ben Ahmedb  a Mines Saint-Etienne, Univ Lyon, Univ Jean Monnet, INSERM, U1059 Sainbiose, Centre  CIS, F-42023 Saint-Etienne, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  b Univ Jean Monnet, INSERM, U1059 Sainbiose and University Hospital of Saint-Etienne, F-  42000 Saint-Etienne,  Abstract  Progressive aneurysmal dilatation is a well-recognized com-  plication in patients with chronic type B aortic dissection (cTBAD), which may  lead to a delayed rupture and create a life-threatening condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' However, our  understanding of such aortic expansion in cTBAD remains weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' In the pre-  sent paper, we propose to use numerical simulations to study the role of growth  and remodeling (G&R) in aneurysmal dilatation after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' We set up a 3D  finite-element model of G&R for aortic dissection within an open-source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Constitutive equations, momentum balance equations, and equations related  to the mechanobiology of the artery were formulated based on the homoge-  nized constrained mixture theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The model was first applied to idealized aor-  tic geometries with cylindrical and toric shapes to demonstrate its feasibility  and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The model was then applied to a patient-specific aortic segment  to show its potential in more relevant and complex patient-specific clinical ap-  plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It was found that the G&R tends to naturally trigger the aneurys-  mal dilatation after dissection, in order to restore its tensional equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Our results indicated that the value of the gain parameter, related to collagen  G&R, plays an important role in the stability of aortic expansion after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  A small gain parameter will induce an excessive aneurysmal degeneration  whilst a large gain parameter helps to recover a stabilized state of the artery  after dissection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Finally, it was found that other mechanobiology-related pa-  rameters, such as the circumferential length of the dissection, as well as the  pressure in the false lumen, may also be determinant for the stability of aneu-  rysmal dilatation after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Both a wide tear and an elevated false lumen  pressure favor an unstable development of aortic expansion after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' As  future work, the present model will be validated through predictions of aneu-  rysmal dilatation in patient-specific clinical cases, in comparison with datasets  followed over a significant period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  INTRODUCTION  Chronic type B aortic dissection (cTBAD) is defined when a tear originates in  the descending aorta and remains 3 months after its onset [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Patients with uncom-  plicated cTBAD are preferentially treated medically with periodic clinical and im-  aging surveillance, regarding the acceptable survival rate generally observed in a  short-term follow-up [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' However, the long-term outcome of such conservative  management remains questionable mainly due to the progressive aneurysmal dila-  tation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Invasive surgical interventions, such as endovascular repair or open sur-  gery are then needed [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Up to now, little is known about the aneurysmal dilatation  after cTBAD, either it is stable with a moderated progression rate or there is an  excessive aneurysmal degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It is yet of crucial importance for surgeons to  be able to assess the risk of aortic expansions in patients with early-stage cTBAD  to choose the optimal treatment approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Patients identified at high risk for aortic  enlargement may therefore benefit from early surgical interventions and reduce  mortality from delayed aneurysm ruptures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Published studies on this topic remain scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It has been widely accepted that  the presence of an excessive aortic diameter, typically greater than 40 mm, and a  patent false lumen are two high-risk factors for late aneurysm development after  cTBAD [6]-[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Besides, older age and elevated mean blood pressure were also  found to promote aneurysmal degeneration in cTBAD [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' reported that  the size, the number, as well as location of tears have significant impacts on the  pressure in the false lumen, and therefore influencing the false lumen expansion [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Recently, Trimarchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' revealed that there are many other factors that may affect  aneurysmal dilatation after cTBAD, including demographic, clinical, pharmaco-  logic, and radiologic variables, such as connective tissues disorders, gender, the  presence of thrombus in the false lumen, etc [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' However, all the above researches  were based on observational studies or clinical trials with data collected over a long  follow-up period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Considering the recent advances in computational mechanics of arteries [11],  [12] and more specifically the growth and remodeling (G&R) models [13]-[17],  numerical models can be an interesting alternative option for studying these influ-  encing factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' However, to the author’s best knowledge, G&R after cTBAD has  never been modeled so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' There is still an important potential for G&R models to  understand vascular adaptation in chronic type B aortic dissection, where the patient  can undergo a long-term process of G&R after breaking the initial mechanical equi-  librium due to tear opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  In this specific context, we developed a 3D finite-element model of vascular ad-  aptation to study the aneurysmal dilatation after cTBAD, within an open-source  code written in python and C++ [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The G&R model of the arterial wall is  based on the homogenized constrained mixture theory (CMT) and the aortic dissec-  tion is modeled through an original two-continuum arterial wall concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' We also  performed a sensitivity analysis to evaluate the influence of several selected mech-  anobiological parameters on the aneurysmal dilatation after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  3  Details of the model are given in this book chapter, by first introducing the math-  ematical framework of the CMT method for G&R with respect to cTBAD, then  describing the two-continuum aortic dissection model, and finally showing poten-  tials of the model, from a simple validation test case to academic applications with  idealized geometries, until a more relevant patient-specific application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Material and Methods  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Constitutive and balance equations  The CMT was first proposed by Humphrey and Rajagopal as a hybrid method to  describe mechano-regulated G&R of arteries [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It was then largely used for mod-  eling aneurysm formation [13]-[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' In this work, we employ the homogenized  CMT [17] to model arterial G&R after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Basic equations formulated under  the homogenized CMT framework are briefly introduced in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Readers  can refer to reference publications for more detailed mathematical formulations and  their interpretations [15], [17], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  First, we assume that the arterial wall can be modeled as an homogenized mixture  made up by a matrix containing a network of elastic fibers, passive reinforcements  represented by 4 collagen fiber families (respectively oriented in circumferential,  axial and diagonal (+/- 45°) direction) and active reinforcements accounting for the  contractility of smooth muscle cells (SMCs) in the circumferential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Let  Ω𝑅 ⊂ ℝ3 and Ω𝑡 ⊂ ℝ3 denote, respectively, the initial traction-free reference con-  figuration at time 𝑡 = 0 and current deformed configuration at time 𝑡 > 0 of the ar-  terial wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' According to homogenized CMT, we assume that all constituents in the  arterial wall deform together with a same deformation gradient 𝐅:  𝐅 = 𝜕𝒙  𝜕𝑿  (1)  where 𝑿 represents a material point in Ω𝑅 and 𝒙 represents the associated spatial  point in Ω𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Moreover, based on the theory of Rodriguez and Hoger [21], this de-  formation gradient tensor 𝐅 can be split into an elastic part and an inelastic part for  each constituent 𝑖 ∈ [𝑒, 𝑐𝑗, 𝑚], such as  𝐅 = 𝐅𝑒𝑙  𝑖 𝐅𝑔𝑟  𝑖  (2)  where 𝑒, 𝑐𝑗, 𝑚 represents respectively the elastic matrix, the 𝑗𝑡ℎ collagen fiber fam-  ily and smooth muscle cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' More precisely, 𝐅𝑒𝑙  𝑖  represents the elastic deformation  tensor related to stresses that balance external mechanical loads over the arterial  wall, while 𝐅𝑔𝑟  𝑖  represents the inelastic deformation tensor related to G&R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' re-  lated to the continuous mass turnover of each constituent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Besides, we assume that  G&R is a fully stress-mediated process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Other non-mechanical effects related to the  mass turnover, such as immune-mediated chemical remodeling, damage, or me-  chanical fatigue, are neglected in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Therefore, the temporal homogenized  mass deposition or degradation rate of each constituent can be expressed as  4  𝜌̇𝑅  𝑖 = 𝜌𝑅  𝑖 𝑘𝜎  𝑖 𝜎𝑖 − 𝜎ℎ  𝑖  𝜎ℎ  𝑖  (3)  where 𝜌𝑅  𝑖  is the reference mass density of constituent 𝑖, related to the reference con-  figuration of the arterial wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The right-hand side term of Equation 3 describes the  mass turnover due to the stress difference between the current stress 𝜎𝑖 and the ho-  meostatic stress 𝜎ℎ  𝑖, where 𝑘𝜎  𝑖  is a regularization parameter (named gain parameter)  with respect to each constituent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  The homogenized CMT consists in the decomposition of the inelastic defor-  mation gradient 𝐅𝑔𝑟  𝑖  through two sub-gradient tensors  𝐅𝑔𝑟  𝑖 = 𝐅𝑔  𝑖𝐅𝑟  𝑖  (4)  where 𝐅𝑔  𝑖 is the growth-related tensor describing volume changes to due mass turn-  over, and 𝐅𝑟  𝑖 is the remodeling-related tensor describing how the prestretch of each  constituent is updated through continuous extant mass degradation and new mass  production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' As suggested by Braeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' [15], we assume that the growth defor-  mation is the same for all constituents in the arterial wall, such as  𝐅𝑔  𝑖 = 𝐅𝑔 = 𝐈 + 𝜌𝑅  𝜌𝑅0  𝒂0  ⊥ ⊗ 𝒂0  ⊥ − 𝒂0  ⊥ ⊗ 𝒂0  ⊥  (5)  where 𝜌𝑅 is the current reference mass density, 𝜌𝑅0 is the initial reference mass den-  sity (at time 𝑡 = 0), 𝐈 is the second order identity and 𝒂0  ⊥ the growth direction [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  The remodeling process of elastin can be generally neglected (𝐅𝑟  𝑒 = 𝐈) considering  its slow mass degradation rate (typically several decades for elastin half-life time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  We assume that the remodeling process of collagen fibers and SMCs occurs at a  constant volume and along the fiber direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' which can be expressed as [22]  𝐅𝑟  𝑖 = 𝜆𝑟  𝑖 𝒂0  𝑖 ⊗ 𝒂0  𝑖 + 1  √𝜆𝑟𝑖 (𝐈 − 𝒂0  𝑖 ⊗ 𝒂0  𝑖 )  (6)  where 𝜆𝑟  𝑖  is the respective remodeling stretch of fiber 𝑖 ∈ [𝑐𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝑚] along its fiber di-  rection 𝒂0  𝑖  with its time evolution 𝜆̇𝑟  𝑖  given by [13]  𝜆̇𝑟  𝑖 = (𝜌̇𝑅  𝑖  𝜌𝑅  𝑖 + 1  𝑇𝑖)  𝜆𝑖  (𝜆𝑒𝑙  𝑖 )  2 (𝜕𝜎𝑖  𝜕𝜆𝑒𝑙  𝑖 )  −1  × (𝜎𝑖 − 𝜎ℎ  𝑖)  (7)  where 𝑇𝑖 is the average mass turnover time during which old mass increment is  degraded and replaced by a new mass increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝜆𝑒𝑙  𝑖  is the elastic stretch of fiber 𝑖  defined as 𝜆𝑒𝑙  𝑖 = √(𝐅𝑒𝑙  𝑖 )  𝑡𝐅𝑒𝑙  𝑖 ∶ 𝒂0  𝑖 ⊗ 𝒂0  𝑖  and 𝜆𝑖 is the total stretch of fiber 𝑖 defined  as 𝜆𝑖 = √(𝐅)𝑡𝐅 ∶ 𝒂0  𝑖 ⊗ 𝒂0  𝑖  Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' considering that the homeostatic configuration of the arterial wall is  achieved at time 𝑡 = 𝑡0 and defining the initial traction-free geometry of the arterial  wall at time 𝑡 = 0 as the same geometry as its homeostatic configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' the initial  5  elastic prestretch of each constituent 𝐆ℎ  𝑖  corresponding to the homeostatic configu-  ration at time 𝑡0 can simply satisfy  𝐅𝑟  𝑖(𝑡0) = (𝐆ℎ  𝑖 )  −1  (8)  due to the fact that 𝐈 = 𝐅(𝑡0) = 𝐅𝑒  𝑖(𝑡0)𝐅𝑔  𝑖(𝑡0)𝐅𝑟  𝑖(𝑡0) = 𝐆ℎ  𝑖 𝐅𝑔  𝑖(𝑡0)𝐅𝑟  𝑖(𝑡0) =  𝐆ℎ  𝑖 𝐅𝑟  𝑖(𝑡0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The detailed expressions of 𝐆ℎ  𝑖  are hereby given for each constituent 𝑖 ∈  [𝑒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝑐𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝑚] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' with respect to a cylindrical coordinate system  𝐆ℎ  𝑒 = 𝐝𝐢𝐚𝐠 ( 1  𝜆𝜃  𝑒 𝜆𝑧𝑒 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='   𝜆𝜃  𝑒 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='   𝜆𝑧  𝑒)  (9)  𝑮ℎ  𝑐𝑖 = 𝜆ℎ  𝑐𝑖𝒂0  𝑐𝑖 ⊗ 𝒂0  𝑐𝑖 +  1  √𝜆ℎ  𝑐𝑖  (𝑰 − 𝒂0  𝑐𝑖 ⊗ 𝒂0  𝑐𝑖)  (10)  𝑮ℎ  𝑚 = 𝜆ℎ  𝑚𝒂0  𝑚 ⊗ 𝒂0  𝑚 +  1  √𝜆ℎ  𝑚 (𝑰 − 𝒂0  𝑚 ⊗ 𝒂0  𝑚)  (11)  where 𝜆𝜃  𝑒  and 𝜆𝑧  𝑒 are the initial deposition stretches of elastin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' respectively in the  circumferential and axial direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' uniform over the whole arterial wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝜆ℎ  𝑐𝑖 and 𝜆ℎ  𝑚  are respectively the initial deposition stretches of collagen fibers (same deposition  stretch for all four directions) and SMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Based on CMT, the strain energy density function of the arterial wall can be  given by  𝛹 = 𝜌𝑅  𝑒𝑊𝑒 + ∑ 𝜌𝑅  𝑐𝑗𝑊𝑐𝑗  4  𝑗=1  + 𝜌𝑅  𝑚𝑊𝑚  (12)  where 𝜌𝑅  𝑒, 𝜌𝑅  𝑐𝑗and 𝜌𝑅  𝑚 are respectively the reference mass densities of the elastic ma-  trix, of the 𝑗𝑡ℎ collagen fiber family and of SMCs, and 𝑊𝑒, 𝑊𝑐𝑗 and 𝑊𝑚 are the  specific strain energy density functions with respect to each constituent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Moreover,  the strain energy density function 𝑊𝑖 of each constituent 𝑖 ∈ [𝑒, 𝑐𝑗, 𝑚], can be ex-  pressed as a function of its elastic deformation tensor 𝐅𝑒𝑙  𝑖 , or equivalently, its elastic  right Cauchy-Green tensor 𝐂𝑒𝑙  𝑖 , defined as 𝐂𝑒𝑙  𝑖 = (𝐅𝑒𝑙  𝑖 )  𝑡𝐅𝑒𝑙  𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' In the present work, the  elastic matrix is considered as a quasi-incompressible Neo-Hookean hyperelastic  material with its specific strain energy density function 𝑊𝑒 given by  𝑊𝑒 = 𝜇𝑒  2 (tr(𝐂̅𝑒𝑙  𝑒 ) − 3) + 𝜅(|𝐅𝑒𝑙  𝑒 | − 1)2  (13)  where 𝜇𝑒 is a material parameter characterizing the shear stiffness of elastin and 𝜅  is an arbitrary but sufficiently high penalty parameter ensuring quasi incompressi-  bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝐂̅𝑒𝑙  𝑒  is the isochoric elastic right Cauchy-Green tensor of elastin, defined as  𝐂̅𝑒𝑙  𝑒 = (𝐅̅𝑒𝑙  𝑒 )𝑡𝐅̅𝑒𝑙  𝑒  and 𝐅̅𝑒𝑙  𝑒 = 𝐅𝑒𝑙  𝑒 |𝐅𝑒𝑙  𝑒 |1/3  ⁄  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The specific strain energy density function  6  of collagen fiber families is modeled by an anisotropic Fung-type exponential func-  tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  𝑊𝑐𝑗 = 𝑘1  𝑐𝑗  2𝑘2  𝑐𝑗 (𝑒𝑘2  𝑐𝑗(𝐼4𝑒𝑙  𝑐𝑗−1)  2  − 1)  (14)  We also use the same anisotropic Fung-type exponential function to model the  passive behavior of SMCs [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' while an additional term is added for the active tone  contribution such as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  𝑊𝑚 = 𝑘1  𝑚  2𝑘2  𝑚 (𝒆𝑘2𝑚(𝐼4𝑒𝑙  𝑚−1)2  − 1) + 𝜎𝑚𝑎𝑥  𝜌𝑅0  (𝜆𝑎𝑐𝑡 + 1  3  (𝜆𝑚𝑎𝑥  𝑚  − 𝜆𝑎𝑐𝑡)3  (𝜆𝑚𝑎𝑥  𝑚  − 𝜆0  𝑚)2 )  (15)  where 𝑘1  𝑐𝑗and 𝑘1  𝑚 are stress-like material parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' and 𝑘2  𝑐𝑗 and 𝑘2  𝑚are dimen-  sionless material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝐼4𝑒𝑙  𝑐𝑗 and 𝐼4𝑒𝑙  𝑚 are pseudo-invariants, which are addi-  tional invariants defined in case of anisotropic materials such as 𝜆𝑒𝑙  𝑖 = 𝐂𝑒𝑙  𝑖 ∶ 𝒂0  𝑖 ⊗  𝒂0  𝑖  with 𝑖 ∈ [𝑐𝑗, 𝑚] [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝜆𝑎𝑐𝑡 is the active stretch in the circumferential direction,  𝜎𝑚𝑎𝑥 is the maximum active Cauchy stress, 𝜌𝑅0 is the reference total mass density  of the arterial wall at time 𝑡 = 0, and 𝜆𝑚𝑎𝑥  𝑚  and 𝜆0  𝑚 are the active stretches respec-  tively at maximum and zero active stress for SMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  The second Piola-Kirchhoff stress tensor 𝐒 and the fourth order elasticity tensor  of the arterial wall ℂ are then deduced by performing the first and second deriva-  tives of the strain energy function 𝛹 with respect to the total Green-Lagrange strain  𝐄  𝐒 = 𝜕𝛹  𝜕𝐄 = 𝜑𝑒𝐒𝑒 + ∑ 𝜑𝑐𝑗𝐒𝑐𝑗  𝑖  + 𝜑𝑚𝐒𝑚  (16)  ℂ = 𝜕2𝐒  𝜕𝐄𝜕𝐄 = 𝜑𝑒ℂ𝑒 + ∑ 𝜑𝑐𝑗ℂ𝑐𝑗  𝑖  + 𝜑𝑚ℂ𝑚  (17)  where 𝜑𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝐒𝑖 and ℂ𝑖 are the mass fraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' second Piola-Kirchhoff stress and forth  order elasticity tensor with respect to each constituent 𝑖 ∈ [𝑒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝑐𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝑚] in the arterial  wall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' defined as  𝜑𝑖 = 𝜌𝑅  𝑖  𝜌𝑅  (18)  𝐒𝑖 = 𝜌𝑅  𝜕𝑊𝑖  𝜕𝐄  (19)  ℂ𝑖 = 𝜌𝑅  𝜕2𝑊𝑖  𝜕𝐄𝜕𝐄  (20)  with 𝜌𝑅 = 𝜌𝑅  𝑒 + 𝜌𝑅  𝑐𝑗 + 𝜌𝑅  𝑚 the reference total mass density of the arterial wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Fi-  nally, assuming that the G&R occurs at a slow time scale and can be considered as  a quasi-static process, the dynamics effects such as inertia or viscoelasticity of the  7  arterial wall can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' the momentum balance equations of the  arterial wall can be simply written as  ∇ ∙ 𝝈 + 𝜌𝒃 = 𝟎  (21)  𝜌 is the spatial density of the arterial wall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' related to its reference density 𝜌𝑅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' as 𝜌 =  𝜌𝑅 |𝐅|  ⁄  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝒃 is the body force vector given in the spatial configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝝈 is the Cau-  chy stress derived from the previous second Piola-Kirchoff stress as  𝝈 = 𝟏  |𝐅| 𝐅𝐒𝐅𝑡  (22)  The boundary conditions applied on the arterial wall can be Dirichlet boundary  conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' assigning the predefined displacement field over the mesh nodes or  Robin boundary conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' which are applied over the mesh surface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' such as  𝝈 ∙ 𝒏 = 𝑃𝑇𝐿𝒏 + 𝒑𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 + 𝑭𝑠𝑝𝑟𝑖𝑛𝑔  (23)  where 𝑃𝑇𝐿 denotes the true luminal pressure of the artery due to blood flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' applied  on the inner surface of the media layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝒏 is the outward pointing unit vector normal  to the arterial inner media surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝒑𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 is the pressure in the false lumen after  cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝑭𝑠𝑝𝑟𝑖𝑛𝑔 is an additional spring-based elastic force, related to the two-con-  tinnumm arterial wall concept proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Details of the two-continuum  arterial wall concept, as well as, 𝑭𝑠𝑝𝑟𝑖𝑛𝑔, 𝒑𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛, will be given in the next sec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Dissection model  In this section, we will firstly present the two-continuum arterial wall concept,  dedicated to the modeling of G&R in the case of cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It is worth noting that the  mechanism of the initial tear formation or the subsequent tear progression is cur-  rently not of primary interest in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Under this specific context, we propose  to model the initial healthy arterial wall without dissection with two continuum bod-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' As shown in Figure 1a, the arterial wall is made up of two layers, respectively  the inner media layer and the outer adventitia layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The two layers are perfectly  connected by high stiffness elastic springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' More precisely, each spring connects  two adjacent mesh surfaces 𝑆𝑎 and 𝑆𝑚, respectively at the inner surface of the ad-  ventitia and the outer surface of media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The force applied on each surface is com-  puted on the relative displacement of the two connecting surfaces, as given in Equa-  tion (24)  𝑭𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 = {−𝑘𝑠𝑝𝑟𝑖𝑛𝑔(𝑑𝑆𝑎 − 𝑑𝑆𝑚)𝒏𝑆𝑎,  𝑎𝑑𝑣𝑒𝑛𝑡𝑖𝑡𝑖𝑎  𝑘𝑠𝑝𝑟𝑖𝑛𝑔(𝑑𝑆𝑎 − 𝑑𝑆𝑚)𝒏𝑆𝑚,  𝑚𝑒𝑑𝑖𝑎  (24)  where 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 is the stiffness of the interfacial spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝑑𝑆𝑎 and 𝑑𝑆𝑚 the nodal-aver-  aged displacement of the mesh surface located respectively in the inner adventitia  8  and outer media surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝒏𝑆𝑎 and 𝒏𝑆𝑚 are the outward pointing unit vectors nor-  mal to the respective mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It is worth noting that displacement of one layer is trans-  mitted to the other layer with low distortions due to the high stiffness of the springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Finally, we assume that this initial configuration of the arterial wall is at its home-  ostatic state with a reference luminal pressure 𝑃𝑇𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' To validate this new concept, a  simple validation test case has been proposed in this work, in comparison with the  conventional arterial wall model with a single continuum body [13], [15], [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 1 - (a) Illustration of the two-continuum arterial wall model, with the adventitia layer  and the media layer connected by high stiffness elastic springs, representing a healthy arterial  wall under its homeostatic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' (b) Initial configuration of the arterial wall after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The  false lumen is created by breaking interfacial elastic springs in a selected region between the  media and adventitia layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Based on this new concept of a two-continuum arterial wall, the initial configu-  ration of cTBAD with false lumen can be obtained by simply vanishing interfacial  springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' As shown in Figure 1b, the false lumen (FL), as well as the free surfaces  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' inner side of adventitia layer and outer side of media layer), is created by break-  ing springs in a selected region where tears are assumed to be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The force  induced by the pressure in the false lumen is then applied on each mesh surface,  𝑆𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛, of the newly created free surfaces  𝒑𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 = 𝑃𝐹𝐿𝒏𝑆𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛  (25)  where 𝑃𝐹𝐿 is the constant pressure in the false lumen and 𝒏𝑆𝑑𝑖𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛 is the outward  pointing unit vector normal to the free mesh surfaces in the false lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Finally, it  is worth noting that the presence of such pressurized false lumen will break the  mechanical equilibrium of the arterial initial homeostatic state, and therefore trig-  ging the G&R of the arterial wall over a long period of time in case of cTBAD, until  the achievement of a new preferred mechanical state or eventually an excessive an-  eurysmal degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  adventitia  adventitia  false lumen (FL)  media  media  VM  W  WW  W  true lumen (TL  WW  W  (a)  (b)9  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Finite element implementation  The proposed model was implemented within an open-source finite element  code, written in Python and C++ [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Three different steps, are defined in the  model: 1st step: Computation of the healthy arterial wall at homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The ini-  tial arterial wall is loaded with a constant luminal pressure, 𝑃𝑇𝐿, on its en-  tire inner surface of media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 2nd step: Opening of the arterial wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Interfacial springs are removed in  a selected region between the adventitia layer and media layer, creating the  initial dissection tear of cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The same luminal pressure in the true  lumen, 𝑃𝑇𝐿, is maintained as in the previous step, applied on the entire in-  ner surface of the media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' In the meantime, the false lumen will be loaded  with a constant pressure, 𝑃𝐹𝐿, applied on the newly created surfaces related  to the tear-open region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 3rd step: Adaptation of the arterial wall with G&R after cTBAD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=', after  the creation of a pressurized false lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Note that there is only one time increment in the first two steps while the third  step is composed of several time increments to obtain relevant results of a long-term  arterial wall adaptation after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' For each time increment, the same set of mo-  mentum balance and constitutive equations is solved with the Newton-Raphson  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Finally, at the end of each time increment, we obtain the displacement field  and the associated stress-strain information on each mesh node of the arterial wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Numerical applications  In order to show the potentials of the present dissection model, four different  simulations were performed in this paper: Validation test case: It consists of a simple test case to validate the  spring-connected two-continuum arterial wall concept proposed in this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The validation was achieved through the comparison with refer-  ence results in the literature [13], [15], [16], [17] where the conventional  single continuum arterial wall model was employed to simulate the an-  eurysm formation in response to external stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a cylindrical artery: The second application is to study  the G&R after cTBAD in the case of an idealized cylindrical artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a toric artery: In this third application, the dissection  model is applied to an idealized toric artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a patient-specific artery: In this last application, we  demonstrate the feasibility of our dissection model for further more  complex and relevant clinical patient-specific applications, by model-  ing the G&R on a dissected patient-specific human descending aortic  segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Validation test case  An idealized two-layered cylindrical artery is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The geometry was the  same as that has been used in the work of Braeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' [15], with an inner arterial  radius of 10 mm and a constant arterial thickness of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='41 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Besides, we assume  that each layer of the arterial wall, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' adventitia layer and media layer, has the same  thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='705 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Moreover, it should be mentioned that a constant gap of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='01 mm is defined between the two layers, allowing the presence of interfacial  connecting springs with respect to the two-continuum arterial wall concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The  mesh was hexahedral and composed of 6 × 40 × 25 elements (thickness × circum-  ferential × length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Finally, the whole geometry is assumed to be at a homeostatic  state, related to a reference luminal pressure of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='3 kPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Following Braeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' [15], we apply a sudden degradation of the elastic matrix  such as  𝐷̇ 𝑒 = − 𝜌𝑅  𝑒  𝑇𝑒 − 𝐷𝑑𝑎𝑚  𝑡𝑑𝑎𝑚  𝜌𝑅0  𝑒 𝑒  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5(  𝑧  𝐿𝑑𝑎𝑚)  2  −  𝑡  𝑡𝑑𝑎𝑚  (26)  where 𝐿𝑑𝑎𝑚 and 𝑡𝑑𝑎𝑚 are respectively the spatial and the temporal damage spread  parameter, 𝐷𝑑𝑎𝑚 is the maximum damage, 𝜌𝑅0  𝑒  is the initial reference mass density  of elastin, 𝑧 is the axial position of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Noting that due to the symmetry of  the problem, only half of the cylinder is modeled in this simulation with 𝑧 varying  from 0 mm to 90 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The first term at the right-hand side (RHS) of Equation 26  describes the natural elastin degradation by aging effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The second term of the  RHS is related to sudden external stimuli, causing a maximum elastin degradation  at the center of the cylinder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 𝑧 = 0 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Material properties used in this valida-  tion test case are taken from the 3D model of Braeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' [15] and are summarized  in Table 1 together with other simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The simulation results ob-  tained in this simulation are compared to the reference results in the literature [13],  by using three different values of collagen gain parameter 𝑘𝜎  𝑐𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Material properties used in the validation test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Symbol  Value  Unit  𝜇𝑒  72  J ∙ kg−1  𝜅  720  J ∙ kg−1  𝑘1  𝑐𝑗  568  J ∙ kg−1  𝑘2  𝑐𝑗  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='2  −  𝑘1  𝑚  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='6  J ∙ kg−1  𝑘2  𝑚  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='4  −  𝜌𝑅0  𝑒  241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  kg ∙ m−3  11  𝜌𝑅0  𝑐1 = 𝜌𝑅0  𝑐2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='1  kg ∙ m−3  𝜌𝑅0  𝑐3 = 𝜌𝑅0  𝑐4  260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='4  kg ∙ m−3  𝜌𝑅0  𝑚  157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  kg ∙ m−3  𝜆𝑧𝑒  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='25  −  𝜆𝜃  𝑒  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='34  −  𝜆ℎ  𝑐𝑗  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='062  −  𝜆ℎ  𝑚  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='1  −  𝑇𝑒  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 16  𝑦ears  𝑇𝑐𝑗  101  days  𝑇𝑚  101  days  𝐿𝑑𝑎𝑚  8  mm  𝑡𝑑𝑎𝑚  40  days  𝐷𝑚𝑎𝑥  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  −  𝜎𝑎𝑐𝑡𝑚𝑎𝑥  54  kPa  𝜆0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8  −  𝜆𝑚  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='4  −  𝑘𝑠𝑝𝑟𝑖𝑛𝑔  1000  kPa ∙ mm−1  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a cylindrical artery  After the validation of the two-continuum arterial wall model, we first apply the  G&R after cTBAD in the case of an idealized two-layered cylindrical artery as  shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The same geometry as in the previous validation case was used,  except that the length of the artery is reduced from 90 mm to 50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The mesh was  hexahedral and composed of 6 × 60 × 20 elements (thickness × circumferential ×  length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Similarly, we assume that this geometry was related to a homeostatic state  under an inner true lumen pressure of 100 mmHg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The initial tear of the dissection  is created by breaking springs in regions where 𝑥 ≤ 10 and 𝑧 ≤ 50, to model a rep-  resentative initial tear of cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' A pure sliding boundary condition is assigned on  the cross-section at two extremities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The outer surface of the adventitia is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The  reference pressure in the false lumen is assumed to be the same as in the true lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  The same material properties as reported in Table 1 has been used, considering ad-  ditionally the layered distribution of different material constituents as suggested by  Mousavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' [16] for human ascending thoracic aorta, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=', the media has 97% of  12  the total elastin, 15% of the total axial and diagonal collagen fibers, and 100% of  the total SMCs, while the adventitia has 3% of the total elastin, 85% of the total  axial and diagonal collagen and 100 % of the total circumferential collagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 2 – Schematic representation of the idealized cylindrical artery on which the presented  dissection model is applied for modeling G&R after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a toric artery  In order to further verify the applicability of the present dissection model, we  employed here an idealized toric geometry, as shown in Figure 3, to simulate the  arterial G&R after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 3 - Schematic representation of the idealized toric artery on which the present dissection  model was applied for modeling G&R after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  The geometry was a fourth of a torus with an arch radius of 65 mm, similar to  the ones already used in the literature [16], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The arterial section is defined with  an inner radius of 18 mm and an outer radius of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='38 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The thickness of the  artery is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='38 mm, including a constant gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='01 mm between two equal-thick-  ness adventitia and media layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The mesh was hexahedral and composed of 6 ×  60 × 20 elements (thickness × circumferential × length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Once again, we assume  that this geometry was related to a homeostatic state, under a constant inner true  lumen pressure of 80 mmHg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Similarly, a pure sliding boundary condition is as-  signed to cross-sections at the two extremities while the outer surface of the adven-  titia is let free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Mechanical parameters, as well as simulation parameters, are re-  ported in Table 2, based on Laubrie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The initial tear of the dissection is  (0,0,0)  10mm  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='42mm  50mm65mm  18mml  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='38mm  (0,0,0)  Y13  created by breaking springs in the regions defined by √𝑥2 + 𝑦2 ≥ 70 and  𝑎𝑟𝑐𝑡𝑎𝑛(𝑥 𝑦  ⁄ ) ≤ 60°, to model a representative initial tear of cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Material properties and simulation parameters used for the toric artery simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Symbol  Value  Unit  𝜇𝑒  80  J ∙ kg−1  𝜅  800  J ∙ kg−1  𝑘1  𝑐𝑗  292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  J ∙ kg−1  𝑘2  𝑐𝑗  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='6  −  𝑘1  𝑚  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8  J ∙ kg−1  𝑘2  𝑚  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  −  𝜌𝑅0  𝑒  (media)  169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  kg ∙ m−3  𝜌𝑅0  𝑐1 = 𝜌𝑅0  𝑐2  (media)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='6  kg ∙ m−3  𝜌𝑅0  𝑐3 = 𝜌𝑅0  𝑐4  (media)  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='4  kg ∙ m−3  𝜌𝑅0  𝑚  (media)  735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  kg ∙ m−3  𝜌𝑅0  𝑒  (adventitia)  565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  kg ∙ m−3  𝜌𝑅0  𝑐1 = 𝜌𝑅0  𝑐2  (adventitia)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  kg ∙ m−3  𝜌𝑅0  𝑐3 = 𝜌𝑅0  𝑐4  (adventitia)  194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  kg ∙ m−3  𝜌𝑅0  𝑚  (adventitia)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  kg ∙ m−3  𝜆ℎ  𝑐𝑗  11  −  𝜆ℎ  𝑚  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='1  −  𝑇𝑒  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' 16  𝑦ears  𝑇𝑐𝑗  101  days  𝑇𝑚  101  days  𝜎𝑎𝑐𝑡𝑚𝑎𝑥  54  kPa  𝜆0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8  −  𝜆𝑚  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='4  −  𝑘𝑠𝑝𝑟𝑖𝑛𝑔  1000  kPa ∙ mm−1  14  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a patient-specific artery  In this last test case, our dissection model was applied to a patient-specific human  descending thoracic aortic segment, as shown in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=" It was taken from a pa-  tient-specific aortic arch geometry, reconstructed from a patient's CT scan [16], as  shown in Figure 4b." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The exact location of the modeled aortic segment is shown in  Figure 4b with the blue mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The thickness of the adventitia and media was as-  sumed to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Besides, the two layers were separated with a constant gap of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='01 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The mesh was hexahedral and composed of 6 × 48 × 42 elements (thick-  ness × circumferential × length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Similar to previous test cases, we assumed that  this initial geometry was related to a homeostatic state, with an inner true lumen  pressure of 80 mmHg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The reference pressure in the false lumen was assumed to be  the same as in the true lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' However, this value may change as a sensitivity study  was performed on the false lumen pressure in this patient-specific case test, which  is detailed in the results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Moreover, it should be mentioned that the pure  sliding boundary condition was applied on the cross-section at the two extremities  and the outer surface of the adventitia was let free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The same material properties  and simulation parameters as summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Finally, it is worth noting that  the non-uniform prestretches were used in this patient-specific artery, which was  computed based on an iterative method previously developed by Laubrie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 4 - (a) Schematic representation of the patient-specific human descending thoracic aor-  tic segment on which the presented dissection model was applied for modeling G&R after  cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' (b) Illustration of the modeled aortic segment location, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=', regions covered by the  blue mesh, with respect to the whole patient-specific aortic arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  In order to better describe the initial dissection tear of cTBAD, we introduce a  numerical parameter, 𝛼, for each circumferential cross-section of the arterial seg-  ment, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Note that the unit vector normal to this cross-section is  computed from the arterial centerline points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The unit vertical direction of the cross-  section is approximately defined as the averaged projection vector of this section on  the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=" After the definition of directions, we define 𝛼 for each interfacial con-  necting spring  α = 𝑙  𝐷  (27)  (0,0,0)  (a)  (b)15  where 𝑙 is the length of averaged spring positions, projected on the cross-section's  vertical direction." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The tear opening criterion is thus defined as 𝛼 ≥ 𝛼𝑚𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Therefore,  with 𝛼𝑚𝑖𝑛 = 0, all interfacial springs between the adventitia and media layers will  be broken, creating a full separation of the two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' While if 𝛼𝑚𝑖𝑛 is equal to 1,  no interfacial springs will be broken and thus no presence of the tear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' In this patient-  specific simulation, the effect of the tear opening length to the G&R after cTBAD  was considered, by varying the values of 𝛼𝑚𝑖𝑛 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8 (a narrow tear) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5 (a  wide tear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 5 - Description of the initial tear opening criteria defined on each circumferential cross-  section of the aortic segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Computational details  All simulations were performed on a Macbook Pro with Intel Core i5 and 8 Go  of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The computation time for each simulation takes around 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The  low computation resources prove the computational efficiency of our dissection  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Results  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Validation test case  Results of the validation test case are shown in Figure 6, illustrating the aneurys-  mal expansion of the arterial wall due to elastin loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The evolution of the maximum  inner radius of the aneurysm is shown, in comparison with the reference results [13],  over a period of 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The results indicated that the current two-continuum ar-  terial wall model is in good agreement with the conventional single continuum ar-  terial wall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The aneurysmal expansion tends to recover its stability with a  large gain parameter while a small gain parameter promotes an uncontrolled expan-  sion of the aneurysm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' With this validation test case, we justified the use of such a  two-continuum arterial wall concept for G&R problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  α=1  1  D  D  0 =16  Figure 6 - Evolution of the maximum arterial inner radius over 10 years in response to an initial  sudden elastin loss for both the two-continuum arterial wall model (solid lines) and the refer-  ence single-continuum arterial wall model (dash lines), with three different values of gain pa-  rameter related to collagen G&R [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a cylindrical artery  We first show the reference simulation results in the case of a cylindrical artery,  as illustrated in Figure 7, with respect to a reference value of gain parameter 𝑘𝜎  𝑐𝑗 =  𝑘𝜎  𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It can be seen that the dissected part of the artery, especially the outer  adventitia layer, continues to dilate due to the effect of G&R after the initial tear  opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' This aneurysmal dilatation tends to be unstable, with an increasing growth  rate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Besides, the maximum stress, which is located at the vicinity of the  tear edge, also increases rapidly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Previous studies on G&R, which modeled aneurysm progression but disregarded  effects of the dissection, reported that gain parameters have a determinant effect on  the stability of aneurysmal dilatation[15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' To investigate the effect of this gain  parameter in the specific context of cTBAD, we considered three different values  of gain parameters, ranging between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='15, with results illustrated in Fig-  ure 8, showing the temporal evolution of the maximum outer diameter of the dis-  sected cylindrical artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It can be seen that the same tendency as previously re-  ported in the literature has been observed for G&R after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' A small gain  parameter tends to induce an unstable aneurysmal degeneration while a large gain  parameter tends to favor the stability of aneurysmal dilatation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  mm  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  kci=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='2  kci = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='09  kci = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='13  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='9  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='6  Maximum  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='3  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  0  2  4  6  8  10  Time[years]17  Figure 7 - Reference results of the cylindrical artery with respect to a reference value of gain  parameter 𝑘𝜎  𝑐𝑗 = 𝑘𝜎𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05, showing geometrical and equivalent von Mises stress evolutions  after cTBAD over 9 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 8 - Influence of the gain parameter to aneurysmal dilatation after cTBAD, showing the  temporal evolution of the maximum outer diameter of the dissected cylindrical artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a toric artery  Reference simulation results in the case of a toric artery, with a reference value  of gain parameter 𝑘𝜎  𝑐𝑗 = 𝑘𝜎  𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05, are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Similarly, the dissected  part of the artery undergoes an unstable aneurysmal dilatation over 6 years after the  initial tear opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The maximum stress also increases over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Besides, it is  interesting to note that for this toric artery case, the stress seems to increase over the  whole dissected adventitia layer, and is not limited to the vicinity of the tear edge  as observed in the previous cylindrical artery case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  The effect of the gain parameter has also been investigated in this dissected toric  artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The temporal evolution of the maximum outer diameter of the dissected toric  t=0years  t=6years  OyM [MPa]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='12  t=3years  t = 9 yearsImm  lartery  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='3  Maximum outerradius of dissected  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05  kci  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='10  kct  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='15  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='2  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='7  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='2  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='6  0  2  4  6  9  11  Time [years]18  artery is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It can be seen that the results obtained are also in  agreement with previous findings of the gain parameter: a large gain parameter fa-  vors a stable growth of aneurysm while a small gain parameter promotes an exces-  sive enlargement of aneurysm after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 9 - Reference results of the toric artery with respect to a reference value of gain parameter  𝑘𝜎  𝑐𝑗 = 𝑘𝜎𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05, showing geometrical and equivalent von Mises stress evolutions after  cTBAD over 6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 10 - Influence of the gain parameter to aneurysmal dilatation after cTBAD, showing the  temporal evolution of the maximum outer diameter of the dissected toric artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Application to a patient-specific artery  The reference simulation results in the case of a patient-specific artery, more  precisely, a patient-specific human descending aortic segment, are shown in Figure  11, with respect to a reference value of gain parameter 𝑘𝜎  𝑐𝑗 = 𝑘𝜎  𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The re-  sults illustrate the geometrical and stress evolutions of the aortic segment, and also  t =0years  t=3years  OvM [MPa]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='51  t=6years[mm]  Maximum outer radius of dissected artery [  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='7  kci  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='10  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='15  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='3  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='1  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8  0  2  4  6  9  11  Time [years]19  the circumferential cross-section at its dissected extremity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' We found that the dis-  sected aortic segment undergoes continuous aneurysmal dilatation overtime after  the initial tear opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Besides, it can be seen that there is a significant increase of  stress over time, mostly on the dissected part of the outer adventitia layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 11 - Reference results of the patient-specific artery with respect to a reference value of  gain parameter 𝑘𝜎  𝑐𝑗 = 𝑘𝜎  𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05, showing geometrical and equivalent von Mises stress evolu-  tions after cTBAD over 6 years, as well as the dilated circumferential cross-section at its dis-  sected maximum extremity, respectively at (a) 0 year of G&R, (b) 3 year of G&R and 6 year  of G&R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Similarly, the effect of the gain parameter on aneurysmal dilatation after cTBAD  was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Results obtained confirm the same tendency as observed in pre-  vious test cases: a large gain parameter tends to stabilize the aneurysmal dilatation  of the dissected artery while a small gain parameter promotes an excessive aneurys-  mal degeneration after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 12 - Influence of the gain parameter to the aneurysmal dilatation after cTBAD, show-  ing the temporal evolution of the maximum outer diameter of the dissected patient-specific  artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  t= 0years  t =3years  t =6years  OvM [MPa]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='19  (a)  (b)outer radius of dissected aorta[mm]  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='1  kct = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='05  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  kei  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='10  k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='15  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='7  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  Maximum  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='3  0  2  3  5  7  8  Time[years]20  Apart from the gain parameter, it has also been reported in the literature that the  tear size may have a significant influence on aneurysmal dilatation after cTBAD  [9], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Being aware that effects of the tear size is very complex, which depend  not only on the position of the tear but also on its irregular shape, involving gener-  ally measures in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' In this paper, note that we try to study only the  cicumferential length of the tear, with four different values of 𝛼𝑚𝑖𝑛 ranging from  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=', from narrow circumferential tear to wide circumferential tear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Re-  sults obtained are shown in Figure 13, describing the temporal evolution of the max-  imum outer diameter of the dissected artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It can be seen that a wider tear pro-  motes an uncontrolled aneurysmal dilatation after cTBAD, while a narrow tear  tends to favor a stable aneurysmal dilatation with a moderate growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Figure 13 - Influence of the initial circumferential opening length of dissecting tear to the an-  eurysmal dilatation after cTBAD, showing the temporal evolution of the maximum outer di-  ameter of the dissected patient-specific artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Finally, in this patient-specific simulation, we also investigated the effect of the  pressure as it has been identified as a high-risk factor in cTBAD, especially in the  false lumen where the pressure may impact directly on the stress distribution of the  most weakened outer adventitia layer [6], [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' In order to evaluate the effect  of the pressure in the false lumen, three different values of false lumen pressure are  considered, respectively higher than the true lumen pressure (𝑝𝐹𝐿 𝑝𝑇𝐿  ⁄  > 1), equal  to the true lumen pressure (𝑝𝐹𝐿 𝑝𝑇𝐿  ⁄  = 1), and lower than the true lumen pressure  (𝑝𝐹𝐿 𝑝𝑇𝐿  ⁄  < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Note that pressure in the true lumen is assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Re-  sults are shown in Figure 14, showing the temporal evolution of the maximum outer  diameter of the dissected artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It can be seen that a higher pressure promotes an-  eurysmal dilatation after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' More precisely, regarding the aneurysmal dilata-  tion rate at 3 years, a 10% pressure increase in the false lumen induces an "enlarg-  ing" growth of the dissected artery (≥ 3mm/year) while for false lumen pressure  equal or below the true lumen pressure, aneurysmal dilatation can be considered as  "stable" (< 3mm/year) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  mm  Taorta  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='4  amin =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  dissected  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='9  amin =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='6  αmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='7  αmin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='8  outer radius of  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='6  Maximum  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='2  0  2  3  5  7  8  Time[years]21  Figure 14 - Influence of the pressure in the false lumen to the aneurysmal dilatation after  cTBAD, showing the temporal evolution of the maximum outer diameter of the dissected pa-  tient-specific artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Discussion  cTBAD is associated with poor long term outcomes, mainly as a result of exces-  sive aneurysmal dilatations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' By consequence, a considerable part of patients with  cTBAD will require ultimately surgical interventions such as endovascular repair  or open surgery [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' However, there is a serious lack of risk assessment tools  because our current understanding of the aneurysmal dilatation mechanism after  cTBAD remains weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  In this chapter, we proposed a numerical approach to study the role of G&R in  aneurysmal dilatation in cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' We found that the G&R process triggers naturally  the aneurysmal dilatation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Moreover, it was found that with a large gain parameter  related to collagen G&R, the aneurysmal dilatation tends to be stable while with a  small gain parameter, there would be an excessive aneurysmal degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It is  interesting to note that the results obtained are in agreement with clinical evidence  reported by Juvonen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' [6], where older patients present a higher risk of aneurys-  mal rupture in case of cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' In fact, this gain parameter describes the capacity of  arteries to restore its tensional equilibrium state in case of a disturbance of its mech-  anobiological equilibrium [14] and it has been reported that age may affect this gain  parameter with older patients generally having an impaired stress-regulated repair  mechanism compared to young patients [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Based on sensitivity analysis performed on patient-specific simulations, we  found that the circumferential tear length also has a significant influence on the  G&R process after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' A wide tear promotes an unstable development of an-  eurysmal dilatation while a narrow tear reduces the risk of uncontrolled aneurysmal  dilatation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The reason could be twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' First, a wide initial opening tear means  mm  aorta  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='2  dissected  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='7  radiusof  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='5  outer  PFL/PTL= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='1  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='3  PFL/PTL = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  laximum  PFL/PTL= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='9  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='0  0  2  3  4  6  Time[years]22  naturally a larger initial dissected arterial diameter once the false lumen is pressur-  ized compared to a narrow initial tear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Secondly, the consequence of a larger initial  dissected arterial diameter is that the deformation and stress will be much higher,  especially in the dissected outer adventitia layer, accelerating the G&R process of  the artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Indeed, the results remained very limited, considering the irregular three-  dimensional tear shape and other complex mechanobiological phenomena ne-  glected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' However, it provides a mechanical proof that the tear size is also an im-  portant influencing factor that needs to be considered in the risk assessment of pa-  tients with cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Finally, our results indicated that the pressure in the false lumen has a determi-  nant role in the aneurysmal progression rate of the dissected artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' We found that  a relative 10% increase of pressure in the false lumen, compared to that of the true  lumen, is sufficient to promote an "unstable" growth of the dissecting aneurysm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  This can be critical for patients, as surgical interventions are usually recommended  for such situations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Despite the above promising results obtained, the current model still has some  shortcomings that could be addressed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' First, in the present model, the  tear configuration remains fixed after its initial creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' However, with the contin-  uous aneurysmal dilatation and accumulation of stress especially near the edge of  the tear, the initial tear may propagate due to high-stress concentration and thus alter  the G&R process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Therefore, integration of the tear propagation models, such as  that reported in the literature [32]-[35], could be an essential step to build a more  reliable dissection model for evaluating the G&R effect to aneurysmal dilatation  after cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Secondly, the dissection model is relatively simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Intraluminal thrombus,  which was often reported to have an important role in dissecting aneurysm pathol-  ogies and rupture [36], [37], was literally neglected in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Besides, the effects  of surrounding tissues have also been neglected although it has already been re-  ported in the literature that the surrounding connective tissues or vertebral column  may impact regional adaptation of aortic walls by changing the local wall stress  distribution or even more directly the aneurysmal aortic shape [38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Moreover,  potential dynamic effects of the blood flow inside the arterial wall are currently not  taken into consideration, despite its non-negligible effect directly on the wall stress  distribution, as reported in the literature [26], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Finally, the initial tear configuration was restricted to its circumferential length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Location of the tear, number of the tears or other dimensions related to the tears are  currently neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' For further patient-specific simulations, these parameters  should be carefully considered as they may directly impact the pressure in both the  false lumen and the true lumen [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Conclusions  23  In summary, we introduced an efficient 3D finite-element model, based on an  open-source in-house code, to model the aneurysmal dilatation due to G&R after  cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' We showed the potential of this dissection model to simulate G&R process  after cTBAD, from simple test cases with idealized arterial geometries to a more  relevant case with a patient-specific geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The effects of different parameters  on aneurysmal dilatation were assessed through a comprehensive sensitivity analy-  sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' It was found that the gain parameter related to collagen G&R as well as the  circumferential initial tear length, has an undeniable impact on the stability of the  dissecting aneurysm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' Moreover, our results indicated that the stability of the dis-  secting aneurysm is very sensitive to the intraluminal false lumen pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' A rela-  tive pressure increase of 10% in the false lumen may induce an excessive aneurys-  mal degeneration in patients with cTBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content='  Future work is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The first one is coupling the present dissection model  with tear propagation models, for applications to more reliable patient-specific sim-  ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
+page_content=' The second one is to account for a more accurate configuration of opening  tears while considering the potential dynamic effects of the blood flow inside the  dissected arteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQfqAGQ/content/2301.02547v1.pdf'}
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diff --git a/otE1T4oBgHgl3EQf1wXJ/content/tmp_files/2301.03472v1.pdf.txt b/otE1T4oBgHgl3EQf1wXJ/content/tmp_files/2301.03472v1.pdf.txt
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@@ -0,0 +1,1414 @@
+MNRAS 000, 1–13 (2022)
+Preprint 10 January 2023
+Compiled using MNRAS LATEX style file v3.0
+The evolution of phase space densities in star-forming regions
+George A. Blaylock-Squibbs⋆ and Richard J. Parker†
+Department of Physics and Astronomy, The University of Sheffield, Hounsfield Road, Sheffield, S3 7RH
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The multi-dimensional phase space density (both position and velocity) of star-forming regions may encode information on
+the initial conditions of star and planet formation. Recently, a new metric based on the Mahalanobis distance has been used
+to show that hot Jupiters are more likely to be found around exoplanet host-stars in high 6D phase space density, suggesting
+a more dynamic formation environment for these planets. However, later work showed that this initial result may be due to a
+bias in the age of hot Jupiters and the kinematics of their host stars. We test the ability of the Mahalanobis distance and density
+to differentiate more generally between star-forming regions with different morphologies by applying it to static regions that
+are either substructured or smooth and centrally concentrated. We find that the Mahalanobis distance is unable to distinguish
+between different morphologies, and that the initial conditions of the N-body simulations cannot be constrained using only
+the Mahalanobis distance or density. Furthermore, we find that the more dimensions in the phase space the less effective the
+Mahalanobis density is at distinguishing between different initial conditions. We show that a combination of the mean three-
+dimensional (x, y, z) Mahalanobis density and the Q-parameter for a region can constrain its initial virial state. However this is
+due to the discriminatory power of the Q-parameter and not from any extra information imprinted in the Mahalanobis density.
+We therefore recommend continued use of multiple diagnostics for determining the initial conditions of star-forming regions,
+rather than relying on a single multi-dimensional metric.
+Key words: galaxies: star formation – methods: statistical – methods: numerical
+1 INTRODUCTION
+Star formation is observed to take place along filaments within giant
+molecular clouds (Palmeirim et al. 2013; Schisano et al. 2014; André
+2017). The initial formation and distribution of these filaments is
+likely due to supersonic turbulence within GMCs (Larson 1981). It is
+along these filaments that cores can form, with further fragmentation
+of these cores leading to stars forming in groups containing 10s to
+1000s of members (Lada & Lada 2003; Bastian et al. 2009).
+One of the foundational questions of star formation is whether
+star formation is a universal process or not. Are the initial condi-
+tions of star-forming regions dependant on the environment, where
+differences in the stellar density, IMF and stellar multiplicity are due
+to the initial conditions of the star-forming region? Or does star for-
+mation happen in a similar way everywhere, and any differences we
+observe in these regions is stochastic in nature?
+There are two main proposed modes of star-formation, monolithic
+and hierarchical. In monolithic formation modes the gas is already
+contained within the final volume of the region before stars begin
+to form, whereas in hierarchical formation the gas extends beyond
+the final volume of the region (Longmore et al. 2014; Williams et al.
+2022). In the hierarchical mode stars are forming while at the same
+time the gas is collapsing.
+The kinds of star-forming regions these modes produce is of in-
+⋆ E-mail: gablaylock-squibbs1@sheffield.ac.uk
+† Royal Society Dorothy Hodgkin fellow
+terest not only in the context of star formation but also the way in
+which the final star-forming regions that form may influence the ar-
+chitecture of the planetary systems that are produced within them
+(Adams 2010; Parker 2020).
+Star-formation is a rapid process, occurring within a few cross-
+ing times (Elmegreen 2000), which is often less than 1 Myr. Dur-
+ing this process, stars are forming and moving (Alcock & Parker
+2019), further muddying the formation picture. And whilst observa-
+tions of the earliest stages have improved greatly with e.g. ALMA,
+observations of star-forming regions are often at older ages, where
+significant dynamical evolution may have taken place (Klessen &
+Kroupa 2001; Allison et al. 2010; Parker et al. 2014; Daffern-Powell
+& Parker 2020; Schoettler et al. 2019). Dynamical evolution alters
+the spatial and kinematic distributions of young stars, erasing the
+signature of the initial conditions, but can be used as a proxy for age
+and used to converge on a likely set of initial conditions for a given
+star-forming region (Parker et al. 2014). To enable comparisons of
+observations and simulations, we need to be able to quantify param-
+eters of the star-formation regions, such as the degree of substructure
+and mass segregation (Cartwright & Whitworth 2004; Allison et al.
+2009; Sánchez & Alfaro 2009; Alfaro & González 2016; González
+& Alfaro 2017; Jaffa et al. 2017; Buckner et al. 2019; Arnold et al.
+2022; Joncour et al. 2018; Kuhn et al. 2014; Gouliermis et al. 2014).
+Early methods such as the auto-correlation function and two-point
+correlation function compared the number of excess star pairings
+to a random distribution of stars as a function of scale (Gomez
+et al. 1993; Larson 1995). These methods where used extensively
+© 2022 The Authors
+arXiv:2301.03472v1  [astro-ph.GA]  9 Jan 2023
+
+2
+G. A. Blaylock-Squibbs & R. J. Parker
+to determine the degree of substructure, with early work suggest-
+ing breaks in the two-point correlation function corresponded to the
+Jeans length (Simon 1997) (though see Bate et al. (1998)) and the
+size of the widest stellar binaries in the regions in question (Kraus &
+Hillenbrand 2008; Joncour et al. 2017).
+Subsequent work made extensive use of minimum spanning trees
+(MSTs) to quantify structures in star-forming regions. Cartwright
+& Whitworth (2004) introduced the Q-parameter to quantify spatial
+substructure, and Allison et al. (2009) introduced the ΛMSR method
+to quantify mass segregation.
+Parker et al. (2014) showed that the initial conditions of a region
+can be inferred from the spatial information, if a suitable number of
+metrics are combined, including the relative stellar surface density
+around the most massive stars (Maschberger & Clarke 2011; Küpper
+et al. 2011).
+However, the majority of the above methods are designed to oper-
+ate on two or three-dimensional spatial data, whereas recent obser-
+vational data (e.g. from Gaia and associated ground-based surveys)
+has provided high resolution spatial and kinematic data (6D).
+Recently, in an attempt to quantify the phase space densities
+of exoplanet host stars Winter et al. (2020) developed the Maha-
+lanobis density, a new application of the Mahalanobis distance (Ma-
+halanobis 1936).
+The Mahalanobis distance has been used in astronomy for classi-
+fying objects, for example in Siegal & Griffiths (1974) it is used to
+analyse and classify different types of asteroid impact craters and in
+Jakimiec et al. (1991) it was used to classify sunspots into groups.
+Due to the differing dimensions, and units of very different scale
+(i.e. length in pc and velocity in kms−1) making multivariate com-
+parisons can be difficult. However the Mahalanobis distance makes
+multivariate comparisons possible over wide ranges of dynamical
+scales by rescaling the axes and removing the units. Winter et al.
+(2020) used this method to develop the Mahalanobis density and use
+it to propose the hypothesis that host stars in high phase space den-
+sities are more likely to have hot Jupiter planets (M > 50 M� and
+a < 0.2 au) around them compared to the lower phase space densi-
+ties. However, Mustill et al. (2022) show that this result may be due
+to a bias from the peculiar velocities of the stars. When the peculiar
+velocities of the stars are accounted for, there is no longer an excess
+of hot Jupiters in high 6D (x, y, z, Vx, Vy, Vz) phase space densities.
+Irrespective of the ongoing debate surrounding the application of
+the Mahalanobis distance to exoplanet host stars, in this paper we
+aim to test this metric when applied to both synthetic static regions
+and assess its performance in quantifying phase space structures of
+N-body simulations of star-forming regions.
+The paper is structured as follows. In § 2 we present the meth-
+ods used. In § 3.1 we present the results of testing the Mahalanobis
+distance’s ability to differentiate between different morphologies. In
+§ 3.2 we show how the Mahalanobis distance and density change
+with time in N-body simulations. In § 3.3 we compare the Maha-
+lanobis distance and density to other methods for quantifying struc-
+ture in star-forming regions. In § 4 we present a discussion of our
+results and we conclude in § 5.
+2 METHODS
+In this section we describe the set-up of the N-body simulations,
+including the initial spatial and kinematic distributions of stars. We
+then describe the methods used to quantify the spatial and kinematic
+distributions in our simulated star-forming regions.
+(a) 1 pc radius Df = 1.6
+(b) 1 pc radius Df = 3.0
+Figure 1. Examples of fractal regions with 1000 stars, on the left-hand side
+is a highly substructure region of fractal dimension Df = 1.6 and on the right-
+hand side is a region with far less substructure with fractal dimension Df =
+3.0.
+2.1 N-Body Simulations
+The N-body simulations are run using the Kira integrator, part of
+the Starlab1 package (Portegies Zwart et al. 1999, 2001). Each
+simulation has a population of 1000 stars and is run for a simulated
+time of 10 Myr. Our choice of 1000 stars comes from Lada & Lada
+(2003) where they find the following relation,
+Ncl ∝ M−2
+cl
+(1)
+where Ncl is the number of clusters and Mcl is the mass of the cluster.
+This power-law is obeyed for star clusters between masses of 10 <
+Mcl/M⊙ < 105 and our choice of 1000 stars puts our simulations
+close to the middle of this distribution.
+Observations of star forming regions show that within these com-
+plexes there are filaments of denser gas, inside of which prestellar
+cores are observed (Myers 2009; André 2017). These filaments are
+thought to be caused by supersonic turbulence, which is likely to be
+responsible for the substructure we observe in these regions (Larson
+1981; Kraljic et al. 2014). To mimic this substructure we make use
+of a box-fractal generator to initialise our simulations with fractal
+dimensions of Df = 1.6 and Df = 3.0 (Goodwin & Whitworth 2004;
+Daffern-Powell & Parker 2020).
+We run subvirial simulations to match observations that indicate
+prestellar cores may be subvirial with respect to one another and
+may then virialise and form bound smooth, centrally concentrated
+star clusters (Foster et al. 2015; Kuznetsova et al. 2015; Parker et al.
+2016). We also run a set of simulations that are initially supervirial.
+We do this to mimic the observations that some young star-forming
+regions (∼ 1−5 Myr) are supervirial and therefore expanding (Bravi
+et al. 2018; Kuhn et al. 2019; Kounkel et al. 2022).
+Figure 1 shows example clusters with the fractal dimensions Df =
+1.6 and Df = 3.0.
+We use the following to define the virial ratio,
+αvir = T
+|Ω|,
+(2)
+where T is the total kinetic energy of the region and Ω is the total
+potential energy of the region. By using this equation we can scale
+the initial velocities to our desired virial ratio either αvir = 0.1 for
+subvirial regions or αvir = 0.9 for supervirial regions. The initial
+conditions for the simulations are summarised in table 1.
+We assign masses using the Maschberger IMF with a lower mass
+1 https://www.sns.ias.edu/~starlab/index.html
+MNRAS 000, 1–13 (2022)
+
+1.0
+0.5
+J
+(d)
+":
+4
+0.0
++.*
+y
+-0.5
+Df = 1.6
+-1.0
+-0.5
+0.0
+0.5
+-1.0
+1.0
+x (pc)1.0
+0.5
+(pc)
+0.0
+y
+-0.5
+：
+D = 3.0
+-1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x (pc)Evolution of phase space densities in SFR
+3
+Table 1. This table shows the different initial conditions of the simulations.
+For each of these initial conditions 10 simulations are run for 10 Myr. From
+left to right the columns are, the initial fractal dimension of the region, the
+number of stars, the initial virial ratio and the initial radius of the simulations
+in pc.
+Fractal Dimension
+N⋆
+Virial Ratio
+Radius (pc)
+Df = 1.6
+1000
+0.1, 0.9
+1, 5
+Df = 3.0
+1000
+0.1, 0.9
+1, 5
+limit of 0.01 M⊙, upper mass limit of 50.0 M⊙ and a mean mass of
+0.2 M⊙ (Maschberger 2013).
+The probability distribution function of the Maschberger IMF is,
+p(m) ∝
+�m
+µ
+�−α �
+1+
+�m
+µ
+�1−α�−β
+,
+(3)
+where µ is the mean stellar mass, α = 2.3 is the high mass exponent
+and β = 1.4 is the low mass exponent (Salpeter 1955).
+2.2 Smooth Centrally Concentrated Regions
+We generate smooth, centrally concentrated regions with radial den-
+sity profiles in which the stars are randomly distributed using the
+following relation,
+n ∝ r−α,
+(4)
+where r is the distance from the centre of the region and α is the
+radial density index and has values α = 0.0,1.0,2.0 and 2.5. We also
+generate Plummer spheres, regions with a three-dimensional density
+distribution of the form,
+ρp(r) = 3M0
+4πa3
+�
+1+ r2
+a2
+�− 5
+2
+,
+(5)
+where M0 is the total mass of the region, r is the distance from the
+centre of the region and a is the Plummer radius (Plummer 1911;
+Kroupa 2008).
+2.3 Generating Fractal Regions
+We follow Goodwin & Whitworth (2004) and Cartwright & Whit-
+worth (2004) to generate substructured regions using the box-fractal
+method. Other examples of this method can be found in (e.g. Allison
+et al. (2009), Parker & Goodwin (2015), Daffern-Powell & Parker
+(2020).
+The method proceeds as follows. A single star is placed at the
+centre of a cube whose side length is chosen to be NDiv. This cube
+is then subdivided down into N3
+Div sub-cubes. A star is then placed
+at the centre of each of the sub-cubes. Each of these sub-cubes then
+has a probability of being subdivided again given by N(3−Df)
+Div
+, where
+Df is the fractal dimension. Cubes that are not subdivided have their
+stars removed along with any previous generations of stars that came
+before them. A small amount of noise is added to each of the stars
+to prevent them having a regular looking appearance.
+These steps are repeated until the desired number of stars is
+reached or exceeded in the latest generation. Once this condition is
+met all previous generations of stars are removed, then the remain-
+ing stars are randomly removed until the desired number of stars is
+reached. By removing the stars in this manner we end up with stars
+distributed inside of a spherical volume.
+The velocities of the first generation of stars are picked from a
+Gaussian with mean zero, with each subsequent generation of stars
+inheriting the velocity of the previous generation plus a random
+component. This results in stars that are close to each other hav-
+ing similar velocities and stars far apart from one another having
+very different velocities. The velocities are related to the length scale
+of the region with the following relation V(L) ∝ L3−Df where L is
+the length scale of interest and Df is the desired fractal dimension
+(Parker & Wright 2018).
+2.4 The Mahalanobis Distance
+The Mahalanobis distance is a metric that measures distances be-
+tween points to the average in the distribution in N dimensional
+phase spaces (Mahalanobis 1936).
+The Mahalanobis distance does this by removing any correlations
+in the data by multiplying the distances between points and the av-
+erage of the region by the inverse of the covariance matrix; this also
+has the effect of re-scaling the data.
+Once this re-scaling has been done the Euclidean distances are
+found in the phase space; this is the Mahalanobis distance, Md.
+Each point in a dataset is described using a vector where each
+element is a measured parameter of that point,
+⃗x = (x1,x2,x3,...,xN)T ,
+(6)
+where x1,x2,x3,...,xn are the parameters. For example, if each point
+has the three parameters, (x, y, z) then this is simply its physical
+position in 3D space.
+The Mahalanobis distance between a point in a distribution and
+the mean of that distribution in an N dimensional phase space is
+defined as,
+Md(⃗x,⃗µ) =
+�
+(⃗x−⃗µ)T S−1 (⃗x−⃗µ),
+(7)
+where the ⃗x is the point vector, ⃗µ is a vector of the averages of the
+parameters of interest and S−1 is the inverse of the covariance matrix
+for all the parameters in the region.
+The Mahalanobis distance, Md has been also used to define a pa-
+rameter space density called the Mahalanobis density, ρm,N (Winter
+et al. 2020)2.
+To calculate the Mahalanobis density we first must define the Ma-
+halanobis distance between points in the phase space (i.e. distance
+between⃗x and⃗y). We follow Winter et al. (2020) and use,
+md(⃗x,⃗y) =
+�
+(⃗x−⃗y)T S−1 (⃗x−⃗y),
+(8)
+where ⃗x is the vector describing the measurements of one point, ⃗y
+is the vector describing the measurements of another and S−1 is the
+inverse of the covariance matrix of all the parameters of interest.
+The calculation of the Mahalanobis density proceeds as follows.
+First we find the Mahalanobis distance to the Nth nearest neighbour,
+then we divide the nearest neighbour number by the volume whose
+side length is defined as the Mahalanobis distance to the Nth nearest
+neighbour. The Mahalanobis density is then defined as,
+ρm,N = Nm−Dp
+d,N ,
+(9)
+2 We have used different notation for the Mahalanobis distance to avoid con-
+fusion with the fractal dimension and also the number of dimensions. We
+instead use Md instead of D as used in Winter et al. (2020) to avoid confu-
+sion with the fractal dimension. We also change the number of dimensions
+in the phase space from D to Dp again to avoid confusion with the fractal
+dimension, which we represent as Df.
+MNRAS 000, 1–13 (2022)
+
+4
+G. A. Blaylock-Squibbs & R. J. Parker
+where ρm,N is the Mahalanobis density, N is the nearest neighbour
+number, md,N is the Mahalanobis distance to the Nth nearest neigh-
+bour and Dp is the number of dimensions in the phase space (Winter
+et al. 2020). The Mahalanobis densities are then normalised so that
+the median Mahalanobis density is unity.
+In this work we apply the Mahalanobis density to two differ-
+ent phase spaces, the positional phase space (3D) and the position-
+velocity phase space (6D). For this work we find the Mahalanobis
+distance to the 20th nearest neighbour in the phase space (i.e. N =
+20), the same as in Winter et al. (2020).
+2.5 Local Surface density ratio
+The local surface density ratio ΣLDR was introduced in Maschberger
+& Clarke (2011) to quantify the differences between the surface den-
+sities of subsets of stars within their host regions and for this work
+we choose the 10 most massive stars as the subset of interest. The
+algorithm proceeds as follows. For each star we find the distance to
+its Nth nearest neighbour, then we calculate the circular area whose
+radius is the distance to the Nth nearest neighbour. To find the sur-
+face density of the stars we divide the nearest neighbour number by
+this area. We use a nearest neighbour number of 5 for this work.
+The ratio is defined as,
+ΣLDR =
+˜Σsubset
+˜Σall
+(10)
+where ˜Σsubset is the median surface density found for the 10 most
+massive stars and ˜Σall is the median surface density found for the
+entire region. Therefore, if ΣLDR > 1 the 10 most massive stars are
+found in areas of higher than average stellar surface density, and con-
+versely, if ΣLDR < 1 then they are located in areas of lower than aver-
+age surface density. The significance of any difference is quantified
+using a two-sample Kolmogorov-Smirnov test. Where if p << 0.01
+we reject the null hypothesis that the 10 most massive stars share the
+same underlying distribution of surface densities compared to the
+entire region.
+2.6 Mass Segregation Ratio
+The mass segregation ratio ΛMSR was first introduced in Allison
+et al. (2009) to quantify the degree of mass segregation in a star-
+forming region. The definition of mass segregation in this case is that
+the most massive stars are closer to each other than expected from
+the average separation of all of the stars in the region. The method
+makes use of minimum spanning trees (MSTs) which are graphs of
+points connected to each other in such a way that the total length of
+the tree is minimised and that all points are connected to at least one
+other point with no closed loops.
+This method generates a minimum spanning tree for the chosen
+subset of stars, for this work we use the 10 most massive stars. It
+will then pick 10 random stars from the region and make an MST
+for these random stars. We do this 200 times to calculate the mean
+edge length of the randomly chosen trees. The ratio is calculated
+using the following equation,
+ΛMSR =
+�
+laverage
+�
+l10
++σ5/6/l10
+−σ1/6/l10
+,
+(11)
+where
+�
+laverage
+�
+is the average edge length found for all the randomly
+constructed MSTs and l10 is the edge length of the subset’s MST. It
+is important to note that the random MSTs we construct can also
+contain members of the chosen subset.
+If the ratio is > 1 then the region’s 10 most massive stars are mass
+segregated, if the ratio is ∼ 1 then the most massive stars are not
+mass segregated and if the ratio is less than 1 they are inversely mass
+segregated (the most massive stars are further apart than the average
+stars in the region). In this work we mark the value 1 to show the
+boundary between mass segregation and inverse mass segregation.
+However we follow Parker & Goodwin (2015) and only take ratio
+values above 2 to be signs of mass segregation, to avoid false posi-
+tives.
+We follow Parker (2018) and calculate the uncertainty using the
+randomly constructed MSTs. First we order the lengths of the ran-
+dom MSTs and find the values that lie 1/6 and 5/6 of the way
+through this list. This gives us values which correspond to a 66 per
+cent deviation from the median MST length found.
+2.7 Q-Parameter
+The Q-Parameter was introduced in Cartwright & Whitworth (2004)
+to quantify and distinguish between different cluster morphologies.
+The Q-parameter also makes use of MSTs and proceeds as fol-
+lows. First the normalised correlation length is found. This is the
+mean separation between all stars in a region which is then divided
+by the region’s radius to normalise it.
+The mean edge length of the region is found by constructing an
+MST for the region and then finding the mean edge length. The mean
+edge length is normalised by diving it by
+NtotalA
+Ntotal−1, where Ntotal is the
+number of stars in the region and A is the area of the region. We use
+the circular area (see Schmeja & Klessen (2006); Parker (2018) for
+a discussion on normalisation techniques), with the radius defined
+as the distance from the centre of mass to the most distant star. The
+Q-parameter is then defined as,
+Q = ¯m
+¯s ,
+(12)
+where ¯m is the normalised mean edge length of the MST and ¯s is
+the normalised correlation length between stars. Regions with sub-
+structured morphologies have Q < 0.8 whereas regions with smooth,
+centrally concentrated morphologies have Q > 0.8.
+3 RESULTS
+We show the results of the Mahalanobis distance applied to both
+static and N-body simulations of star-forming regions with various
+initial conditions. We present the 3D and 6D Mahalanobis densities
+calculated in the N-body simulations and compare the evolution of
+the Mahalanobis density over time to the other methods for quanti-
+fying spatial and kinematic distributions in star-forming regions.
+3.1 Static Regions
+We first find the Mahalanobis distances between stars and the aver-
+age point in the region, ⃗µ, and calculate the average Mahalanobis
+distance in their respective regions ( ¯Md) for sets of synthetic and
+static star clusters, with each set having a different structural parame-
+ter. Each region in the set consists of 1000 stars. We calculate ¯Md for
+substructured regions with fractal dimensions Df = 1.6,2.0,2.6,3.0
+and clusters with radial density profile indexes, α = 0.0,1.0,2.0,2.5.
+We also show the results of a set of 100 Plummer spheres which have
+a radial density profile described by equation 5 with a radial density
+index of 2.5 (see § 2.2).
+MNRAS 000, 1–13 (2022)
+
+Evolution of phase space densities in SFR
+5
+Figure 2. Mean of the mean Mahalanobis distances calculated in the 3D
+phase space for sets of 100 different star-forming regions plotted against the
+structural parameter used to make the sets. The red triangles are the smooth,
+centrally concentrated radial regions, the purple star (on top of the red trian-
+gle with the structural parameter equal to 2.5) is the Plummer sphere and the
+black crosses are the fractal regions. The error bars show a single standard
+deviation.
+Figure 2 shows the mean of the means for the Md (the Maha-
+lanobis distance of each star to its region’s averages in the 3D phase
+space) for the 100 clusters in each set of initial conditions. We first
+calculate the mean Mahalanobis distance in each of the 100 regions
+in the set and then we calculate the mean of those means. The error
+bars show the standard deviation of the mean of the mean Maha-
+lanobis distances found in each of the regions in a particular set.
+Figure 2 clearly shows that ¯Md is degenerate across a wide range
+of morphologies and we are therefore unable to use ¯Md calculated
+in the 3D phase space to differentiate between the different mor-
+phologies. There is much more spread in the values for the Plummer
+sphere compared to the radial and fractal models, with the fractals
+having the smallest spread of ¯Md and the radial regions sitting be-
+tween the two. This is due to Plummer spheres being formally infi-
+nite in extent, so the calculation occasionally has to normalise over
+very distant stars.
+3.2 N-body Results
+Figure 3 shows the mean of the mean Mahalanobis distances (calcu-
+lated in both 3D and 6D) found for 10 different N-body simulations
+with initial fractal dimension Df = 1.6 and 1 pc radii. The left hand
+panel shows the results for subvirial (collapsing) regions, and the
+right hand panel shows the results for the supervirial (expanding) re-
+gions. In both the sub- and supervirial cases there is a decrease in
+the Mahalanobis distance over time for both the 3D and 6D phase
+spaces. For the initially subvirial simulations the 3D Mahalanobis
+distance swiftly decreases at the start and then continues to decrease
+for the rest of the simulation but at a slower rate. For the supervirial
+regions we see a less pronounced decrease in ¯Md compared to the
+initially subvirial regions. The ¯Md calculated in the 6D phase space
+shows more modest decrease over time for both initially sub- and
+supervirial simulations.
+Figure 4 shows the mean Mahalanobis density ( ¯ρm,20) calculated
+in both the 3D and 6D phase spaces for two sets of 10 (one sub-
+virial and the other supervirial) simulations with an initial fractal
+dimension of Df = 1.6 and initial radii of 1 pc. The highest Ma-
+halanobis densities are calculated in the 3D phase space (x, y, z)
+for the supervirial simulations, however these large final values are
+only present for a few of the simulations. In the 3D phase space
+the Mahalanobis density increases in the first 2-4 Myr, after which
+the density stays the same for the rest of the run time. This is most
+likely due to the early dynamical interactions of stars; as they move
+closer to each other the stars’ Mahalanobis densities will increase. In
+Appendix A we show the relationship between the Mahalanobis dis-
+tance and density for the high density simulations with and without
+substructure. We show this for both the 3D and 6D phase spaces.
+Initially, in the subvirial simulations, the 6D Mahalanobis density
+decreases for the first 1 Myr and then stays the same until around 5
+Myr where it then starts to increase again. For the supervirial sim-
+ulations this initial decrease in the 6D phase space happens more
+rapidly than in the subvirial simulations. It also does not reach the
+low densities that the subvirial regions attain. In the supervirial re-
+gions, the stars will expand together in co-moving groups, therefore
+they will have similar positions but can still have velocities that ex-
+hibit kinematic substructure. In contrast, in the subvirial simulations
+the stars interact more and erase this substructure. The difference
+in the velocities explains why the 6D Mahalanobis density is much
+lower than the 3D density. Some of the supervirial regions attain
+higher 6D Mahalanobis densities compared to the subvirial regions
+at the end of the 10 Myr.
+We also calculate the Mahalanobis density and distance for low
+density simulations for simulations with Df = 1.6 and Df = 3.0,
+where the initial radii regions are 5 pc (with a mean number density
+of around 3 stars pc−3 and a mean stellar mass density of around
+1.6 M⊙ pc−3). We find that the evolution of ¯Md is almost identical
+to the its evolution in the high density simulations, both in 3D and
+6D phase spaces.
+We show the evolution of the 3D and 6D Mahalanobis densities
+for these more diffuse simulations in Figure 5, which shows ¯ρm,20
+plotted against time for the same substructured regions with fractal
+dimension Df = 1.6 with initial radii of 5 pc. For the 3D phase space
+we see the same trends as in Figure 4 but a slight difference for the
+6D phase space. Now ¯ρm,20 decreases in the subvirial simulations
+as the regions evolve, and the supervirial simulations show a steady
+Mahalanobis density after 1 Myr.
+We show the evolution of the Mahalanobis distance (measured be-
+tween the points and the mean values of the phase in each snapshot),
+and Mahalanobis density in the simulations that have no primordial
+substructure (i.e. they are uniform spheres at t = 0 Myr).
+Figure 6 shows the Mahalanobis distance over time for regions
+with a fractal dimension Df = 3.0 with radii of 1 pc. For the 3D
+phase space there is a decrease over time for the sub- and supervirial
+simulations.
+Comparing these results to Fig. 3 we see that the Df = 3.0 re-
+gions’ Mahalanobis distances decrease at a slower rate compared to
+regions with initially more substructure. This behaviour also results
+in a slightly greater 3D Mahalanobis distance being measured for
+regions with fractal dimension Df = 3.0 at 10 Myr. However, this is
+not seen in the 6D phase space Mahalanobis distances which show
+little change over the 10 Myr in the simulations. Also, very little
+difference is seen when comparing the 6D Mahalanobis distances
+between the sub- and supervirial simulations.
+Figure 7 shows the 3D and 6D ¯ρm,20 against time for the sim-
+ulations with no primordial substructure (with an initial fractal di-
+mension Df = 3.0 and radii 1 pc). The left-hand panel shows ¯ρm,20
+against time for the subvirial simulations. It shows the same increase
+in Mahalanobis density as the Df = 1.6 simulations but we don’t see
+MNRAS 000, 1–13 (2022)
+
+4
+3
+2
+M
+1
+0
+Radial
+Fractal
+-1
+Plummer
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+α / Df6
+G. A. Blaylock-Squibbs & R. J. Parker
+(a) Df = 1.6, subvirial, 1 pc
+(b) Df = 1.6, supervirial, 1 pc
+Figure 3. Plots of the mean Mahalanobis distance calculated in both the 3D and 6D phase spaces against time for regions with both high initial volume
+densities and high degrees of substructure (i.e. fractal dimension Df = 1.6 with radii of 1 pc) consisting of 1000 stars. The shaded areas show the range of mean
+Mahalanobis distances found across all 10 of the simulations at the current time. The solid lines show the mean of the mean Mahalanobis distances across all 10
+simulations. The blue area and solid blue line shows the 3D phase space and the black dashed line and the grey area show the same but for the 6D phase space.
+(a) Df = 1.6, subvirial, 1 pc
+(b) Df = 1.6, supervirial, 1 pc
+Figure 4. The mean Mahalanobis density against time for each of the 10 subvirial (left-hand panel) and supervirial simulations (right-hand panel) with fractal
+dimension D = 1.6 and radii of 1 pc. The simulations consist of 1000 stars. The blue shaded area shows the minimum and maximum mean Mahalanobis density
+(in the 3D phase space) found across all 10 of the simulations at the current time. The solid blue line shows the mean of the means for the Mahalanobis density
+in the 3D phase space. The grey shaded area and the dashed black line shows the same but for the Mahalanobis density calculated in the 6D phase space. The
+mean number densities of the 10 simulations is around 314 stars pc−1 with a mean stellar mass density of around 201 M⊙ pc−3.
+the same initial decrease in the 6D ¯ρm,20 that we see in the Df = 1.6
+simulations. At around 0.5 Myr a decrease in the 3D Mahalanobis
+density is seen, then a second period of increasing Mahalanobis den-
+sity is seen around 0.9 Myr before attaining a steady Mahalanobis
+density for the rest of the simulations’ run time. This initial increase
+is due to the region collapsing and stars moving closer to each other
+which raises the 3D Mahalanobis density. What stops it increasing
+further is likely the dynamical interactions causing stars to move fur-
+ther away from each other. Once this initial dynamical stage settles
+down the density can increase again due to stars being close to each
+other near the centre of the region.
+The right-hand panel of Figure 7 shows the 3D and 6D Maha-
+lanobis density calculated for regions that are initially supervirial.
+The ‘bump’ like feature is much smaller for the 3D phase space cal-
+culations than in the initially subvirial simulations. The decrease in
+the bump compared to the subvirial simulations is due to the fact
+that the stars are constantly and continuously moving away from
+each other, meaning that the slight increase that is still present is due
+to small groupings of stars clumping together. We see similar be-
+haviour for the 6D phase space in both the subvirial and supervirial
+simulations.
+We show the 3D and 6D Mahalanobis densities for simulations
+with Df = 3.0 and initial radii of 5 pc in Figure 8. These low den-
+sity simulations have mean number density of around 1 star pc−3, or
+MNRAS 000, 1–13 (2022)
+
+100
+6D
+3D
+10 :
+M
+1
+0
+0
+1
+10
+Time (Myr)100
+10 :
+1
+0
+0
+10
+Time (Myr)100
+6D
+3D
+10
+1
+0
+0
+1
+10
+Time (Myr)100
+10
+pm,20
+1
+0
+0
+7
+10
+Time (Myr)Evolution of phase space densities in SFR
+7
+(a) Df = 1.6, subvirial, 5 pc
+(b) Df = 1.6, supervirial, 5 pc
+Figure 5. Plots showing the mean Mahalanobis density against time for each of the 10 subvirial (left-hand panels) and supervirial (right-hand panels) simulations
+with fractal dimension Df = 1.6 and radii of 5 pc. These simulations have a low initial stellar number density with a mean around 3 stars pc−3 and a mean stellar
+mass density of around 1.6 M⊙ pc−3. The shaded blue area shows the minimum and maximum mean Mahalanobis density found across all 10 simulations in
+the 3D phase space. The solid blue line shows the mean of the means Mahalanobis density against time. The shaded grey area and the dashed black line show
+the same but for the 6D phase space.
+(a) Df = 3.0, subvirial, 1 pc
+(b) Df = 3.0, supervirial, 1 pc
+Figure 6. Plots of the mean Mahalanobis distance from each star to the average in simulations without primordial substructure, i.e. a fractal dimension of
+Df = 3.0, over time. The shaded blue area and solid blue line show the minimum and maximum mean Mahalanobis distance found across the 10 simulations
+and the the solid blue line shows the mean of the means across all 10 simulations, respectively. The Mahalanobis distance is calculated in the 3D phase space
+for the blue area and line and calculated in the 6D phase space, shown by the grey shaded area and the black dashed line.
+mean stellar mass density of around 0.7 M⊙ pc−3. We find similar
+results to the more dense subvirial simulations simulations, where
+the 3D Mahalanobis density clearly traces the ‘bump’ of the col-
+lapse, which then decreases as stars move apart. The time the ‘bump’
+occurs is delayed by several Myr compared to the higher density re-
+gions, due to the longer dynamical time scales.
+3.3 Comparison to other methods of quantifying structure
+We now plot the 3D and 6D Mahalanobis densities against other
+measures of quantifying structure in star-forming regions. Figure 9
+shows ¯ρm,20 plotted against the established methods of ΛMSR, Q and
+ΣLDR for the simulations with initial fractal dimension Df = 1.6 and
+radius 1 pc. For the initially subvirial simulations the ¯ρm,20 values
+stay below 12 for the first 5 Myr whereas the supervirial simulations
+can achieve much higher values. This is due to the stars in supervirial
+regions forming small groupings as the region expands which causes
+an increase in the Mahalanobis density. In contrast, for the subvirial
+regions more stars are interacting with each other, which erases spa-
+tial and kinematic substructure and also ejects stars (Schoettler et al.
+2020). As we are measuring the mean Mahalanobis density we are
+sensitive to a small number of stars being ejected which we see as a
+decrease in the mean Mahalanobis density for the subvirial simula-
+tions.
+MNRAS 000, 1–13 (2022)
+
+100
+6D
+3D
+10 :
+1
+0
+0
+1
+10
+Time (Myr)100:
+10 :
+1 :
+0
+0
+L
+10
+Time (Myr)100
+6D
+3D
+10
+pm,20
+0
+0
+1
+10
+Time (Myr)100
+10
+pm,20
+0
+0
+1
+10
+Time (Myr)8
+G. A. Blaylock-Squibbs & R. J. Parker
+(a) Df = 3.0, subvirial, 1 pc
+(b) Df = 3.0, supervirial, 1 pc
+Figure 7. The mean Mahalanobis density against time for each of the 10 subvirial (left-hand panel) and supervirial (right-hand panel) simulations without
+primordial substructure (fractal dimension Df = 3.0) and radii of 1 pc. The shaded blue area and solid blue line show the range of mean Mahalanobis densities
+calculated in the 3D phase space and the mean of the means found across all 10 simulations, respectively. The grey shaded area and the dashed black line show
+the same but for the 6D phase space.
+(a) Df = 3.0, subvirial, 5 pc
+(b) Df = 3.0, supervirial, 5 pc
+Figure 8. The mean Mahalanobis density against time for each of the 10 subvirial (left-hand panel) and supervirial (right-hand panel) simulations without
+primordial substructure (fractal dimension Df = 3.0) with radii of 5 pc. The shaded blue area and solid blue line show the range of mean Mahalanobis densities
+calculated in the 3D phase space and the mean of the means found across all 10 simulations, respectively. The grey shaded area and the dashed black line show
+the same but for the 6D phase space.
+We first show the Mahalanobis density versus the amount of mass
+segregation as defined by ΛMSR Allison et al. (2009) in Figure 9(a)
+and (b). For the subvirial simulations mass segregation is detected
+for 6 of the 10 simulations at 1 Myr and only one simulation has
+mass segregation present at 5 Myr, with ΛMSR > 2. The reason for
+the dissipation in the amount of mass segregation is due to the ejec-
+tion of massive stars from unstable Trapezium-like systems (Allison
+et al. 2010; Allison & Goodwin 2011; Parker & Goodwin 2015). In
+the supervirial simulations (see panel (b)) one region becomes mass
+segregated at 1 Myr and another at 5 Myr. If the cluster splits in two,
+with the most massive stars located in one of the halves then ΛMSR
+can increase to the value we see in Figure 9 (b) of around 5.5. As
+discussed in Parker et al. (2014), this is because the massive stars
+generally do not interact with each other as they do in the subvirial
+simulations where there is more mixing resulting in any structure in
+the phase spaces being erased.
+The supervirial simulations display a wider spread in the Maha-
+lanobis densities meaning that the plot of ¯ρm,20 versus ΛMSR can be
+used to distinguish between different initial virial states after at least
+5 Myr of dynamical evolution.
+The clearest distinction between different times in the simulations
+comes when ¯ρm,20 is combined with the Q-parameter. Figure 9(c)
+and (d) show this clearly for both the subvirial simulations and the
+supervirial simulations. The plots also show that, as expected, the
+supervirial simulations maintain substructure for longer. With some
+MNRAS 000, 1–13 (2022)
+
+100
+6D
+3D
+10
+0
+0
+1
+10
+Time (Myr)100
+10:
+,20
+0
+0
+1
+10
+Time (Myr)100
+6D
+3D
+10:
+0
+0
+1
+10
+Time (Myr)100
+10 :
+,20
+0
+0
+1
+10
+Time (Myr)Evolution of phase space densities in SFR
+9
+of the regions maintaining traces of substructure for 5 Myr as mea-
+sured using Q (i.e. Q < 0.8).
+Panels (e) and (f) of Fig. 9 show ¯ρm,20 plotted against the relative
+local surface density ratio, ΣLDR. We find that for both subvirial and
+supervirial simulations there is an increase in the local surface den-
+sity of the 10 most massive stars compared to all stars in the region.
+Interestingly the simulations with the highest local surface density
+around the 10 most massive stars do not necessarily have the highest
+Mahalanobis densities. This is likely due to the local surface density
+being calculated on the plane of the sky whereas the Mahalanobis
+density is being calculated for the full 3D phase space.
+The supervirial regions display high Mahalanobis densities at
+later times, and we can use this, and the different evolution of the
+Q-parameter and ΛMSR to distinguish between initial conditions af-
+ter several Myr of dynamical evolution.
+Using the 6D Mahalanobis density does not improve the diagnos-
+tic ability of the metric. We find that the range of ¯ρm,20 decreases
+when calculated in 6D. Meaning we see that the ¯ρm,20 values overlap
+making differentiating between different snapshots and virial states
+impractical.
+We now show the same plots but for simulations with little to no
+primordial spatial or kinematic substructure. Figure 10 shows the
+mean 3D and 6D Mahalanobis densities plotted against the estab-
+lished methods for simulations that have an initial fractal dimen-
+sion Df = 3.0 and radius 1 pc. The Mahalanobis densities for these
+simulations increase over time, with supervirial simulations having
+higher Mahalanobis densities compared to the subvirial simulations
+after 10 Myr of evolution.
+Panels (a) and (b) show ¯ρm,20 plotted against ΛMSR for the 10
+simulations. For the subvirial simulations we see mass segregation
+detected in three of the 10 simulations, for the supervirial simula-
+tions we detect mass segregation in two of the 10 simulations (recall
+that our threshold for declaring mass segregation is ΛMSR > 2.)
+As for the highly substructured simulations (Df = 1.6), the plot
+of the mean Mahalanobis density when combined with the Q-
+parameter gives the clearest distinction between the different snap-
+shots. For panels (e) and (f) we show ¯ρm,20 against ΣLDR. We can see
+that the 10 most massive stars can end up in a wide range of local
+surface density ratios The subvirial simulations have a wider range
+of values, with ΣLDR between 0.1 and 10, whereas the supervirial
+simulations all finish with ΣLDR > 1.
+The grey markers in Figure 10 show the 6D Mahalanobis densities
+against the established methods. We find once again that the spread
+in the Mahalanobis densities has decreased, making differentiating
+between different times or virial states impractical.
+4 DISCUSSION
+We have been motivated to test the Mahalanobis density due to its
+recent applications in quantifying the phase space of exoplanet host
+stars. In Winter et al. (2020) they propose that hot Jupiters are more
+likely to be found around host stars that are in high 6D phase space
+density, as measured using the 6D Mahalanobis density. However,
+this was questioned by Mustill et al. (2022) who show the peculiar
+velocities introduce a bias that once accounted for results in no sig-
+nificant excess of hot Jupiters around host stars in high 6D phase
+space density. The aim of this work is not to make a scientific as-
+sessment of the Mahalanobis density in its use in planet formation
+specifically but simply to see how it changes over time when look-
+ing at simple N-body simulations of star-forming regions to see what
+information, if any, it may be able to give us about the initial condi-
+tions (i.e. virial state, density and initial morphology).
+Due to the simplicity of our simulations there are a number of
+important caveats that must be taken into account. First, there is no
+galactic potential or tidal force acting on our simulated star-forming
+regions. The presence of an external Galactic tidal field would likely
+increase the dissolution of the star-forming region by causing out-
+lying stars to become unbound, which would in turn increase the
+potency of the Galactic tidal field at later ages.
+Two more important caveats are that we do not simulate any gas,
+and therefore there is no gas potential and also that our systems are
+fully isolated. The most important caveat that disallows direct com-
+parison to the works of Winter et al. (2020) and Mustill et al. (2022)
+is that we do not simulate planets in our simulations and so how rep-
+resentative our simulations are of real regions with exoplanet host
+stars is uncertain.
+In § 3.3 we show that the 6D Mahalanobis density in isolation
+cannot be used to reliably infer the initial conditions of star-forming
+regions due to overlap in the sub- and supervirial values. However,
+when the Mahalanobis density in the 3D phase space is combined
+with either ΛMSR, ΣLDR or Q then the initial virial conditions can be
+inferred. This is most clear to see in Figure 10.
+The regions that are initially supervirial attain higher final phase
+space densities than subvirial regions.
+This is somewhat counter intuitive, but is due to the fact that as the
+region expands small groupings of stars can form which will have
+similar positions and therefore higher phase space densities. The
+Mahalanobis distances between these groupings is reduced when
+we multiply by the inverse of the covariance matrix. In the sub-
+virial cases we see lower 3D phase space densities due to stars being
+ejected and ending up in relative isolation compared to the rest of
+the region. As we are using the mean Mahalanobis density we are
+sensitive to only a few stars being ejected.
+We cannot determine the initial conditions using the 6D Maha-
+lanobis density. One would assume that the more data we have (and
+therefore dimensions in the phase space), the more clearly we would
+see the distinction between sub- and supervirial simulations. How-
+ever, somewhat counter intuitively, adding more dimensions to the
+phase space effectively ’washes’ out any information that would al-
+low us to determine the initial conditions of our star-forming regions.
+For example, in our Df = 1.6 supervirial simulations, as the stars dy-
+namically evolve they may get further apart spatially but kinemati-
+cally they may be quite similar. If two stars that are very far apart end
+up moving in the same general direction the velocity and positional
+phases spaces will effectively cancel each other out and therefore
+removing any information about the initial conditions of the region
+(i.e. different positions but similar velocities).
+We find that the 6D Mahalanobis density for all simulations is
+similar at 10 Myr for both sub- and supervirial regions independent
+of the fractal dimension Df and the initial radii of the region. We
+therefore suggest that the Mahalanobis distance, and its associated
+density, are not suitable for quantifying the initial conditions of star
+formation, nor any subsequent dynamical evolution.
+5 CONCLUSION
+We present N-body simulations with different initial fractal dimen-
+sions, virial states and initial radii and quantify the 3D and 6D phase
+space density using the Mahalanobis distance. We compare the per-
+formance of the Mahalanobis density to more established meth-
+ods for quantifying structure in star-forming regions, namely ΛMSR,
+MNRAS 000, 1–13 (2022)
+
+10
+G. A. Blaylock-Squibbs & R. J. Parker
+(a) ¯ρm,20 vs ΛMSR, subvirial, 1 pc
+(b) ¯ρm,20 vs ΛMSR, supervirial, 1 pc
+(c) ¯ρm,20 vs Q, subvirial, 1 pc
+(d) ¯ρm,20 vs Q, supervirial, 1 pc
+(e) ¯ρm,20 vs ΣLDR, subvirial, 1 pc
+(f) ¯ρm,20 vs ΣLDR, supervirial, 1 pc
+Figure 9. The mean Mahalanobis density calculated for 3D and 6D phase spaces plotted against other methods of quantifying structure for 10 subvirial and
+supervirial simulations which are initially substructured with fractal dimension Df = 1.6 and 1 pc radii. The left-hand panels show the results for the subvirial
+regions and the right-hand panels show the results for the supervirial regions. The initial values at 0 Myr are represented by the black circles, the blue crosses
+show 1 Myr and the red triangles show 5 Myr. The grey open circles show the comparison of the 6D Mahalanobis density at 0 Myr, the open grey crosses show
+it for 1 Myr and the open grey triangles for 5 Myr. From top to bottom the rows show the different methods which ¯ρm,20 is plotted against, with the top row
+showing ΛMSR, second row showing Q and the bottom row showing ΣLDR.
+MNRAS 000, 1–13 (2022)
+
+25.0
+O Myr
+1 Myr
+20.0.
+5 Myr
+15.0
+pm,
+10.0
+5.0 -
+83
+33
+0.0
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+6.0
+AMSR,1025.0
+20.0
+20
+15.0
+10.0
+A
+5.0 
+A
+X
+83
+0.0 
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+6.0
+AMSR,1025.0
+20.0
+15.0
+10.0
+5.0
+8
+A
+3
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+Q25.0
+20.0
+15.0
+10.0
+5.0
+8
+A
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+Q25.0
+20.0
+,20
+15.0
+10.0
+Po
+5.0
+10
+100
+#
+0.0
+0
+7
+10
+100
+ZLDR25.0
+20.0
+15.0
+E
+10.0
+P
+5.0
+10
+A
+0.0
+0
+1
+10
+100
+ZLDREvolution of phase space densities in SFR
+11
+(a) ¯ρm,20 vs ΛMSR, subvirial, 1 pc
+(b) ¯ρm,20 vs ΛMSR, supervirial, 1 pc
+(c) ¯ρm,20 vs Q, subvirial, 1 pc
+(d) ¯ρm,20 vs Q, supervirial, 1 pc
+(e) ¯ρm,20 vs ΣLDR, subvirial, 1 pc
+(f) ¯ρm,20 vs ΣLDR, supervirial, 1 pc
+Figure 10. The mean Mahalanobis density calculated for the 3D and 6D phase spaces plotted against other methods of quantifying substructure for 10 subvirial
+and supervirial simulations which have little to no initial substructured with fractal dimension Df = 3.0 and 1 pc radii. The left-hand panels show the subvirial
+results and the right-hand panels show the supervirial regions. The initial values at 0 Myr are represented by the black circles, the blue pluses show 1 Myr and
+the red triangles show 5 Myr. We show the mean 6D Mahalanobis densities at 0 Myr, 1 Myr and 5 Myr with grey open circles, grey open crosses and grey open
+triangles, respectively. From top to bottom the rows show the different methods which ¯ρm,20 is plotted against, with the top row showing ΛMSR, second row
+showing Q and the bottom row showing ΣLDR.
+MNRAS 000, 1–13 (2022)
+
+10.0-
+8.0
+o Myr
+2
+6.0-
+1 Myr
+X
+5 Myr
+4.0 
+2.0 
+0.0 -
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+6.0
+ΛMSR,1010.0
+8.0
+2
+6.0
+4.0 
+2.0 -
+△
+0.0
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+6.0
+AMSR,1010.0-
+8.0
+6.0-
+4.0 
+2.0-
+0.0
+0.0
+0.5
+1.0
+1.5
+Q10.0-
+8.0 -
+6.0-
+IQ
+4.0-
+2.0 
+4
+0.0 -
+0.0
+0.5
+1.0
+1.5
+Q10.0-
+8.0
+2
+6.0
+"
+4.0
+2.0
+A
+0.0
+0
+10
+ZLDR10.0:
+8.0
+2
+6.0 -
+4.0
+2.0
+4
+A
+0.0
+0
+10
+ZLDR12
+G. A. Blaylock-Squibbs & R. J. Parker
+ΣLDR and Q. We also applied the Mahalanobis distance in 3D phase
+space to sets of static synthetic regions of different morphologies to
+test its ability to discriminate between different morphologies. Our
+conclusions are as follows:
+(i) The Mahalanobis distance in the 3D phase space is degenerate
+across a wide range of morphologies commonly observed in star-
+forming regions, associations and clusters, and so it cannot be used
+to differentiate between different morphologies.
+(ii) The 3D Mahalanobis densities, ρm,20 can be used to distin-
+guish between the high and low stellar density regions with large
+amounts of substructure (Df = 1.6). The low stellar density regions
+regions show similar behaviour but delayed by around 0.6 Myr
+compared to the high volume density regions due to the dynamical
+timescales being longer. This effect is even more pronounced in the
+simulations with little to no initial substructure (Df = 3.0) where we
+see the ‘bump’ occurring several Myr later than in the higher stellar
+density simulations corresponding to the subvirial collapse.
+(iii) We show that the Mahalanobis density calculated in the 3D
+phase space can be used with the Q-parameter, ΛMSR or ΣLDR to
+infer information about a region’s initial virial state.
+(iv) When using the 6D Mahalanobis densities we see no sig-
+nificant differences between any of the simulations. Adding more
+parameters (adding more dimensions) to the phase space suppresses
+any changes in the Mahalanobis density over time.
+We therefore advise against using the Mahalanobis distance as a
+method to quantify the morphology of star-forming regions due to
+its degeneracy across both substructured regions and also smooth,
+centrally concentrated regions.
+When applied to spatial and kinematic phase space (6D), all of its
+discriminatory power is washed out (similar to the issues encoun-
+tered when applying the Q-parameter to kinematic data, Cartwright
+(2009)), and we advocate using combinations of spatial and kine-
+matic metrics instead.
+ACKNOWLEDGEMENTS
+Plots have been generated using Matplotlib 3.3.4 (Hunter 2007). Nu-
+merical results calculated using Numpy 1.20.1 and SciPy 1.9.0 (Har-
+ris et al. 2020; Virtanen et al. 2020). RJP acknowledges support from
+the Royal Society in the form of a Dorothy Hodgkin fellowship. We
+wish to thank the anonymous reviewer for their comments which
+have improved the paper.
+DATA AVAILABILITY
+Data is available on reasonable request to the corresponding author.
+REFERENCES
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+APPENDIX A: MAHALANOBIS DISTANCE VERSUS
+MAHALANOBIS DENSITY
+In Figure A1 we show the relation between the Mahalanobis distance
+and density across the two different phase spaces investigated and
+the two different initial virial states.
+We see that in the positional phase space (3D, the coloured mark-
+ers) there is significant overlap in both the Mahalanobis distance and
+density, making differentiating between different snapshots imprac-
+tical. For the supervirial regions we see less overlap in the Maha-
+lanobis density between the snapshots. However, there is significant
+overlap between the sub- and supervirial simulations, meaning that
+neither the Mahalanobis distance nor density can reliably distinguish
+between different virial states.
+In both Figure A1 and Figure A2 we show the position-velocity
+phase space (6D) with the grey open markers.
+Figure A2 shows the mean Mahalanobis distance plotted against
+the mean Mahalanobis density for high density (radii of 1 pc) region
+with little or no substructure (Df = 3.0).
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–13 (2022)
+
+14
+G. A. Blaylock-Squibbs & R. J. Parker
+(a) ¯ρm,20 vs ¯Md, subvirial, 1pc
+(b) ¯ρm,20 vs ¯Md, supervirial, 1 pc
+Figure A1. The mean Mahalanobis density ( ¯ρm,20) plotted against the mean Mahalanobis distance ( ¯Md) for highly substructured regions with fractal dimensions
+Df = 1.6 and initial radii of 1 pc. Each region contains 1000 stars. The black circles the values at 0 Myr, the blue plus signs are the values at 1 Myr and the red
+triangles are the values at 5 Myr. The grey open circles, crosses and triangles show the same information but for the Mahalanobis distance and density calculated
+in the 6D phase space.
+(a) ¯ρm,20 vs ¯Md, subvirial, 1 pc
+(b) ¯ρm,20 vs ¯Md, supervirial, 1 pc
+Figure A2. The mean Mahalanobis density ( ¯ρm,20) plotted against the mean Mahalanobis distance ( ¯Md) for substructured regions with fractal dimensions
+Df = 3.0 and scales 1 pc for different snapshots. Each region contains 1000 stars. The black circles the values at 0 Myr, the blue crosses are the values at 1 Myr
+and the red triangles the values at 5 Myr. The grey open circles, crosses and triangles show the same information but for the Mahalanobis distance and density
+calculated in the 6D phase space.
+MNRAS 000, 1–13 (2022)
+
+10.0-
+O Myr
+x
+1 Myr
+5 Myr
+8.0 -
+IQ
+4.0-
+2.0-
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Md10.0
+8.0 
+26.0
+i
+4.0
+A
+2.0 
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Md10.0-
+O Myr
+x
+1 Myr
+5 Myr
+8.0 -
+A
+E
+4.0-
+2.0 
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Md10.0
+8.0 
+26.0
+4.0
+2.0
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Md
\ No newline at end of file
diff --git a/otE1T4oBgHgl3EQf1wXJ/content/tmp_files/load_file.txt b/otE1T4oBgHgl3EQf1wXJ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9c8420523a4b3edcc7939ed173b7a97dbd48b300
--- /dev/null
+++ b/otE1T4oBgHgl3EQf1wXJ/content/tmp_files/load_file.txt
@@ -0,0 +1,1076 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf,len=1075
+page_content='MNRAS 000, 1–13 (2022)  Preprint 10 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  The evolution of phase space densities in star-forming regions  George A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Blaylock-Squibbs⋆ and Richard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker†  Department of Physics and Astronomy, The University of Sheffield, Hounsfield Road, Sheffield, S3 7RH  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  The multi-dimensional phase space density (both position and velocity) of star-forming regions may encode information on  the initial conditions of star and planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Recently, a new metric based on the Mahalanobis distance has been used  to show that hot Jupiters are more likely to be found around exoplanet host-stars in high 6D phase space density, suggesting  a more dynamic formation environment for these planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' However, later work showed that this initial result may be due to a  bias in the age of hot Jupiters and the kinematics of their host stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We test the ability of the Mahalanobis distance and density  to differentiate more generally between star-forming regions with different morphologies by applying it to static regions that  are either substructured or smooth and centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We find that the Mahalanobis distance is unable to distinguish  between different morphologies, and that the initial conditions of the N-body simulations cannot be constrained using only  the Mahalanobis distance or density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Furthermore, we find that the more dimensions in the phase space the less effective the  Mahalanobis density is at distinguishing between different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We show that a combination of the mean three-  dimensional (x, y, z) Mahalanobis density and the Q-parameter for a region can constrain its initial virial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' However this is  due to the discriminatory power of the Q-parameter and not from any extra information imprinted in the Mahalanobis density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We therefore recommend continued use of multiple diagnostics for determining the initial conditions of star-forming regions,  rather than relying on a single multi-dimensional metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Key words: galaxies: star formation – methods: statistical – methods: numerical  1 INTRODUCTION  Star formation is observed to take place along filaments within giant  molecular clouds (Palmeirim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Schisano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' André  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The initial formation and distribution of these filaments is  likely due to supersonic turbulence within GMCs (Larson 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' It is  along these filaments that cores can form, with further fragmentation  of these cores leading to stars forming in groups containing 10s to  1000s of members (Lada & Lada 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Bastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  One of the foundational questions of star formation is whether  star formation is a universal process or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Are the initial condi-  tions of star-forming regions dependant on the environment, where  differences in the stellar density, IMF and stellar multiplicity are due  to the initial conditions of the star-forming region?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Or does star for-  mation happen in a similar way everywhere, and any differences we  observe in these regions is stochastic in nature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  There are two main proposed modes of star-formation, monolithic  and hierarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In monolithic formation modes the gas is already  contained within the final volume of the region before stars begin  to form, whereas in hierarchical formation the gas extends beyond  the final volume of the region (Longmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In the hierarchical mode stars are forming while at the same  time the gas is collapsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The kinds of star-forming regions these modes produce is of in-  ⋆ E-mail: gablaylock-squibbs1@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='uk  † Royal Society Dorothy Hodgkin fellow  terest not only in the context of star formation but also the way in  which the final star-forming regions that form may influence the ar-  chitecture of the planetary systems that are produced within them  (Adams 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Star-formation is a rapid process, occurring within a few cross-  ing times (Elmegreen 2000), which is often less than 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Dur-  ing this process, stars are forming and moving (Alcock & Parker  2019), further muddying the formation picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' And whilst observa-  tions of the earliest stages have improved greatly with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' ALMA,  observations of star-forming regions are often at older ages, where  significant dynamical evolution may have taken place (Klessen &  Kroupa 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Allison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Daffern-Powell  & Parker 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Schoettler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Dynamical evolution alters  the spatial and kinematic distributions of young stars, erasing the  signature of the initial conditions, but can be used as a proxy for age  and used to converge on a likely set of initial conditions for a given  star-forming region (Parker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' To enable comparisons of  observations and simulations, we need to be able to quantify param-  eters of the star-formation regions, such as the degree of substructure  and mass segregation (Cartwright & Whitworth 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Allison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Sánchez & Alfaro 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Alfaro & González 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' González  & Alfaro 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Jaffa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Buckner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Joncour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Gouliermis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Early methods such as the auto-correlation function and two-point  correlation function compared the number of excess star pairings  to a random distribution of stars as a function of scale (Gomez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Larson 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' These methods where used extensively  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='03472v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='GA]  9 Jan 2023  2  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Blaylock-Squibbs & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker  to determine the degree of substructure, with early work suggest-  ing breaks in the two-point correlation function corresponded to the  Jeans length (Simon 1997) (though see Bate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (1998)) and the  size of the widest stellar binaries in the regions in question (Kraus &  Hillenbrand 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Joncour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Subsequent work made extensive use of minimum spanning trees  (MSTs) to quantify structures in star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Cartwright  & Whitworth (2004) introduced the Q-parameter to quantify spatial  substructure, and Allison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2009) introduced the ΛMSR method  to quantify mass segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Parker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2014) showed that the initial conditions of a region  can be inferred from the spatial information, if a suitable number of  metrics are combined, including the relative stellar surface density  around the most massive stars (Maschberger & Clarke 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Küpper  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  However, the majority of the above methods are designed to oper-  ate on two or three-dimensional spatial data, whereas recent obser-  vational data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' from Gaia and associated ground-based surveys)  has provided high resolution spatial and kinematic data (6D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Recently, in an attempt to quantify the phase space densities  of exoplanet host stars Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2020) developed the Maha-  lanobis density, a new application of the Mahalanobis distance (Ma-  halanobis 1936).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The Mahalanobis distance has been used in astronomy for classi-  fying objects, for example in Siegal & Griffiths (1974) it is used to  analyse and classify different types of asteroid impact craters and in  Jakimiec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (1991) it was used to classify sunspots into groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Due to the differing dimensions, and units of very different scale  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' length in pc and velocity in kms−1) making multivariate com-  parisons can be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' However the Mahalanobis distance makes  multivariate comparisons possible over wide ranges of dynamical  scales by rescaling the axes and removing the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  (2020) used this method to develop the Mahalanobis density and use  it to propose the hypothesis that host stars in high phase space den-  sities are more likely to have hot Jupiter planets (M > 50 M� and  a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='2 au) around them compared to the lower phase space densi-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' However, Mustill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2022) show that this result may be due  to a bias from the peculiar velocities of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' When the peculiar  velocities of the stars are accounted for, there is no longer an excess  of hot Jupiters in high 6D (x, y, z, Vx, Vy, Vz) phase space densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Irrespective of the ongoing debate surrounding the application of  the Mahalanobis distance to exoplanet host stars, in this paper we  aim to test this metric when applied to both synthetic static regions  and assess its performance in quantifying phase space structures of  N-body simulations of star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In § 2 we present the meth-  ods used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='1 we present the results of testing the Mahalanobis  distance’s ability to differentiate between different morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In  § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='2 we show how the Mahalanobis distance and density change  with time in N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='3 we compare the Maha-  lanobis distance and density to other methods for quantifying struc-  ture in star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In § 4 we present a discussion of our  results and we conclude in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2 METHODS  In this section we describe the set-up of the N-body simulations,  including the initial spatial and kinematic distributions of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We  then describe the methods used to quantify the spatial and kinematic  distributions in our simulated star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  (a) 1 pc radius Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6  (b) 1 pc radius Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Examples of fractal regions with 1000 stars, on the left-hand side  is a highly substructure region of fractal dimension Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and on the right-  hand side is a region with far less substructure with fractal dimension Df =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='1 N-Body Simulations  The N-body simulations are run using the Kira integrator, part of  the Starlab1 package (Portegies Zwart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 1999, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Each  simulation has a population of 1000 stars and is run for a simulated  time of 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Our choice of 1000 stars comes from Lada & Lada  (2003) where they find the following relation,  Ncl ∝ M−2  cl  (1)  where Ncl is the number of clusters and Mcl is the mass of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  This power-law is obeyed for star clusters between masses of 10 <  Mcl/M⊙ < 105 and our choice of 1000 stars puts our simulations  close to the middle of this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Observations of star forming regions show that within these com-  plexes there are filaments of denser gas, inside of which prestellar  cores are observed (Myers 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' André 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' These filaments are  thought to be caused by supersonic turbulence, which is likely to be  responsible for the substructure we observe in these regions (Larson  1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Kraljic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' To mimic this substructure we make use  of a box-fractal generator to initialise our simulations with fractal  dimensions of Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 (Goodwin & Whitworth 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Daffern-Powell & Parker 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We run subvirial simulations to match observations that indicate  prestellar cores may be subvirial with respect to one another and  may then virialise and form bound smooth, centrally concentrated  star clusters (Foster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Kuznetsova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We also run a set of simulations that are initially supervirial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We do this to mimic the observations that some young star-forming  regions (∼ 1−5 Myr) are supervirial and therefore expanding (Bravi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Kounkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Figure 1 shows example clusters with the fractal dimensions Df =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We use the following to define the virial ratio,  αvir = T  |Ω|,  (2)  where T is the total kinetic energy of the region and Ω is the total  potential energy of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' By using this equation we can scale  the initial velocities to our desired virial ratio either αvir = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='1 for  subvirial regions or αvir = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='9 for supervirial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The initial  conditions for the simulations are summarised in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We assign masses using the Maschberger IMF with a lower mass  1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='sns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content='edu/~starlab/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='html  MNRAS 000, 1–13 (2022)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  J  (d)  ":  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' *  y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  x (pc)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  (pc)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  ：  D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  x (pc)Evolution of phase space densities in SFR  3  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This table shows the different initial conditions of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  For each of these initial conditions 10 simulations are run for 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' From  left to right the columns are, the initial fractal dimension of the region, the  number of stars, the initial virial ratio and the initial radius of the simulations  in pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Fractal Dimension  N⋆  Virial Ratio  Radius (pc)  Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='9  1, 5  Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='9  1, 5  limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='01 M⊙, upper mass limit of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 M⊙ and a mean mass of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='2 M⊙ (Maschberger 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The probability distribution function of the Maschberger IMF is,  p(m) ∝  �m  µ  �−α �  1+  �m  µ  �1−α�−β  ,  (3)  where µ is the mean stellar mass, α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='3 is the high mass exponent  and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='4 is the low mass exponent (Salpeter 1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='2 Smooth Centrally Concentrated Regions  We generate smooth, centrally concentrated regions with radial den-  sity profiles in which the stars are randomly distributed using the  following relation,  n ∝ r−α,  (4)  where r is the distance from the centre of the region and α is the  radial density index and has values α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We also  generate Plummer spheres, regions with a three-dimensional density  distribution of the form,  ρp(r) = 3M0  4πa3  �  1+ r2  a2  �− 5  2  ,  (5)  where M0 is the total mass of the region, r is the distance from the  centre of the region and a is the Plummer radius (Plummer 1911;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Kroupa 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='3 Generating Fractal Regions  We follow Goodwin & Whitworth (2004) and Cartwright & Whit-  worth (2004) to generate substructured regions using the box-fractal  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Other examples of this method can be found in (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Allison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2009), Parker & Goodwin (2015), Daffern-Powell & Parker  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The method proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A single star is placed at the  centre of a cube whose side length is chosen to be NDiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This cube  is then subdivided down into N3  Div sub-cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A star is then placed  at the centre of each of the sub-cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Each of these sub-cubes then  has a probability of being subdivided again given by N(3−Df)  Div  , where  Df is the fractal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Cubes that are not subdivided have their  stars removed along with any previous generations of stars that came  before them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A small amount of noise is added to each of the stars  to prevent them having a regular looking appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  These steps are repeated until the desired number of stars is  reached or exceeded in the latest generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Once this condition is  met all previous generations of stars are removed, then the remain-  ing stars are randomly removed until the desired number of stars is  reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' By removing the stars in this manner we end up with stars  distributed inside of a spherical volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The velocities of the first generation of stars are picked from a  Gaussian with mean zero, with each subsequent generation of stars  inheriting the velocity of the previous generation plus a random  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This results in stars that are close to each other hav-  ing similar velocities and stars far apart from one another having  very different velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The velocities are related to the length scale  of the region with the following relation V(L) ∝ L3−Df where L is  the length scale of interest and Df is the desired fractal dimension  (Parker & Wright 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='4 The Mahalanobis Distance  The Mahalanobis distance is a metric that measures distances be-  tween points to the average in the distribution in N dimensional  phase spaces (Mahalanobis 1936).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The Mahalanobis distance does this by removing any correlations  in the data by multiplying the distances between points and the av-  erage of the region by the inverse of the covariance matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' this also  has the effect of re-scaling the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Once this re-scaling has been done the Euclidean distances are  found in the phase space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' this is the Mahalanobis distance, Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Each point in a dataset is described using a vector where each  element is a measured parameter of that point,  ⃗x = (x1,x2,x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=',xN)T ,  (6)  where x1,x2,x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=',xn are the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For example, if each point  has the three parameters, (x, y, z) then this is simply its physical  position in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The Mahalanobis distance between a point in a distribution and  the mean of that distribution in an N dimensional phase space is  defined as,  Md(⃗x,⃗µ) =  �  (⃗x−⃗µ)T S−1 (⃗x−⃗µ),  (7)  where the ⃗x is the point vector, ⃗µ is a vector of the averages of the  parameters of interest and S−1 is the inverse of the covariance matrix  for all the parameters in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The Mahalanobis distance, Md has been also used to define a pa-  rameter space density called the Mahalanobis density, ρm,N (Winter  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2020)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  To calculate the Mahalanobis density we first must define the Ma-  halanobis distance between points in the phase space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' distance  between⃗x and⃗y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We follow Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2020) and use,  md(⃗x,⃗y) =  �  (⃗x−⃗y)T S−1 (⃗x−⃗y),  (8)  where ⃗x is the vector describing the measurements of one point, ⃗y  is the vector describing the measurements of another and S−1 is the  inverse of the covariance matrix of all the parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The calculation of the Mahalanobis density proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  First we find the Mahalanobis distance to the Nth nearest neighbour,  then we divide the nearest neighbour number by the volume whose  side length is defined as the Mahalanobis distance to the Nth nearest  neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The Mahalanobis density is then defined as,  ρm,N = Nm−Dp  d,N ,  (9)  2 We have used different notation for the Mahalanobis distance to avoid con-  fusion with the fractal dimension and also the number of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We  instead use Md instead of D as used in Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2020) to avoid confu-  sion with the fractal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We also change the number of dimensions  in the phase space from D to Dp again to avoid confusion with the fractal  dimension, which we represent as Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2022)  4  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Blaylock-Squibbs & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker  where ρm,N is the Mahalanobis density, N is the nearest neighbour  number, md,N is the Mahalanobis distance to the Nth nearest neigh-  bour and Dp is the number of dimensions in the phase space (Winter  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The Mahalanobis densities are then normalised so that  the median Mahalanobis density is unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  In this work we apply the Mahalanobis density to two differ-  ent phase spaces, the positional phase space (3D) and the position-  velocity phase space (6D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For this work we find the Mahalanobis  distance to the 20th nearest neighbour in the phase space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' N =  20), the same as in Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5 Local Surface density ratio  The local surface density ratio ΣLDR was introduced in Maschberger  & Clarke (2011) to quantify the differences between the surface den-  sities of subsets of stars within their host regions and for this work  we choose the 10 most massive stars as the subset of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The  algorithm proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For each star we find the distance to  its Nth nearest neighbour, then we calculate the circular area whose  radius is the distance to the Nth nearest neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' To find the sur-  face density of the stars we divide the nearest neighbour number by  this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We use a nearest neighbour number of 5 for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The ratio is defined as,  ΣLDR =  ˜Σsubset  ˜Σall  (10)  where ˜Σsubset is the median surface density found for the 10 most  massive stars and ˜Σall is the median surface density found for the  entire region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Therefore, if ΣLDR > 1 the 10 most massive stars are  found in areas of higher than average stellar surface density, and con-  versely, if ΣLDR < 1 then they are located in areas of lower than aver-  age surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The significance of any difference is quantified  using a two-sample Kolmogorov-Smirnov test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Where if p << 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='01  we reject the null hypothesis that the 10 most massive stars share the  same underlying distribution of surface densities compared to the  entire region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 Mass Segregation Ratio  The mass segregation ratio ΛMSR was first introduced in Allison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2009) to quantify the degree of mass segregation in a star-  forming region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The definition of mass segregation in this case is that  the most massive stars are closer to each other than expected from  the average separation of all of the stars in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The method  makes use of minimum spanning trees (MSTs) which are graphs of  points connected to each other in such a way that the total length of  the tree is minimised and that all points are connected to at least one  other point with no closed loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  This method generates a minimum spanning tree for the chosen  subset of stars, for this work we use the 10 most massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' It  will then pick 10 random stars from the region and make an MST  for these random stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We do this 200 times to calculate the mean  edge length of the randomly chosen trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The ratio is calculated  using the following equation,  ΛMSR =  �  laverage  �  l10  +σ5/6/l10  −σ1/6/l10  ,  (11)  where  �  laverage  �  is the average edge length found for all the randomly  constructed MSTs and l10 is the edge length of the subset’s MST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' It  is important to note that the random MSTs we construct can also  contain members of the chosen subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  If the ratio is > 1 then the region’s 10 most massive stars are mass  segregated, if the ratio is ∼ 1 then the most massive stars are not  mass segregated and if the ratio is less than 1 they are inversely mass  segregated (the most massive stars are further apart than the average  stars in the region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In this work we mark the value 1 to show the  boundary between mass segregation and inverse mass segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  However we follow Parker & Goodwin (2015) and only take ratio  values above 2 to be signs of mass segregation, to avoid false posi-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We follow Parker (2018) and calculate the uncertainty using the  randomly constructed MSTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' First we order the lengths of the ran-  dom MSTs and find the values that lie 1/6 and 5/6 of the way  through this list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This gives us values which correspond to a 66 per  cent deviation from the median MST length found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='7 Q-Parameter  The Q-Parameter was introduced in Cartwright & Whitworth (2004)  to quantify and distinguish between different cluster morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The Q-parameter also makes use of MSTs and proceeds as fol-  lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' First the normalised correlation length is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This is the  mean separation between all stars in a region which is then divided  by the region’s radius to normalise it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The mean edge length of the region is found by constructing an  MST for the region and then finding the mean edge length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The mean  edge length is normalised by diving it by  NtotalA  Ntotal−1, where Ntotal is the  number of stars in the region and A is the area of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We use  the circular area (see Schmeja & Klessen (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker (2018) for  a discussion on normalisation techniques), with the radius defined  as the distance from the centre of mass to the most distant star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The  Q-parameter is then defined as,  Q = ¯m  ¯s ,  (12)  where ¯m is the normalised mean edge length of the MST and ¯s is  the normalised correlation length between stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Regions with sub-  structured morphologies have Q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='8 whereas regions with smooth,  centrally concentrated morphologies have Q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  3 RESULTS  We show the results of the Mahalanobis distance applied to both  static and N-body simulations of star-forming regions with various  initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We present the 3D and 6D Mahalanobis densities  calculated in the N-body simulations and compare the evolution of  the Mahalanobis density over time to the other methods for quanti-  fying spatial and kinematic distributions in star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='1 Static Regions  We first find the Mahalanobis distances between stars and the aver-  age point in the region, ⃗µ, and calculate the average Mahalanobis  distance in their respective regions ( ¯Md) for sets of synthetic and  static star clusters, with each set having a different structural parame-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Each region in the set consists of 1000 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We calculate ¯Md for  substructured regions with fractal dimensions Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  and clusters with radial density profile indexes, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We also show the results of a set of 100 Plummer spheres which have  a radial density profile described by equation 5 with a radial density  index of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5 (see § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2022)  Evolution of phase space densities in SFR  5  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Mean of the mean Mahalanobis distances calculated in the 3D  phase space for sets of 100 different star-forming regions plotted against the  structural parameter used to make the sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The red triangles are the smooth,  centrally concentrated radial regions, the purple star (on top of the red trian-  gle with the structural parameter equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5) is the Plummer sphere and the  black crosses are the fractal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The error bars show a single standard  deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Figure 2 shows the mean of the means for the Md (the Maha-  lanobis distance of each star to its region’s averages in the 3D phase  space) for the 100 clusters in each set of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We first  calculate the mean Mahalanobis distance in each of the 100 regions  in the set and then we calculate the mean of those means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The error  bars show the standard deviation of the mean of the mean Maha-  lanobis distances found in each of the regions in a particular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Figure 2 clearly shows that ¯Md is degenerate across a wide range  of morphologies and we are therefore unable to use ¯Md calculated  in the 3D phase space to differentiate between the different mor-  phologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' There is much more spread in the values for the Plummer  sphere compared to the radial and fractal models, with the fractals  having the smallest spread of ¯Md and the radial regions sitting be-  tween the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This is due to Plummer spheres being formally infi-  nite in extent, so the calculation occasionally has to normalise over  very distant stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='2 N-body Results  Figure 3 shows the mean of the mean Mahalanobis distances (calcu-  lated in both 3D and 6D) found for 10 different N-body simulations  with initial fractal dimension Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and 1 pc radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The left hand  panel shows the results for subvirial (collapsing) regions, and the  right hand panel shows the results for the supervirial (expanding) re-  gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In both the sub- and supervirial cases there is a decrease in  the Mahalanobis distance over time for both the 3D and 6D phase  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the initially subvirial simulations the 3D Mahalanobis  distance swiftly decreases at the start and then continues to decrease  for the rest of the simulation but at a slower rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the supervirial  regions we see a less pronounced decrease in ¯Md compared to the  initially subvirial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The ¯Md calculated in the 6D phase space  shows more modest decrease over time for both initially sub- and  supervirial simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Figure 4 shows the mean Mahalanobis density ( ¯ρm,20) calculated  in both the 3D and 6D phase spaces for two sets of 10 (one sub-  virial and the other supervirial) simulations with an initial fractal  dimension of Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and initial radii of 1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The highest Ma-  halanobis densities are calculated in the 3D phase space (x, y, z)  for the supervirial simulations, however these large final values are  only present for a few of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In the 3D phase space  the Mahalanobis density increases in the first 2-4 Myr, after which  the density stays the same for the rest of the run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This is most  likely due to the early dynamical interactions of stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' as they move  closer to each other the stars’ Mahalanobis densities will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In  Appendix A we show the relationship between the Mahalanobis dis-  tance and density for the high density simulations with and without  substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We show this for both the 3D and 6D phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Initially, in the subvirial simulations, the 6D Mahalanobis density  decreases for the first 1 Myr and then stays the same until around 5  Myr where it then starts to increase again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the supervirial sim-  ulations this initial decrease in the 6D phase space happens more  rapidly than in the subvirial simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' It also does not reach the  low densities that the subvirial regions attain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In the supervirial re-  gions, the stars will expand together in co-moving groups, therefore  they will have similar positions but can still have velocities that ex-  hibit kinematic substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In contrast, in the subvirial simulations  the stars interact more and erase this substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The difference  in the velocities explains why the 6D Mahalanobis density is much  lower than the 3D density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Some of the supervirial regions attain  higher 6D Mahalanobis densities compared to the subvirial regions  at the end of the 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We also calculate the Mahalanobis density and distance for low  density simulations for simulations with Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0,  where the initial radii regions are 5 pc (with a mean number density  of around 3 stars pc−3 and a mean stellar mass density of around  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 M⊙ pc−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We find that the evolution of ¯Md is almost identical  to the its evolution in the high density simulations, both in 3D and  6D phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We show the evolution of the 3D and 6D Mahalanobis densities  for these more diffuse simulations in Figure 5, which shows ¯ρm,20  plotted against time for the same substructured regions with fractal  dimension Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 with initial radii of 5 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the 3D phase space  we see the same trends as in Figure 4 but a slight difference for the  6D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Now ¯ρm,20 decreases in the subvirial simulations  as the regions evolve, and the supervirial simulations show a steady  Mahalanobis density after 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We show the evolution of the Mahalanobis distance (measured be-  tween the points and the mean values of the phase in each snapshot),  and Mahalanobis density in the simulations that have no primordial  substructure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' they are uniform spheres at t = 0 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Figure 6 shows the Mahalanobis distance over time for regions  with a fractal dimension Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 with radii of 1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the 3D  phase space there is a decrease over time for the sub- and supervirial  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Comparing these results to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 3 we see that the Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 re-  gions’ Mahalanobis distances decrease at a slower rate compared to  regions with initially more substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This behaviour also results  in a slightly greater 3D Mahalanobis distance being measured for  regions with fractal dimension Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 at 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' However, this is  not seen in the 6D phase space Mahalanobis distances which show  little change over the 10 Myr in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Also, very little  difference is seen when comparing the 6D Mahalanobis distances  between the sub- and supervirial simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Figure 7 shows the 3D and 6D ¯ρm,20 against time for the sim-  ulations with no primordial substructure (with an initial fractal di-  mension Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 and radii 1 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The left-hand panel shows ¯ρm,20  against time for the subvirial simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' It shows the same increase  in Mahalanobis density as the Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 simulations but we don’t see  MNRAS 000, 1–13 (2022)  4  3  2  M  1  0  Radial  Fractal  1  Plummer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0  α / Df6  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Blaylock-Squibbs & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker  (a) Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6, subvirial, 1 pc  (b) Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6, supervirial, 1 pc  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Plots of the mean Mahalanobis distance calculated in both the 3D and 6D phase spaces against time for regions with both high initial volume  densities and high degrees of substructure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' fractal dimension Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 with radii of 1 pc) consisting of 1000 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The shaded areas show the range of mean  Mahalanobis distances found across all 10 of the simulations at the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The solid lines show the mean of the mean Mahalanobis distances across all 10  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The blue area and solid blue line shows the 3D phase space and the black dashed line and the grey area show the same but for the 6D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  (a) Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6, subvirial, 1 pc  (b) Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6, supervirial, 1 pc  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The mean Mahalanobis density against time for each of the 10 subvirial (left-hand panel) and supervirial simulations (right-hand panel) with fractal  dimension D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and radii of 1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The simulations consist of 1000 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The blue shaded area shows the minimum and maximum mean Mahalanobis density  (in the 3D phase space) found across all 10 of the simulations at the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The solid blue line shows the mean of the means for the Mahalanobis density  in the 3D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The grey shaded area and the dashed black line shows the same but for the Mahalanobis density calculated in the 6D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The  mean number densities of the 10 simulations is around 314 stars pc−1 with a mean stellar mass density of around 201 M⊙ pc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  the same initial decrease in the 6D ¯ρm,20 that we see in the Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' At around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5 Myr a decrease in the 3D Mahalanobis  density is seen, then a second period of increasing Mahalanobis den-  sity is seen around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='9 Myr before attaining a steady Mahalanobis  density for the rest of the simulations’ run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This initial increase  is due to the region collapsing and stars moving closer to each other  which raises the 3D Mahalanobis density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' What stops it increasing  further is likely the dynamical interactions causing stars to move fur-  ther away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Once this initial dynamical stage settles  down the density can increase again due to stars being close to each  other near the centre of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The right-hand panel of Figure 7 shows the 3D and 6D Maha-  lanobis density calculated for regions that are initially supervirial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The ‘bump’ like feature is much smaller for the 3D phase space cal-  culations than in the initially subvirial simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The decrease in  the bump compared to the subvirial simulations is due to the fact  that the stars are constantly and continuously moving away from  each other, meaning that the slight increase that is still present is due  to small groupings of stars clumping together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We see similar be-  haviour for the 6D phase space in both the subvirial and supervirial  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We show the 3D and 6D Mahalanobis densities for simulations  with Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 and initial radii of 5 pc in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' These low den-  sity simulations have mean number density of around 1 star pc−3, or  MNRAS 000, 1–13 (2022)  100  6D  3D  10 :  M  1  0  0  1  10  Time (Myr)100  10 :  1  0  0  10  Time (Myr)100  6D  3D  10  1  0  0  1  10  Time (Myr)100  10  pm,20  1  0  0  7  10  Time (Myr)Evolution of phase space densities in SFR  7  (a) Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6, subvirial, 5 pc  (b) Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6, supervirial, 5 pc  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Plots showing the mean Mahalanobis density against time for each of the 10 subvirial (left-hand panels) and supervirial (right-hand panels) simulations  with fractal dimension Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and radii of 5 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' These simulations have a low initial stellar number density with a mean around 3 stars pc−3 and a mean stellar  mass density of around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 M⊙ pc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The shaded blue area shows the minimum and maximum mean Mahalanobis density found across all 10 simulations in  the 3D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The solid blue line shows the mean of the means Mahalanobis density against time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The shaded grey area and the dashed black line show  the same but for the 6D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  (a) Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0, subvirial, 1 pc  (b) Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0, supervirial, 1 pc  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Plots of the mean Mahalanobis distance from each star to the average in simulations without primordial substructure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' a fractal dimension of  Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0, over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The shaded blue area and solid blue line show the minimum and maximum mean Mahalanobis distance found across the 10 simulations  and the the solid blue line shows the mean of the means across all 10 simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The Mahalanobis distance is calculated in the 3D phase space  for the blue area and line and calculated in the 6D phase space, shown by the grey shaded area and the black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  mean stellar mass density of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='7 M⊙ pc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We find similar  results to the more dense subvirial simulations simulations, where  the 3D Mahalanobis density clearly traces the ‘bump’ of the col-  lapse, which then decreases as stars move apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The time the ‘bump’  occurs is delayed by several Myr compared to the higher density re-  gions, due to the longer dynamical time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='3 Comparison to other methods of quantifying structure  We now plot the 3D and 6D Mahalanobis densities against other  measures of quantifying structure in star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Figure 9  shows ¯ρm,20 plotted against the established methods of ΛMSR, Q and  ΣLDR for the simulations with initial fractal dimension Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and  radius 1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the initially subvirial simulations the ¯ρm,20 values  stay below 12 for the first 5 Myr whereas the supervirial simulations  can achieve much higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This is due to the stars in supervirial  regions forming small groupings as the region expands which causes  an increase in the Mahalanobis density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In contrast, for the subvirial  regions more stars are interacting with each other, which erases spa-  tial and kinematic substructure and also ejects stars (Schoettler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' As we are measuring the mean Mahalanobis density we are  sensitive to a small number of stars being ejected which we see as a  decrease in the mean Mahalanobis density for the subvirial simula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2022)  100  6D  3D  10 :  1  0  0  1  10  Time (Myr)100:  10 :  1 :  0  0  L  10  Time (Myr)100  6D  3D  10  pm,20  0  0  1  10  Time (Myr)100  10  pm,20  0  0  1  10  Time (Myr)8  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Blaylock-Squibbs & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker  (a) Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0, subvirial, 1 pc  (b) Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0, supervirial, 1 pc  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The mean Mahalanobis density against time for each of the 10 subvirial (left-hand panel) and supervirial (right-hand panel) simulations without  primordial substructure (fractal dimension Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0) and radii of 1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The shaded blue area and solid blue line show the range of mean Mahalanobis densities  calculated in the 3D phase space and the mean of the means found across all 10 simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The grey shaded area and the dashed black line show  the same but for the 6D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  (a) Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0, subvirial, 5 pc  (b) Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0, supervirial, 5 pc  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The mean Mahalanobis density against time for each of the 10 subvirial (left-hand panel) and supervirial (right-hand panel) simulations without  primordial substructure (fractal dimension Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0) with radii of 5 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The shaded blue area and solid blue line show the range of mean Mahalanobis densities  calculated in the 3D phase space and the mean of the means found across all 10 simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The grey shaded area and the dashed black line show  the same but for the 6D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We first show the Mahalanobis density versus the amount of mass  segregation as defined by ΛMSR Allison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2009) in Figure 9(a)  and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the subvirial simulations mass segregation is detected  for 6 of the 10 simulations at 1 Myr and only one simulation has  mass segregation present at 5 Myr, with ΛMSR > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The reason for  the dissipation in the amount of mass segregation is due to the ejec-  tion of massive stars from unstable Trapezium-like systems (Allison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Allison & Goodwin 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker & Goodwin 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In  the supervirial simulations (see panel (b)) one region becomes mass  segregated at 1 Myr and another at 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' If the cluster splits in two,  with the most massive stars located in one of the halves then ΛMSR  can increase to the value we see in Figure 9 (b) of around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' As  discussed in Parker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2014), this is because the massive stars  generally do not interact with each other as they do in the subvirial  simulations where there is more mixing resulting in any structure in  the phase spaces being erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The supervirial simulations display a wider spread in the Maha-  lanobis densities meaning that the plot of ¯ρm,20 versus ΛMSR can be  used to distinguish between different initial virial states after at least  5 Myr of dynamical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The clearest distinction between different times in the simulations  comes when ¯ρm,20 is combined with the Q-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Figure 9(c)  and (d) show this clearly for both the subvirial simulations and the  supervirial simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The plots also show that, as expected, the  supervirial simulations maintain substructure for longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' With some  MNRAS 000, 1–13 (2022)  100  6D  3D  10  0  0  1  10  Time (Myr)100  10:  ,20  0  0  1  10  Time (Myr)100  6D  3D  10:  0  0  1  10  Time (Myr)100  10 :  ,20  0  0  1  10  Time (Myr)Evolution of phase space densities in SFR  9  of the regions maintaining traces of substructure for 5 Myr as mea-  sured using Q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Panels (e) and (f) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 9 show ¯ρm,20 plotted against the relative  local surface density ratio, ΣLDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We find that for both subvirial and  supervirial simulations there is an increase in the local surface den-  sity of the 10 most massive stars compared to all stars in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Interestingly the simulations with the highest local surface density  around the 10 most massive stars do not necessarily have the highest  Mahalanobis densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This is likely due to the local surface density  being calculated on the plane of the sky whereas the Mahalanobis  density is being calculated for the full 3D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The supervirial regions display high Mahalanobis densities at  later times, and we can use this, and the different evolution of the  Q-parameter and ΛMSR to distinguish between initial conditions af-  ter several Myr of dynamical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Using the 6D Mahalanobis density does not improve the diagnos-  tic ability of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We find that the range of ¯ρm,20 decreases  when calculated in 6D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Meaning we see that the ¯ρm,20 values overlap  making differentiating between different snapshots and virial states  impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We now show the same plots but for simulations with little to no  primordial spatial or kinematic substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Figure 10 shows the  mean 3D and 6D Mahalanobis densities plotted against the estab-  lished methods for simulations that have an initial fractal dimen-  sion Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 and radius 1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The Mahalanobis densities for these  simulations increase over time, with supervirial simulations having  higher Mahalanobis densities compared to the subvirial simulations  after 10 Myr of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Panels (a) and (b) show ¯ρm,20 plotted against ΛMSR for the 10  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the subvirial simulations we see mass segregation  detected in three of the 10 simulations, for the supervirial simula-  tions we detect mass segregation in two of the 10 simulations (recall  that our threshold for declaring mass segregation is ΛMSR > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=')  As for the highly substructured simulations (Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6), the plot  of the mean Mahalanobis density when combined with the Q-  parameter gives the clearest distinction between the different snap-  shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For panels (e) and (f) we show ¯ρm,20 against ΣLDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We can see  that the 10 most massive stars can end up in a wide range of local  surface density ratios The subvirial simulations have a wider range  of values, with ΣLDR between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='1 and 10, whereas the supervirial  simulations all finish with ΣLDR > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The grey markers in Figure 10 show the 6D Mahalanobis densities  against the established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We find once again that the spread  in the Mahalanobis densities has decreased, making differentiating  between different times or virial states impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  4 DISCUSSION  We have been motivated to test the Mahalanobis density due to its  recent applications in quantifying the phase space of exoplanet host  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2020) they propose that hot Jupiters are more  likely to be found around host stars that are in high 6D phase space  density, as measured using the 6D Mahalanobis density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' However,  this was questioned by Mustill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2022) who show the peculiar  velocities introduce a bias that once accounted for results in no sig-  nificant excess of hot Jupiters around host stars in high 6D phase  space density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The aim of this work is not to make a scientific as-  sessment of the Mahalanobis density in its use in planet formation  specifically but simply to see how it changes over time when look-  ing at simple N-body simulations of star-forming regions to see what  information, if any, it may be able to give us about the initial condi-  tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' virial state, density and initial morphology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Due to the simplicity of our simulations there are a number of  important caveats that must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' First, there is no  galactic potential or tidal force acting on our simulated star-forming  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The presence of an external Galactic tidal field would likely  increase the dissolution of the star-forming region by causing out-  lying stars to become unbound, which would in turn increase the  potency of the Galactic tidal field at later ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Two more important caveats are that we do not simulate any gas,  and therefore there is no gas potential and also that our systems are  fully isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The most important caveat that disallows direct com-  parison to the works of Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2020) and Mustill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' (2022)  is that we do not simulate planets in our simulations and so how rep-  resentative our simulations are of real regions with exoplanet host  stars is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='3 we show that the 6D Mahalanobis density in isolation  cannot be used to reliably infer the initial conditions of star-forming  regions due to overlap in the sub- and supervirial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' However,  when the Mahalanobis density in the 3D phase space is combined  with either ΛMSR, ΣLDR or Q then the initial virial conditions can be  inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This is most clear to see in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  The regions that are initially supervirial attain higher final phase  space densities than subvirial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  This is somewhat counter intuitive, but is due to the fact that as the  region expands small groupings of stars can form which will have  similar positions and therefore higher phase space densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The  Mahalanobis distances between these groupings is reduced when  we multiply by the inverse of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' In the sub-  virial cases we see lower 3D phase space densities due to stars being  ejected and ending up in relative isolation compared to the rest of  the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' As we are using the mean Mahalanobis density we are  sensitive to only a few stars being ejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We cannot determine the initial conditions using the 6D Maha-  lanobis density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' One would assume that the more data we have (and  therefore dimensions in the phase space), the more clearly we would  see the distinction between sub- and supervirial simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' How-  ever, somewhat counter intuitively, adding more dimensions to the  phase space effectively ’washes’ out any information that would al-  low us to determine the initial conditions of our star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  For example, in our Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 supervirial simulations, as the stars dy-  namically evolve they may get further apart spatially but kinemati-  cally they may be quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' If two stars that are very far apart end  up moving in the same general direction the velocity and positional  phases spaces will effectively cancel each other out and therefore  removing any information about the initial conditions of the region  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' different positions but similar velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We find that the 6D Mahalanobis density for all simulations is  similar at 10 Myr for both sub- and supervirial regions independent  of the fractal dimension Df and the initial radii of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We  therefore suggest that the Mahalanobis distance, and its associated  density, are not suitable for quantifying the initial conditions of star  formation, nor any subsequent dynamical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  5 CONCLUSION  We present N-body simulations with different initial fractal dimen-  sions, virial states and initial radii and quantify the 3D and 6D phase  space density using the Mahalanobis distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We compare the per-  formance of the Mahalanobis density to more established meth-  ods for quantifying structure in star-forming regions, namely ΛMSR,  MNRAS 000, 1–13 (2022)  10  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Blaylock-Squibbs & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker  (a) ¯ρm,20 vs ΛMSR, subvirial, 1 pc  (b) ¯ρm,20 vs ΛMSR, supervirial, 1 pc  (c) ¯ρm,20 vs Q, subvirial, 1 pc  (d) ¯ρm,20 vs Q, supervirial, 1 pc  (e) ¯ρm,20 vs ΣLDR, subvirial, 1 pc  (f) ¯ρm,20 vs ΣLDR, supervirial, 1 pc  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The mean Mahalanobis density calculated for 3D and 6D phase spaces plotted against other methods of quantifying structure for 10 subvirial and  supervirial simulations which are initially substructured with fractal dimension Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and 1 pc radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The left-hand panels show the results for the subvirial  regions and the right-hand panels show the results for the supervirial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The initial values at 0 Myr are represented by the black circles, the blue crosses  show 1 Myr and the red triangles show 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The grey open circles show the comparison of the 6D Mahalanobis density at 0 Myr, the open grey crosses show  it for 1 Myr and the open grey triangles for 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' From top to bottom the rows show the different methods which ¯ρm,20 is plotted against, with the top row  showing ΛMSR, second row showing Q and the bottom row showing ΣLDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2022)  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content='0  0  1  10  100  ZLDREvolution of phase space densities in SFR  11  (a) ¯ρm,20 vs ΛMSR, subvirial, 1 pc  (b) ¯ρm,20 vs ΛMSR, supervirial, 1 pc  (c) ¯ρm,20 vs Q, subvirial, 1 pc  (d) ¯ρm,20 vs Q, supervirial, 1 pc  (e) ¯ρm,20 vs ΣLDR, subvirial, 1 pc  (f) ¯ρm,20 vs ΣLDR, supervirial, 1 pc  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The mean Mahalanobis density calculated for the 3D and 6D phase spaces plotted against other methods of quantifying substructure for 10 subvirial  and supervirial simulations which have little to no initial substructured with fractal dimension Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 and 1 pc radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The left-hand panels show the subvirial  results and the right-hand panels show the supervirial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The initial values at 0 Myr are represented by the black circles, the blue pluses show 1 Myr and  the red triangles show 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We show the mean 6D Mahalanobis densities at 0 Myr, 1 Myr and 5 Myr with grey open circles, grey open crosses and grey open  triangles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' From top to bottom the rows show the different methods which ¯ρm,20 is plotted against, with the top row showing ΛMSR, second row  showing Q and the bottom row showing ΣLDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  MNRAS 000, 1–13 (2022)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content='0  0  10  ZLDR12  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Blaylock-Squibbs & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker  ΣLDR and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We also applied the Mahalanobis distance in 3D phase  space to sets of static synthetic regions of different morphologies to  test its ability to discriminate between different morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Our  conclusions are as follows:  (i) The Mahalanobis distance in the 3D phase space is degenerate  across a wide range of morphologies commonly observed in star-  forming regions, associations and clusters, and so it cannot be used  to differentiate between different morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  (ii) The 3D Mahalanobis densities, ρm,20 can be used to distin-  guish between the high and low stellar density regions with large  amounts of substructure (Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The low stellar density regions  regions show similar behaviour but delayed by around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 Myr  compared to the high volume density regions due to the dynamical  timescales being longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' This effect is even more pronounced in the  simulations with little to no initial substructure (Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0) where we  see the ‘bump’ occurring several Myr later than in the higher stellar  density simulations corresponding to the subvirial collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  (iii) We show that the Mahalanobis density calculated in the 3D  phase space can be used with the Q-parameter, ΛMSR or ΣLDR to  infer information about a region’s initial virial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  (iv) When using the 6D Mahalanobis densities we see no sig-  nificant differences between any of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Adding more  parameters (adding more dimensions) to the phase space suppresses  any changes in the Mahalanobis density over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We therefore advise against using the Mahalanobis distance as a  method to quantify the morphology of star-forming regions due to  its degeneracy across both substructured regions and also smooth,  centrally concentrated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  When applied to spatial and kinematic phase space (6D), all of its  discriminatory power is washed out (similar to the issues encoun-  tered when applying the Q-parameter to kinematic data, Cartwright  (2009)), and we advocate using combinations of spatial and kine-  matic metrics instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  Plots have been generated using Matplotlib 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='4 (Hunter 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Nu-  merical results calculated using Numpy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='1 and SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='0 (Har-  ris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' RJP acknowledges support from  the Royal Society in the form of a Dorothy Hodgkin fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' We  wish to thank the anonymous reviewer for their comments which  have improved the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  DATA AVAILABILITY  Data is available on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=', 2020, Nature Methods, 17, 261  Williams B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content=', 2022, Monthly Notices of the Royal Astronomical  Society, 514, 578  Winter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content=', Longmore S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content=', Chevance M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=', 2020, Na-  ture, 586, 528  APPENDIX A: MAHALANOBIS DISTANCE VERSUS  MAHALANOBIS DENSITY  In Figure A1 we show the relation between the Mahalanobis distance  and density across the two different phase spaces investigated and  the two different initial virial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  We see that in the positional phase space (3D, the coloured mark-  ers) there is significant overlap in both the Mahalanobis distance and  density, making differentiating between different snapshots imprac-  tical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' For the supervirial regions we see less overlap in the Maha-  lanobis density between the snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' However, there is significant  overlap between the sub- and supervirial simulations, meaning that  neither the Mahalanobis distance nor density can reliably distinguish  between different virial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  In both Figure A1 and Figure A2 we show the position-velocity  phase space (6D) with the grey open markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='  Figure A2 shows the mean Mahalanobis distance plotted against  the mean Mahalanobis density for high density (radii of 1 pc) region  with little or no substructure (Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content=' Blaylock-Squibbs & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Parker  (a) ¯ρm,20 vs ¯Md, subvirial, 1pc  (b) ¯ρm,20 vs ¯Md, supervirial, 1 pc  Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The mean Mahalanobis density ( ¯ρm,20) plotted against the mean Mahalanobis distance ( ¯Md) for highly substructured regions with fractal dimensions  Df = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content='6 and initial radii of 1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' Each region contains 1000 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The black circles the values at 0 Myr, the blue plus signs are the values at 1 Myr and the red  triangles are the values at 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
+page_content=' The grey open circles, crosses and triangles show the same information but for the Mahalanobis distance and density calculated  in the 6D phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content=' The mean Mahalanobis density ( ¯ρm,20) plotted against the mean Mahalanobis distance ( ¯Md) for substructured regions with fractal dimensions  Df = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+page_content=' The black circles the values at 0 Myr, the blue crosses are the values at 1 Myr  and the red triangles the values at 5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf'}
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+The mixed-state entanglement in holographic p-wave
+superconductor model
+Zhe Yang 1,∗ Fang-Jing Cheng 2,† Chao Niu 1,‡ Cheng-Yong Zhang 1,§ and Peng Liu 1¶
+1 Department of Physics and Siyuan Laboratory,
+Jinan University, Guangzhou 510632, China
+2 Department of Astronomy, Beijing Normal University, Beijing 100875, China
+Abstract
+In this paper, we investigate the mixed-state entanglement in a model of p-wave superconduc-
+tivity phase transition using holographic methods. We calculate several entanglement measures,
+including holographic entanglement entropy (HEE), mutual information (MI), and entanglement
+wedge cross-section (EWCS). Our results show that these measures display critical behavior at the
+phase transition points, with the EWCS exhibiting opposite temperature behavior compared to the
+HEE. Additionally, we find that the critical exponents of all entanglement measures are twice those
+of the condensate. Moreover, we find that the EWCS is a more sensitive indicator of the critical
+behavior of phase transitions than the HEE. Furthermore, we uncover a universal inequality in the
+growth rates of EWCS and MI near critical points in thermal phase transitions, such as p-wave
+and s-wave superconductivity, suggesting that MI captures more information than EWCS when a
+phase transition first occurs.
+∗Electronic address: yzar55@stu2021.jnu.edu.cn
+†Electronic address: fjcheng@mail.bnu.edu.cn
+‡Electronic address: niuchaophy@gmail.com
+§Electronic address: zhangcy@email.jnu.edu.cn
+¶Electronic address: phylp@email.jnu.edu.cn; Corresponding author
+1
+arXiv:2301.13574v1  [hep-th]  31 Jan 2023
+
+Contents
+I. Introduction
+2
+II. Holographic setup for p-wave superconductor and Holographic
+information-related quantities
+4
+A. The holographic p-wave superconductor model
+4
+B. The phase diagram of holographic p-wave superconductor model
+6
+C. The holographic quantum information
+8
+III. The computation of the holographic quantum information
+11
+A. The holographic entanglement entropy and mutual information
+11
+B. The minimum entanglement wedge cross section
+13
+IV. The Scaling behavior of the quantum information
+15
+V. The growth rate of the holographic quantum information
+17
+VI. Discussion
+19
+Acknowledgments
+20
+References
+20
+I.
+INTRODUCTION
+Quantum entanglement is the most crucial characteristic of the quantum system and
+lays the key foundation of quantum information theory. Recently, quantum information has
+been attracting heavy attention from numerous fields, such as holographic theory, quantum
+many-body systems, and condensed matter theory. According to recent research, quantum
+information can detect quantum phase transitions and play a key role in spacetime emergence
+[1–5].
+In recent years, a variety of measures of quantum information have been proposed, such
+as entanglement entropy (EE), mutual information (MI), and R´enyi entropy. EE is a widely
+used quantity that describes the entanglement of pure states very well. However, EE is not
+2
+
+suitable for describing the entanglement of the more prevalent mixed states. To address this
+issue, new measures such as entanglement of purification (EOP), reflected entropy, quantum
+discord, and others have been suggested for mixed-state systems [6, 7]. However, calculating
+these measures of quantum information can be challenging, particularly in strongly corre-
+lated systems. The complexity of these calculations increases exponentially with the size of
+the quantum system.
+The gauge/gravity duality theory has been proved powerful tool for studying strongly
+correlated quantum systems by dualizing such systems to classical gravitational systems
+[8–12]. It has been shown that the background geometry of the dual gravitational system
+encodes the quantum information of the dual field theory. For instance, the entanglement
+entropy (EE) is related to the minimum surface in the bulk, also known as the holographic
+entanglement entropy (HEE) [13]. The ability of HEE to detect quantum phase transitions
+and thermal phase transitions has been investigated in [14–17].
+Recently, the entangle-
+ment wedge cross-section (EWCS) has been proposed as a novel measure of mixed-state
+entanglement in holographic systems [18, 19]. Additionally, various types of mixed-state
+entanglement, such as reflected entropy, logarithmic negativity, and odd entropy have been
+linked to the EWCS in holographic systems [20–23]. In conclusion, EWCS is a powerful tool
+for investigating mixed-state entanglement in strongly coupled field theories [24–28].
+Holographic superconductivity is a key topic in the gauge/gravity theory, providing a
+novel approach to studying high-temperature superconductors [29–32]. The symmetry of
+the Cooper pair wave function allows for the classification of superconductors as s-wave, p-
+wave, d-wave, etc. The main characteristics of the phase transition in superconductors are
+spontaneous symmetry breaking and the emergence of order parameters. For instance, an s-
+wave holographic superconductor is thought to be the spontaneous scalarization of the black
+hole, a p-wave holographic superconductor requires a charged vector field in the bulk as the
+vector order parameter, and a d-wave model was built by introducing a charged massive spin
+two field propagating in the bulk [33]. It has been recently shown that quantum information
+can be used to diagnose the phase transition of an s-wave superconductor [16, 26, 34].
+However, research on the effects of mixed-state entanglement in p-wave superconductors is
+currently lacking. Therefore, it would be interesting to investigate the connection between
+the holographic p-wave superconducting phase transition and mixed-state entanglement.
+In this paper, we aim to systematically study the role of mixed-state entanglement dur-
+3
+
+ing the p-wave superconductivity phase transition. The paper is organized as follows: In
+Sec. II, we introduce the holographic p-wave superconductor model, and the concepts of
+holographic quantum information, including holographic HEE, MI and EWCS. We explore
+the characteristics of mixed-state entanglement in Sec. III. In Sec. IV, we provide analyt-
+ical and numerical analysis of the scaling behavior of mixed-state entanglement measures.
+Additionally, we uncover an inequality between EWCS and MI in Sec. V. Finally, in Sec.
+VI, we summarize our findings and conclusions.
+II.
+HOLOGRAPHIC SETUP FOR P-WAVE SUPERCONDUCTOR AND HOLO-
+GRAPHIC INFORMATION-RELATED QUANTITIES
+We begin by presenting the model of a holographic p-wave superconductor and its phase
+diagram. Following that, we introduce HEE, as well as the mixed-state entanglement mea-
+sures MI and EWCS.
+A.
+The holographic p-wave superconductor model
+In the p-wave superconductor model, as the temperature drops to a specific critical value,
+spontaneous symmetry breaking occurs, resulting in a vector order parameter. The system
+then transits from the normal phase (absence of vector hair) to the superconducting phase
+(presence of vector hair). The holographic p-wave model is constructed by introducing a
+complex vector field into Einstein-Maxwell theory with a negative cosmological constant
+[35, 36],
+S = 1
+2κ2
+�
+d4x√−g
+�
+R + 6
+L2 − 1
+4FµνF µν − 1
+2ρ†
+µνρµν − m2ρ†
+µρµ + iqγρµρ†
+νF µν
+�
+,
+(1)
+where κ2 = 8πG is related to the gravitational constant, L the AdS radius that we set as
+1. A is the gauge field and the field strength Fµν = ∇µAν − ∇νAµ. ρµ is a complex vector
+field with mass m and charge q. The tensor ρµν = Dµρν − Dνρµ with covariant derivative
+defined as Dµ = ∇µ − iqAµ. The last term in the action is the non-minimum coupling term
+between the Maxwell field and the complex vector field. In this paper, we only consider the
+4
+
+case without an external magnetic field. The equation of motion (EOM) can be read as,
+∇νFνµ =iq(ρνρ†
+νµ − ρν†ρνµ) + iqγ∇ν(ρνρ†
+µ − ρ†
+νρµ),
+Dνρνµ−m2ρµ + iqγρνFνµ = 0,
+Rµν − 1
+2Rgµν − 3
+L2gµν =1
+2FµλF λ
+ν + 1
+2
+�
+−1
+4FµνF µν − 1
+2ρ†
+µνρµν − m2ρ†
+µρµ + iqγρµρ†
+νF µν
+�
+gµν+
+1
+2{[ρ†
+µλρλ
+ν + m2ρ†
+µρν − iqγ(ρµρ†
+λ − ρ†
+µρλ)F λ
+ν ] + µ ↔ ν}.
+(2)
+We solve the EOM with this ansatz,
+ds2 = 1
+z2
+�
+−p(z)(1 − z)U(z)dt2 +
+1
+p(z)(1 − z)U(z)dz2 + V1(z)dx2 + V2(z)dy2
+�
+,
+Aνdxν = µ(1 − z)a(z)dt,
+ρνdxν = ρx(z)dx,
+(3)
+where p(z) ≡ 1 + z + z2 − µ2z3
+4 . µ is the chemical potential of the dual field theory. The
+radius axis is denoted by z, which ranges from 0 to 1, with z = 0 and z = 1 representing the
+AdS boundary and horizon, respectively. In our ansatz, there are five unknown functions,
+U(z), V1(z), V2(z), a(z), and ρx(z), which can be obtained by solving the EOM. The ansatz
+(3) reduces to the AdS-RN black brane solution when U = V1 = V2 = a = 1 and ρx = 0.
+The expansion of the ρx near the AdS boundary is
+ρx = ρx−z∆− + ρx+z∆+ + · · · ,
+(4)
+where the scaling dimension ∆± = 1±
+√
+1+4m2
+2
+and we set the source ρx− = 0 for the condensate
+arise spontaneously. After solving the EOM, we can obtain the condensate ⟨Jx⟩ by extracting
+the coefficient ρx+ . The condensate ⟨Jx⟩ emerges at a specific temperature when varying
+m2 and q. Consequently, in the dual quantum field, the vector operator acquires a non-zero
+vacuum expectation value and spontaneously breaking the U(1) symmetry and rotational
+symmetry. Therefore, ⟨Jx⟩ can be used as the order parameter of p-wave superconducting
+phase transition.
+The Hawking temperature of this model is ˜T = 12−µ2
+16π . The system is invariant under the
+following rescaling,
+(t, x, y) → α−1(t, x, y),
+(U, V1, V2) → α2(U, V1, V2),
+µ → αµ,
+˜T → α ˜T,
+ρx+ → α∆++1ρx+.
+(5)
+5
+
+0.2
+0.4
+0.6
+0.8
+1.0 T/Tc
+0.1
+0.2
+0.3
+0.4
+〈Jx〉
+2
+5
+q=1.5, m2=3/4
+-8
+-7
+-6
+-5
+-4
+ln(δ(1-T/Tc)
+-2.0
+-1.5
+-1.0
+-0.5
+ln(δ〈Jx〉)
+0.2
+0.4
+0.6
+0.8
+1.0
+T/Tc
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+〈Jx〉
+2
+5
+q=1.2, m2=3/4
+0.85
+0.90
+0.95
+1.00
+1.05
+T/Tc
+-0.1002
+-0.1000
+-0.0998
+-0.0996
+-0.0994
+-0.0992
+Ω
+FIG. 1: Left plot: The second-order phase transition occurs as the temperature falls below the
+critical value. The inset plot illustrates the scaling behavior of the condensate ⟨Jx⟩. Right plot:
+The first-order phase transition occurs when the temperature falls below the critical temperature,
+which represents by the black dashed line. The inset plot illustrates the effective free energy density
+Ω versus the temperature T/Tc, revealing that the superconducting phase is thermodynamically
+favored.
+In this paper, we adopt the chemical potential µ as the scaling unit, which is equivalent to
+treating the dual system as a field theory described by the giant canonical ensemble. The
+dimensionless Hawking temperature T = ˜T/µ.
+B.
+The phase diagram of holographic p-wave superconductor model
+This holographic p-wave superconductor model can exhibit zeroth-order, first-order, and
+second-order phase transitions depending on the values of m and q. For example, a second-
+order phase transition can occur at q = 1.5, m2 = 3/4 with a critical temperature of
+Tc ≈ 0.01791. In Fig. 1, we demonstrate the relationship between the condensate ⟨Jx⟩2/5
+and temperature by plotting the scaling relationship,
+δ(⟨Jx⟩) ∼
+�
+1 − T
+Tc
+�αc
+.
+(6)
+Theoretical calculations predict that the critical exponent is α = 1/2 [35]. Our numerical
+results also indicate that αc ≈ 0.500106.
+A first-order phase transition can occur at q = 1.2 and m2 = 3/4 with a critical tem-
+perature of Tc ≈ 0.003382. To better visualize the phase structure, we plot the effective
+6
+
+m2=0.10
+m2=0.30
+m2=0.50
+m2=0.80
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+q
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+T
+Superconducting phase
+Normal phase
+FIG. 2: The phase diagram of holographic p-wave superconductor model with positive m2. The
+solid lines are the critical points.
+free energy density (as shown in Fig.1).
+The effective free energy density is defined as
+˜Ω = M − Ts, where T is the Hawking temperature, s is the entropy density, and M is the
+mass density of the black brane [37]. The mass density of the black brane can be obtained
+by using the AdS asymptotic behavior of gtt in our ansatz,
+(1 − z)U(z)
+�
+− 1
+4µ2z3 + z2 + z + 1
+�
+z2
+∼ 1
+z2 + Mz + Qz2 + · · · .
+(7)
+The free energy of the superconducting phase is lower than the normal phase when the
+temperature drops below the critical temperature Tc. As a result, the system will abruptly
+transition from the normal phase to the superconducting phase.
+To more thoroughly understand the behavior of p-wave superconductivity, we present
+the phase diagram in Fig. 2. The phase diagram is constructed by identifying the critical
+points, which can be found by examining the emergence of condensation as a perturbation
+near these points. The linearized equations of motion can be transformed into an eigenvalue
+problem that we solve using numerical methods
+1
+32µ2(z − 1)z2(µ2z3 − 4z2 − 4z − 4)(2z2(z2(µ2(4z − 3) − 12)ρ′
+x(z)+
+�
+µ2z4 −
+�
+µ2 + 4
+�
+z3 + 4
+�
+ρ′′
+x(z)) − 6ρx(z)) = −q2ρx(z).
+(8)
+By analyzing the eigenvalues, we can determine the upper or lower bounds of the critical
+points, which correspond to the boundaries of the different phases in the diagram.
+7
+
+x
+y
+z
+x
+y
+z
+FIG. 3: Left plot: the minimum surface for a subsystem (red region). Right plot: the minimum
+cross-section (green surface) of the entanglement wedge.
+C.
+The holographic quantum information
+Quantum entanglement is a fundamental characteristic of quantum systems. EE is a
+well-known measure of entanglement, which quantifies the correlation between a subsystem
+and its complement for pure states. It is defined in terms of the reduced density matrix ρA
+[38],
+SA(|ψ⟩) = −Tr[ρAlog(ρA)],
+ρA = TrB(|ψ⟩⟨ψ|).
+(9)
+The HEE was proposed to be dual to the area of the minimum surface in the gravitational
+system [39]. In this paper, we consider the HEE of the configuration with an infinitely long
+strip along the y-axis (see Fig.
+3).
+HEE typically diverges due to the asymptotic AdS
+boundary. The regulation is implemented by subtracting the divergent term from the HEE.
+It should be noted that HEE is not suitable for describing the mixed-state entanglement.
+For example, EE of the quantum system characterized by the direct product state HA ⊗HB
+is not equal to zero, but the entanglement of the subsystems is vanishing. This is because
+EE contains both quantum and classical correlation. Therefore, as the dual of EE, HEE is
+also affected by thermodynamic entropy in mixed-state systems [40, 41].
+To better solve the problem of mixed-state entanglement measurement, numerous novel
+entanglement measures have been proposed. One popular measure is mutual information
+(MI), which quantifies the correlation between two subsystems A and C that are separated
+by a subsystem B. According to the definition of MI, it is calculated as [42, 43],
+I(a : c) = S(a) + S(c) − min(S(a ∪ c)),
+(10)
+where S(x) denotes the entanglement entropy of subsystem x. Unlike entanglement entropy,
+8
+
+a
+Sc
+Sa
+Sb
+Sa+b+c
+b
+c
+FIG. 4: The illustration of the holographic mutual information.
+MI for direct product states HA ⊗ HB is always zero, making it a more appropriate measure
+for mixed-state entanglement. In the holographic context, the dual of MI is the difference in
+area between red (disconnected configuration) and blue surfaces (connected configuration),
+as shown in Fig. 4. As the subsystem A, C becomes smaller or when the separation B
+becomes larger, MI decreases and eventually reaches zero, indicating a disentangling phase
+transition. However, MI has some limitations as a mixed-state entanglement measure as it
+is directly related to entanglement entropy and can be dominated by it in some cases [25].
+Therefore, it is important to explore other mixed-state entanglement measures.
+Recently, the minimum cross-section of the entanglement wedge (EWCS) is proposed
+as a novel holographic mixed-state entanglement measure [18]. EWCS is considered to be
+the duality of reflected entropy, logarithmic negativity, and odd entropy. The definition of
+EWCS is as follows,
+Ew(ρAB) = min
+ΣAB
+�Area(ΣAB)
+4GN
+�
+.
+(11)
+Fig. 3 is an illustration of EWCS in a bipartite system a∪c divided by b. The area bounded
+by the minimum surface of the disconnected configuration is known as the entanglement
+wedge. It is important to note that entanglement between subsystems only exists when the
+total correlation is not zero, which means the MI does not vanish.
+Although EWCS plays a significant role in measuring the entanglement of mixed-state
+systems, it is still challenging to solve it [24]. First, it is hard to solve the highly nonlinear
+EOM of the minimum surface. Second, the minimum cross-section is obtained by scanning
+a two-dimensional parameter space, which is a hard task. Last but not least, the coordinate
+singularity close to the AdS boundary with the asymptotic AdS can hinder numerical preci-
+sion. We have proposed an efficient algorithm for solving EWCS based on the requirement
+9
+
+θ2
+θ1
+C1(θ1)
+C2(θ2)
+P1
+P2
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+x
+0.2
+0.4
+0.6
+0.8
+1.0
+z
+FIG. 5: The illustration of the numerical algorithm for EWCS.
+that the minimum cross-section is locally orthogonal to the boundaries of the entanglement
+wedge [26]. Fig. 5 shows the illustration of the key concept for our numerical algorithm.
+We consider EWCS of the infinite strip along the y-direction in a homogeneous background
+ds2 = gttdt2 + gzzdz2 + gxxdx2 + gyydy2.
+(12)
+The minimum surfaces of the connected configuration can be represented as C1(θ1) and
+C2(θ2). The minimum surfaces intersect with the cross-section at points p1 and p2, and the
+area of this local minimum surface (the red curve in Fig. 5) is,
+A =
+�
+Cp1,p2
+�
+gxxgyyx′(z)2 + gzzgyydz.
+(13)
+Variating (13), we obtain the EOM determining the local minimum surface,
+x′(z)3
+� gxxg′
+yy
+2gyygzz
++ g′
+xx
+2gzz
+�
++ x′(z)
+�g′
+xx
+gxx
++ g′
+yy
+2gyy
+− g′
+zz
+2gzz
+�
++ x′′(z) = 0.
+(14)
+Remind that the global minimum cross-section is locally orthogonal to the entanglement
+wedge, which implies that
+� ∂
+∂z
+, ∂
+∂θ1
+�
+p1
+= 0,
+� ∂
+∂z
+, ∂
+∂θ2
+�
+p2
+= 0
+(15)
+where ⟨·, ·⟩ represents the vector product with metric gµν. We can normalize the orthogonal
+relation,
+Q1(θ1, θ2) ≡
+⟨ ∂
+∂z,
+∂
+∂θ1⟩
+�
+⟨ ∂
+∂z, ∂
+∂z⟩⟨ ∂
+∂θ1,
+∂
+∂θ1⟩
+������
+p1
+= 0,
+Q2(θ1, θ2) ≡
+⟨ ∂
+∂z,
+∂
+∂θ1⟩
+�
+⟨ ∂
+∂z, ∂
+∂z⟩⟨ ∂
+∂θ2,
+∂
+∂θ2⟩
+������
+p2
+= 0.
+(16)
+10
+
+l=1.50
+l=1.52
+l=1.54
+l=1.56
+l=1.58
+l=1.60
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+T
+Tc
+-1.1
+-1.0
+-0.9
+-0.8
+-0.7
+SE
+q=1.5, m2=3/4
+l=3.00
+l=3.20
+l=3.40
+l=3.60
+l=3.80
+l=4.00
+0.4
+0.6
+0.8
+1.0
+1.2
+T
+Tc
+-0.3
+-0.2
+-0.1
+0.0
+SE
+q=1.2, m2=3/4
+FIG. 6: The holographic entanglement entropy SE vs temperature T/Tc with various strip width
+l. The critical point is indicated by the black dashed line. The stable and metastable states are
+depicted by solid and transparent lines, respectively. Left plot: The second-order phase transition
+at Tc ≈ 0.01791. Right plot: The first-order phase transition at Tc ≈ 0.003382.
+Finding the cross-section located at the minimum surface at (θ1, θ2) where (16) is satisfied,
+we obtain the minimum cross-section. To this end, we adopt the Newton-Raphson method
+to locate the endpoints satisfying the local perpendicular conditions. Based on the above
+techniques, we can study the relationship between the holographic p-wave superconductor
+and the EWCS [26].
+III.
+THE COMPUTATION OF THE HOLOGRAPHIC QUANTUM INFORMA-
+TION
+A.
+The holographic entanglement entropy and mutual information
+In Fig.
+6, we show the relationship between the HEE and temperature T/Tc during
+second-order and first-order phase transitions.
+For q = 1.5 and m2 = 3/4, where the
+second-order phase transition occurs, the HEE increases with increasing temperature. For
+q = 1.2 and m2 = 3/4, where the first-order phase transition occurs, the HEE jumps
+abruptly when crossing the critical point. To understand this behavior, we can examine the
+relationship between HEE and thermodynamic entropy, as when the configuration is large or
+the temperature is high enough, the minimum surface will approach the horizon of the black
+brane and HEE will primarily be determined by thermodynamic entropy. Therefore, we will
+next analyze the thermodynamic entropy behavior of the black brane to better understand
+11
+
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+T
+Tc
+0.02
+0.04
+0.06
+0.08
+0.10
+s
+q=1.5, m2=3/4
+-6.0 -5.8 -5.6 -5.4 -5.2 -5.0
+ln(δ(1-T/Tc))
+-5.4
+-5.2
+-5.0
+-4.8
+-4.6
+ln(δs)
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+T
+Tc
+0.02
+0.04
+0.06
+0.08
+s
+q=1.2, m2=3/4
+FIG. 7: The entropy density s vs temperature T/Tc. The black dashed line represents the entropy
+density of the normal phase. The entropy density of the superconducting phase is represented by
+the purple line. Left plot: The second-order phase transition of holographic p-wave superconductor
+model. The inset plot depicts the logarithm between s and 1− T
+Tc . Right plot: The first-order phase
+transition of holographic p-wave superconductor model. The critical temperature is indicated by
+the red dashed line.
+the behavior of HEE [40, 41].
+The entropy density is given by,
+˜s = 2πA
+κ2
+= 2π
+�
+V1(z)V2(z)
+κ2
+ˆV ,
+(17)
+where A is the area of the horizon and ˆV =
+�
+dxdy is the corresponding area of the region
+in the dual field theory [44].
+Dividing the entropy by the area ˆV and µ2, we have the
+dimensionless entropy density s =
+κ2˜s
+2π ˆV µ2. The plot of the entropy density near the critical
+point can be seen in Fig.
+7.
+The above phenomena show that both HEE and entropy
+density can detect the critical behavior of the holographic p-wave superconducting phase
+transitions. Similar phenomena of the HEE in the superconducting phase transition can see
+in [16, 17, 26, 45].
+MI is one of the mixed-state entanglement measures that can extract the total correlation
+of the systems. Since MI is directly defined by HEE (see (10)), it also can diagnose the
+phase transition. Moreover, a disentangling phase transition occurs when MI is zero and
+entanglement exists only when MI is greater than zero.
+Fig.
+8 illustrates the behavior
+of the disentangling phase transition for various configurations. However, in certain cases,
+MI is determined by the thermodynamic entropy [25, 26, 37]. Therefore, it is necessary to
+investigate other mixed-state entanglement measures.
+12
+
+c=1.00
+c=1.02
+c=1.04
+c=1.06
+c=1.08
+c=1.10
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+T
+Tc
+0.72
+0.74
+0.76
+0.78
+0.80
+0.82
+bc
+a=2
+a=2.00
+a=2.10
+a=2.20
+a=2.30
+a=2.40
+a=2.50
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+T/Tc
+0.53
+0.54
+0.55
+0.56
+0.57
+cc
+b=0.5
+FIG. 8: The critical configuration for disentangling phase transition. The critical temperature is
+indicated by the black dashed line. The solid and translucent lines represent stable and metastable
+states. Left plot: When b is above the bc, the disentangling phase transition occurs. Right plot:
+When c is below cc, the disentangling phase transition occurs.
+B.
+The minimum entanglement wedge cross section
+We begin by examining the EWCS during a second-order phase transition. Fig. 9 shows
+that EWCS can diagnose the critical behavior of holographic p-wave superconducting phase
+transitions. At the critical point of a second-order phase transition, EWCS is continuous, but
+its first derivative is discontinuous. In the superconducting phase, EWCS always decreases
+with increasing temperature.
+However, we find that the EWCS in the normal phase is
+configuration-dependent. In large configurations, it behaves similarly to the HEE, showing
+a monotonically increasing trend with temperature. In contrast, for small configurations,
+the EWCS of the normal phase exhibits a monotonically decreasing trend with temperature,
+opposite to the behavior of the HEE.
+Next, we investigate the behavior of the EWCS during a first-order phase transition.
+Fig. 10 illustrates EWCS behavior during this phase transition, with the inset plot showing
+the derivative of EWCS with respect to temperature (∂TEw) versus temperature T. The
+inset plot illustrates that in normal phase the EWCS decreases with increasing temperature.
+Unlike the HEE, the EWCS of the superconducting phase always decreases with tempera-
+ture. When the temperature falls below a critical point, EWCS abruptly jumps from the
+normal phase to the superconducting phase, this sudden change in EWCS suggests that it
+can capture the first-order phase transition, similar to the HEE and MI.
+In addition to diagnosing the critical points, it is also important to investigate the scaling
+13
+
+c=0.80
+c=0.84
+c=0.88
+c=0.92
+c=0.96
+c=1.00
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1T/Tc
+2.12
+2.14
+2.16
+2.18
+2.20
+Ew
+a=2, b=0.5
+0.95 1.00 1.05 1.10
+T/Tc
+-0.25
+-0.20
+-0.15
+-0.10
+-0.05
+0.00
+∂TEw
+c=0.80
+c=0.84
+c=0.88
+c=0.92
+c=0.96
+c=1.00
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1T/Tc
+5.38
+5.40
+5.42
+5.44
+5.46
+5.48
+5.50
+Ew
+a=1, b=0.2
+0.95 1.00 1.05 1.10
+T/Tc
+-0.08
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+∂TEw
+FIG. 9: The EWCS Ew vs the temperature T/Tc. The inset graph depicts ∂T Ew. The black
+dashed line depicts the critical temperature. Left plot: The ∂T Ew of the normal phase is greater
+than zero when a = 2 and b = 0.5. Right plot: The ∂T Ew of the normal phase is less than zero
+when a = 1 and b = 0.2.
+c=1.80
+c=1.84
+c=1.88
+c=1.92
+c=1.96
+c=2.00
+0.9
+1.0
+1.1
+1.2
+1.3
+T
+Tc
+2.120
+2.125
+2.130
+2.135
+2.140
+2.145
+2.150
+Ew
+a=2, b=0.5
+b=0.500
+b=0.504
+b=0.508
+b=0.512
+b=0.516
+b=0.520
+0.9
+1.0
+1.1
+1.2
+T
+Tc
+2.05
+2.10
+2.15
+Ew
+a=2,c=1.8
+FIG. 10: The EWCS Ew versus the temperature T/Tc in the first-order phase transition. The
+dashed black line represents the critical temperature when Tc ≈ 0.003382. The translucent line
+represents the metastable state, whereas the solid line represents the stable state. Left plot: We
+set a = 2 and b = 0.5 with varying c values. Right plot: We set a = 2 and c = 1.8 while varying b
+values.
+behavior of the holographic quantum information. Next, we analyze the critical behavior
+of the quantum information-related quantities during the p-wave superconductivity phase
+transitions.
+14
+
+l=0.50
+l=1.00
+l=1.50
+l=2.00
+l=3.00
+-9
+-8
+-7
+-6
+-5
+ln(δ(1-
+T
+Tc
+))
+-9
+-8
+-7
+-6
+-5
+ln(δSE)
+-8.0-7.8-7.6-7.4-7.2-7.0-6.8-6.6
+ln(δ(1-
+T
+Tc
+)
+0.97
+0.98
+0.99
+1.00
+1.01
+1.02
+slope
+c=1.50
+c=1.75
+c=2.00
+c=2.25
+c=2.50
+-9.0
+-8.5
+-8.0
+-7.5
+-7.0
+-6.5
+-6.0
+ln(δ(1-
+T
+Tc
+)
+-9
+-8
+-7
+-6
+-5
+ln(δEw)
+a=2, b=0.5
+-8.0-7.8-7.6-7.4-7.2-7.0-6.8-6.6
+ln(δ(1-
+T
+Tc
+))
+0.98
+1.00
+1.02
+1.04
+ln(δEw)
+FIG. 11: The scaling behavior of HEE and EWCS. The inset plot shows the slope of the holographic
+quantum information. Left plot: ln(δSE) versus ln(δ(1− T
+Tc)) with different width of l. Right plot:
+we set a = 2 and b = 0.5 and ln(δEw) versus ln(δ(1 − T
+Tc )) with different values of c.
+IV.
+THE SCALING BEHAVIOR OF THE QUANTUM INFORMATION
+As the critical point marks the bifurcation point between the normal and superconducting
+phases, to study the critical behavior, we compare the quantum information quantities of
+the normal phase to those of the superconducting phase by subtracting the former from the
+latter,
+δSE = Scond
+E
+− Snormal
+E
+,
+δEw = Econd
+w
+− Enormal
+w
+.
+(18)
+We propose the following critical behaviors for the HEE and EWCS,
+δS ∼
+�
+1 − T
+Tc
+�αHEE
+,
+δEw ∼
+�
+1 − T
+Tc
+�αEWCS
+,
+(19)
+where αHEE and αEWCS are the critical exponent of the HEE and EWCS, respectively. We
+plot the critical scaling behavior in Fig. 11, from which we find that both EWCS and HEE
+exhibit excellent scaling behavior near the critical point. More importantly, they both have
+the same critical exponent,
+αHEE ≈ αEWCS ≈ 1.
+(20)
+It is important to note that the vector field ρµ is always zero at temperatures higher than
+the critical temperature. At temperatures slightly below the critical point, the condensate
+vacuum expectation value of ⟨Jx⟩ is small and can be analyzed using perturbation theory.
+We can expand the vector field ρµ and the metric function near the critical point as [46–48],
+15
+
+ρx = ϵρ(1) + ϵ3ρ(3) + ϵ5ρ(5) + · · · ,
+U = 1 + ϵ2U (1) + ϵ4U (4) + · · · ,
+V1 = 1 + ϵ2V (1)
+1
++ ϵ4V (4)
+1
++ · · · .
+(21)
+From (21), we can deduce that the critical exponent of the metric function U, V1, is twice
+that of the condensate ⟨Jx⟩. This can be understood by noting that holographic quantum
+information is represented by geometric objects that depend only on the metric.
+Their
+critical exponent can be written as,
+δ(SE) ∼ δ(Ew) ∼ δ(⟨Jx⟩)2 ∼
+�
+1 − T
+Tc
+�2αc
+.
+(22)
+Therefore, the theoretical critical exponent of holographic quantum information should be
+twice that of the condensate ⟨Jx⟩,
+αHEE = αEWCS = 2αc.
+(23)
+Although EWCS and HEE have the same critical exponent in the critical region, they do
+not tend to the scaling law at the same rate in the critical region. To better investigate this
+phenomenon near the critical point, we define the quasi-critical exponent (QCE) as
+α ≡
+d ln(δS)
+d ln
+�
+1 − T
+Tc
+�.
+(24)
+QCE is a function of ln
+�
+1 − T
+Tc
+�
+. Apparently, the QCE α behavior along ln
+�
+1 − T
+Tc
+�
+can
+measure the extent to which a wide range of S can converge to the scaling law.
+We show the QCE of HEE and EWCS in Fig. 12. From the left plot of Fig. 12 we find
+that the width l has an impact on the scaling behavior of HEE. As the width l increases, the
+scaling behavior of HEE is closer to the theoretical scaling behavior. As the temperature
+moves away from the critical point or the width l decreases, however, the scaling behavior
+of the HEE begins to deviate from the theoretical result.
+The QCE of EWCS is depicted in the right plot of Fig. 12. Comparing the left plot and
+the right plot of Fig. 12, EWCS converges to the theoretical scaling law over a broader range.
+As the separation b decreases, the scaling behavior of EWCS becomes close to the theoretical
+results. This behavior suggests that EWCS, as a measure for mixed-state entanglement, can
+more accurately describe the scaling behavior during superconductivity phase transitions
+than HEE.
+16
+
+l=0.50
+l=2.50
+l=4.50
+l=8.00
+-7
+-6
+-5
+-4
+-3
+ln(δ(1-
+T
+Tc
+))
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+QCE(SE)
+-7.0
+-6.8
+-6.6
+-6.4
+-6.2
+-6.0
+ln(δ(1-T/Tc))
+0.980
+0.985
+0.990
+0.995
+QCE(SE)
+b=0.10
+b=0.25
+b=0.40
+b=0.70
+-7
+-6
+-5
+-4
+-3
+ln(δ(1-
+T
+Tc
+))
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+QCE(Ew)
+a=2, c=1.5
+-7.0
+-6.8
+-6.6
+-6.4
+-6.2
+-6.0
+1-T/Tc
+0.980
+0.985
+0.990
+0.995
+QCE(Ew)
+FIG. 12: The QCE of the HEE and EWCS. The red dashed line represents the twice QCE of the
+condensate ⟨Jx⟩. Left plot: The QCE of the HEE near the critical points. Right plot: The QCE
+of the EWCS near the critical points, when we fix the a = 2 and c = 1.5.
+V.
+THE GROWTH RATE OF THE HOLOGRAPHIC QUANTUM INFORMA-
+TION
+Several important inequalities involving the EWCS have been proposed in the literature
+[19, 49, 50], such as the inequality Ew(ρAC) ≥ 1
+2I(A, c), which states that the EWCS cannot
+be smaller than half of the MI. These inequalities are crucial in the study of mixed-state
+entanglement measures, particularly in testing the validity of holographic duals of certain
+quantum information.
+In this paper, we find a new inequality behavior of EWCS and
+MI related to the superconductivity phase transition: near the phase transition point, the
+relative growth rate of MI along the temperature axis is always greater than that of EWCS.
+When the temperature drops below the critical temperature, the EWCS and the MI of
+the superconducting phases are always larger than those of the normal phases. To take a
+closer look at the relationship between the EWCS and the MI, we define the relative values
+of the MI and the EWCS,
+˜Ew = Ew,cond
+Ew,norm
+,
+˜I = Icond
+Inorm
+.
+(25)
+With this definition, ˜Ew and ˜I are fixed at 1 at the critical point. In Fig. 13, we depict
+the relationship between the ˜I and ˜Ew. Contrary to the inequality [19, 49, 50], the relative
+MI is always larger than the relative values of EWCS in the critical region. To describe this
+relationship quantitatively, we examine the fact that,
+δ(Q) ≃ A(Q)
+�
+1 − T
+Tc
+�α
+,
+(26)
+17
+
+b=0.20
+b=0.47
+b=0.73
+b=1.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1-
+T
+Tc
+1.02
+1.04
+1.06
+1.08
+Relative Value
+a=2, c=2
+0.02 0.04 0.06 0.08 0.10 0.12 0.14
+1-T/Tc
+0.1
+0.2
+0.3
+0.4
+0.5
+Slope
+a=0.4, c=0.2
+a=0.8, c=0.6
+a=1.1, c=0.9
+a=1.5, c=1.3
+0.02
+0.04
+0.06
+0.08
+0.10
+1-
+T
+Tc
+1.0002
+1.0004
+1.0006
+1.0008
+1.0010
+1.0012
+1.0014
+Relative Value
+b=0.1
+0.02 0.04 0.06 0.08 0.10 0.12 0.14
+1-T/Tc
+0.005
+0.010
+0.015
+Slope
+FIG. 13: The relative value of MI and EWCS near the critical point. The dashed lines represent
+˜I and the solid lines represent ˜Ew. The inset plot is the slope of ˜I and ˜Ew. Left plot: We fix the
+subsystems a = c = 2 and change the separation b. Right plot: We fix the separation b = 0.1 and
+change the subsystem a and c.
+where Q stands for any physical quantity possessing critical behaviors. From (26) we find
+that,
+˜Ew = 1 + A( ˜Ew)
+�
+1 − T
+Tc
+�α
+,
+˜I = 1 + A(˜I)
+�
+1 − T
+Tc
+�α
+.
+(27)
+Accordingly, it can be seen that A actually measures the increasing phenomenon of holo-
+graphic quantum information in Fig. 12, and hence we call A the growth rate. We work
+out A( ˜Ew) and A(˜I) for several different configurations and list them in Table I. From these
+numerical results we conclude a new inequality between the EWCS and MI growth rates
+near the critical point,
+A(˜I) > A( ˜Ew).
+(28)
+The growth rate of MI is always greater than that of EWCS near the critical point. Further-
+more, the difference between the growth rates of EWCS and MI increases as the subsystem
+separation b increases.
+Near the critical point, the entanglement of the system changes
+rapidly, and MI is more sensitive to these changes than EWCS. This behavior may be due
+to the fact that MI captures the total correlation of the system, which is more than what
+is captured by EWCS. We have also tested this inequality in other thermal phase transi-
+tion models, such as the holographic s-wave superconductor model, and suggest that this
+inequality (28) may be universal in thermal phase transitions.
+18
+
+TABLE I: The growth rate A( ˜Ew) and A(˜I) at different configurations.
+Configuration
+A( ˜Ew)
+A(˜I)
+a = c = 2, b = 0.20
+0.0426
+0.0741
+a = c = 2, b = 0.47
+0.1162
+0.2947
+a = c = 2, b = 0.73
+0.2094
+0.9855
+a = c = 2, b = 1.00
+0.3212
+10.8712
+a = 0.8, b = 0.2, c = 0.4
+0.00218
+0.00525
+a = 0.8, b = 0.2, c = 0.8
+0.00520
+0.00803
+a = 0.8, b = 0.2, c = 1.2
+0.00862
+0.01336
+a = 0.8, b = 0.2, c = 1.6
+0.01209
+0.01975
+a = 0.5, b = 0.2, c = 0.4
+0.00116
+0.00364
+a = 1.0, b = 0.2, c = 0.9
+0.00794
+0.01182
+a = 1.5, b = 0.2, c = 1.4
+0.02016
+0.03120
+a = 3.0, b = 0.2, c = 2.9
+0.06042
+0.11959
+VI.
+DISCUSSION
+In this study, we investigate mixed-state entanglement measures, including HEE, MI and
+EWCS, in a holographic p-wave superconductor model. The model exhibits both second
+and first-order phase transitions when varying system parameters. We find that HEE and
+EWCS can accurately diagnose the critical behavior of these phase transitions. Additionally,
+we observe that the behavior of HEE is related to thermodynamic entropy as the subsystem
+configuration increases. However, as a mixed-state entanglement measure, EWCS exhibits
+the opposite behavior from HEE in the superconducting phase. Specifically, HEE always
+increases with temperature, whereas EWCS in the superconducting state decreases with
+temperature. In the case of first-order phase transitions, the holographic quantum infor-
+mation experiences sudden changes. However, the EWCS behavior in the normal phase is
+dependent on the subsystem configuration. This behavior demonstrates that EWCS can not
+only detect phase transitions but also capture more information than HEE.
+In addition to diagnosing phase transitions, we also examine the scaling behaviors of the
+condensate and the holographic quantum information. Through analyzing the scaling behav-
+19
+
+ior of various holographic quantum information measures, we find that HEE and EWCS not
+only detect the critical point but also exhibit scaling behaviors. We show both numerically
+and analytically that the critical exponent of holographic quantum information is twice that
+of the condensate. Furthermore, we observe that compared to HEE, EWCS provides a more
+sensitive characterization of the scaling behavior, making it more suitable as a measure for
+mixed-state entanglement in superconductivity phase transitions. Additionally, we propose
+a novel inequality for EWCS and MI in phase transitions and provide numerical evidence
+for this result. The relative growth rate of MI is always larger than that of EWCS near the
+critical point.
+Next, we point out several directions worth further investigation. The investigation of
+topological and quantum phase transitions is an important area of research in condensed
+matter theory [51–53]. In particular, the relationship between HEE and quantum phase tran-
+sitions has been studied under holographic framework in previous works [40, 41]. Further
+research into the mixed-state entanglement in quantum phase transitions and topological
+quantum phase transitions is therefore desirable. Additionally, it would be interesting to
+test the inequality (28) in other thermal phase transition models, such as the d-wave su-
+perconductivity model and the massive gravity model [26, 37]. We are working on these
+directions.
+Acknowledgments
+Peng Liu would like to thank Yun-Ha Zha for her kind encouragement during this work.
+Zhe Yang appreciates Feng-Ying Deng’s support and warm words of encouragement during
+this work. We are also very grateful to Chong-Ye Chen, Mu-Jing Li, and Wei Xiong for their
+helpful discussion and suggestions. This work is supported by the Natural Science Founda-
+tion of China under Grant No. 11905083, 12005077, 11805083, the Science and Technology
+Planning Project of Guangzhou (202201010655) and Guangdong Basic and Applied Basic
+Research Foundation (2021A1515012374).
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf,len=1241
+page_content='The mixed-state entanglement in holographic p-wave  superconductor model  Zhe Yang 1,∗ Fang-Jing Cheng 2,† Chao Niu 1,‡ Cheng-Yong Zhang 1,§ and Peng Liu 1¶  1 Department of Physics and Siyuan Laboratory,  Jinan University, Guangzhou 510632, China  2 Department of Astronomy, Beijing Normal University, Beijing 100875, China  Abstract  In this paper, we investigate the mixed-state entanglement in a model of p-wave superconduc-  tivity phase transition using holographic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We calculate several entanglement measures,  including holographic entanglement entropy (HEE), mutual information (MI), and entanglement  wedge cross-section (EWCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Our results show that these measures display critical behavior at the  phase transition points, with the EWCS exhibiting opposite temperature behavior compared to the  HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Additionally, we find that the critical exponents of all entanglement measures are twice those  of the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Moreover, we find that the EWCS is a more sensitive indicator of the critical  behavior of phase transitions than the HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Furthermore, we uncover a universal inequality in the  growth rates of EWCS and MI near critical points in thermal phase transitions, such as p-wave  and s-wave superconductivity, suggesting that MI captures more information than EWCS when a  phase transition first occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  ∗Electronic address: yzar55@stu2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='cn  †Electronic address: fjcheng@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='bnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='cn  ‡Electronic address: niuchaophy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='com  §Electronic address: zhangcy@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='cn  ¶Electronic address: phylp@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Corresponding author  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='13574v1  [hep-th]  31 Jan 2023  Contents  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Introduction  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Holographic setup for p-wave superconductor and Holographic  information-related quantities  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The holographic p-wave superconductor model  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The phase diagram of holographic p-wave superconductor model  6  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The holographic quantum information  8  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The computation of the holographic quantum information  11  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The holographic entanglement entropy and mutual information  11  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The minimum entanglement wedge cross section  13  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The Scaling behavior of the quantum information  15  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The growth rate of the holographic quantum information  17  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Discussion  19  Acknowledgments  20  References  20  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  INTRODUCTION  Quantum entanglement is the most crucial characteristic of the quantum system and  lays the key foundation of quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Recently, quantum information has  been attracting heavy attention from numerous fields, such as holographic theory, quantum  many-body systems, and condensed matter theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' According to recent research, quantum  information can detect quantum phase transitions and play a key role in spacetime emergence  [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  In recent years, a variety of measures of quantum information have been proposed, such  as entanglement entropy (EE), mutual information (MI), and R´enyi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' EE is a widely  used quantity that describes the entanglement of pure states very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' However, EE is not  2  suitable for describing the entanglement of the more prevalent mixed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' To address this  issue, new measures such as entanglement of purification (EOP), reflected entropy, quantum  discord, and others have been suggested for mixed-state systems [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' However, calculating  these measures of quantum information can be challenging, particularly in strongly corre-  lated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The complexity of these calculations increases exponentially with the size of  the quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The gauge/gravity duality theory has been proved powerful tool for studying strongly  correlated quantum systems by dualizing such systems to classical gravitational systems  [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' It has been shown that the background geometry of the dual gravitational system  encodes the quantum information of the dual field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' For instance, the entanglement  entropy (EE) is related to the minimum surface in the bulk, also known as the holographic  entanglement entropy (HEE) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The ability of HEE to detect quantum phase transitions  and thermal phase transitions has been investigated in [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Recently, the entangle-  ment wedge cross-section (EWCS) has been proposed as a novel measure of mixed-state  entanglement in holographic systems [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Additionally, various types of mixed-state  entanglement, such as reflected entropy, logarithmic negativity, and odd entropy have been  linked to the EWCS in holographic systems [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In conclusion, EWCS is a powerful tool  for investigating mixed-state entanglement in strongly coupled field theories [24–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Holographic superconductivity is a key topic in the gauge/gravity theory, providing a  novel approach to studying high-temperature superconductors [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The symmetry of  the Cooper pair wave function allows for the classification of superconductors as s-wave, p-  wave, d-wave, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The main characteristics of the phase transition in superconductors are  spontaneous symmetry breaking and the emergence of order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' For instance, an s-  wave holographic superconductor is thought to be the spontaneous scalarization of the black  hole, a p-wave holographic superconductor requires a charged vector field in the bulk as the  vector order parameter, and a d-wave model was built by introducing a charged massive spin  two field propagating in the bulk [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' It has been recently shown that quantum information  can be used to diagnose the phase transition of an s-wave superconductor [16, 26, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  However, research on the effects of mixed-state entanglement in p-wave superconductors is  currently lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Therefore, it would be interesting to investigate the connection between  the holographic p-wave superconducting phase transition and mixed-state entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  In this paper, we aim to systematically study the role of mixed-state entanglement dur-  3  ing the p-wave superconductivity phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The paper is organized as follows: In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' II, we introduce the holographic p-wave superconductor model, and the concepts of  holographic quantum information, including holographic HEE, MI and EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We explore  the characteristics of mixed-state entanglement in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' IV, we provide analyt-  ical and numerical analysis of the scaling behavior of mixed-state entanglement measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Additionally, we uncover an inequality between EWCS and MI in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  VI, we summarize our findings and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  HOLOGRAPHIC SETUP FOR P-WAVE SUPERCONDUCTOR AND HOLO-  GRAPHIC INFORMATION-RELATED QUANTITIES  We begin by presenting the model of a holographic p-wave superconductor and its phase  diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Following that, we introduce HEE, as well as the mixed-state entanglement mea-  sures MI and EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The holographic p-wave superconductor model  In the p-wave superconductor model, as the temperature drops to a specific critical value,  spontaneous symmetry breaking occurs, resulting in a vector order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The system  then transits from the normal phase (absence of vector hair) to the superconducting phase  (presence of vector hair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The holographic p-wave model is constructed by introducing a  complex vector field into Einstein-Maxwell theory with a negative cosmological constant  [35, 36],  S = 1  2κ2  �  d4x√−g  �  R + 6  L2 − 1  4FµνF µν − 1  2ρ†  µνρµν − m2ρ†  µρµ + iqγρµρ†  νF µν  �  ,  (1)  where κ2 = 8πG is related to the gravitational constant, L the AdS radius that we set as  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' A is the gauge field and the field strength Fµν = ∇µAν − ∇νAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' ρµ is a complex vector  field with mass m and charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The tensor ρµν = Dµρν − Dνρµ with covariant derivative  defined as Dµ = ∇µ − iqAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The last term in the action is the non-minimum coupling term  between the Maxwell field and the complex vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In this paper, we only consider the  4  case without an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The equation of motion (EOM) can be read as,  ∇νFνµ =iq(ρνρ†  νµ − ρν†ρνµ) + iqγ∇ν(ρνρ†  µ − ρ†  νρµ),  Dνρνµ−m2ρµ + iqγρνFνµ = 0,  Rµν − 1  2Rgµν − 3  L2gµν =1  2FµλF λ  ν + 1  2  �  −1  4FµνF µν − 1  2ρ†  µνρµν − m2ρ†  µρµ + iqγρµρ†  νF µν  �  gµν+  1  2{[ρ†  µλρλ  ν + m2ρ†  µρν − iqγ(ρµρ†  λ − ρ†  µρλ)F λ  ν ] + µ ↔ ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (2)  We solve the EOM with this ansatz,  ds2 = 1  z2  �  −p(z)(1 − z)U(z)dt2 +  1  p(z)(1 − z)U(z)dz2 + V1(z)dx2 + V2(z)dy2  �  ,  Aνdxν = µ(1 − z)a(z)dt,  ρνdxν = ρx(z)dx,  (3)  where p(z) ≡ 1 + z + z2 − µ2z3  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' µ is the chemical potential of the dual field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The  radius axis is denoted by z, which ranges from 0 to 1, with z = 0 and z = 1 representing the  AdS boundary and horizon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In our ansatz, there are five unknown functions,  U(z), V1(z), V2(z), a(z), and ρx(z), which can be obtained by solving the EOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The ansatz  (3) reduces to the AdS-RN black brane solution when U = V1 = V2 = a = 1 and ρx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The expansion of the ρx near the AdS boundary is  ρx = ρx−z∆− + ρx+z∆+ + · · · ,  (4)  where the scaling dimension ∆± = 1±  √  1+4m2  2  and we set the source ρx− = 0 for the condensate  arise spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' After solving the EOM, we can obtain the condensate ⟨Jx⟩ by extracting  the coefficient ρx+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The condensate ⟨Jx⟩ emerges at a specific temperature when varying  m2 and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Consequently, in the dual quantum field, the vector operator acquires a non-zero  vacuum expectation value and spontaneously breaking the U(1) symmetry and rotational  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Therefore, ⟨Jx⟩ can be used as the order parameter of p-wave superconducting  phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The Hawking temperature of this model is ˜T = 12−µ2  16π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The system is invariant under the  following rescaling,  (t, x, y) → α−1(t, x, y),  (U, V1, V2) → α2(U, V1, V2),  µ → αµ,  ˜T → α ˜T,  ρx+ → α∆++1ρx+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (5)  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0 T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  〈Jx〉  2  5  q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5, m2=3/4  8  7  6  5  4  ln(δ(1-T/Tc)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  ln(δ〈Jx〉)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='30  〈Jx〉  2  5  q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2, m2=3/4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='05  T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0994  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0992  Ω  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 1: Left plot: The second-order phase transition occurs as the temperature falls below the  critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The inset plot illustrates the scaling behavior of the condensate ⟨Jx⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot:  The first-order phase transition occurs when the temperature falls below the critical temperature,  which represents by the black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The inset plot illustrates the effective free energy density  Ω versus the temperature T/Tc, revealing that the superconducting phase is thermodynamically  favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  In this paper, we adopt the chemical potential µ as the scaling unit, which is equivalent to  treating the dual system as a field theory described by the giant canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The  dimensionless Hawking temperature T = ˜T/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The phase diagram of holographic p-wave superconductor model  This holographic p-wave superconductor model can exhibit zeroth-order, first-order, and  second-order phase transitions depending on the values of m and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' For example, a second-  order phase transition can occur at q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5, m2 = 3/4 with a critical temperature of  Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='01791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 1, we demonstrate the relationship between the condensate ⟨Jx⟩2/5  and temperature by plotting the scaling relationship,  δ(⟨Jx⟩) ∼  �  1 − T  Tc  �αc  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (6)  Theoretical calculations predict that the critical exponent is α = 1/2 [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Our numerical  results also indicate that αc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='500106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  A first-order phase transition can occur at q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2 and m2 = 3/4 with a critical tem-  perature of Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='003382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' To better visualize the phase structure, we plot the effective  6  m2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  m2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='30  m2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  m2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='12  T  Superconducting phase  Normal phase  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 2: The phase diagram of holographic p-wave superconductor model with positive m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The  solid lines are the critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  free energy density (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The effective free energy density is defined as  ˜Ω = M − Ts, where T is the Hawking temperature, s is the entropy density, and M is the  mass density of the black brane [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The mass density of the black brane can be obtained  by using the AdS asymptotic behavior of gtt in our ansatz,  (1 − z)U(z)  �  − 1  4µ2z3 + z2 + z + 1  �  z2  ∼ 1  z2 + Mz + Qz2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (7)  The free energy of the superconducting phase is lower than the normal phase when the  temperature drops below the critical temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' As a result, the system will abruptly  transition from the normal phase to the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  To more thoroughly understand the behavior of p-wave superconductivity, we present  the phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The phase diagram is constructed by identifying the critical  points, which can be found by examining the emergence of condensation as a perturbation  near these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The linearized equations of motion can be transformed into an eigenvalue  problem that we solve using numerical methods  1  32µ2(z − 1)z2(µ2z3 − 4z2 − 4z − 4)(2z2(z2(µ2(4z − 3) − 12)ρ′  x(z)+  �  µ2z4 −  �  µ2 + 4  �  z3 + 4  �  ρ′′  x(z)) − 6ρx(z)) = −q2ρx(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (8)  By analyzing the eigenvalues, we can determine the upper or lower bounds of the critical  points, which correspond to the boundaries of the different phases in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  7  x  y  z  x  y  z  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 3: Left plot: the minimum surface for a subsystem (red region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot: the minimum  cross-section (green surface) of the entanglement wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The holographic quantum information  Quantum entanglement is a fundamental characteristic of quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' EE is a  well-known measure of entanglement, which quantifies the correlation between a subsystem  and its complement for pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' It is defined in terms of the reduced density matrix ρA  [38],  SA(|ψ⟩) = −Tr[ρAlog(ρA)],  ρA = TrB(|ψ⟩⟨ψ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (9)  The HEE was proposed to be dual to the area of the minimum surface in the gravitational  system [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In this paper, we consider the HEE of the configuration with an infinitely long  strip along the y-axis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  HEE typically diverges due to the asymptotic AdS  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The regulation is implemented by subtracting the divergent term from the HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  It should be noted that HEE is not suitable for describing the mixed-state entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  For example, EE of the quantum system characterized by the direct product state HA ⊗HB  is not equal to zero, but the entanglement of the subsystems is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' This is because  EE contains both quantum and classical correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Therefore, as the dual of EE, HEE is  also affected by thermodynamic entropy in mixed-state systems [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  To better solve the problem of mixed-state entanglement measurement, numerous novel  entanglement measures have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' One popular measure is mutual information  (MI), which quantifies the correlation between two subsystems A and C that are separated  by a subsystem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' According to the definition of MI, it is calculated as [42, 43],  I(a : c) = S(a) + S(c) − min(S(a ∪ c)),  (10)  where S(x) denotes the entanglement entropy of subsystem x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Unlike entanglement entropy,  8  a  Sc  Sa  Sb  Sa+b+c  b  c  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 4: The illustration of the holographic mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  MI for direct product states HA ⊗ HB is always zero, making it a more appropriate measure  for mixed-state entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In the holographic context, the dual of MI is the difference in  area between red (disconnected configuration) and blue surfaces (connected configuration),  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' As the subsystem A, C becomes smaller or when the separation B  becomes larger, MI decreases and eventually reaches zero, indicating a disentangling phase  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' However, MI has some limitations as a mixed-state entanglement measure as it  is directly related to entanglement entropy and can be dominated by it in some cases [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Therefore, it is important to explore other mixed-state entanglement measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Recently, the minimum cross-section of the entanglement wedge (EWCS) is proposed  as a novel holographic mixed-state entanglement measure [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' EWCS is considered to be  the duality of reflected entropy, logarithmic negativity, and odd entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The definition of  EWCS is as follows,  Ew(ρAB) = min  ΣAB  �Area(ΣAB)  4GN  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (11)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 3 is an illustration of EWCS in a bipartite system a∪c divided by b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The area bounded  by the minimum surface of the disconnected configuration is known as the entanglement  wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' It is important to note that entanglement between subsystems only exists when the  total correlation is not zero, which means the MI does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Although EWCS plays a significant role in measuring the entanglement of mixed-state  systems, it is still challenging to solve it [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' First, it is hard to solve the highly nonlinear  EOM of the minimum surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Second, the minimum cross-section is obtained by scanning  a two-dimensional parameter space, which is a hard task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Last but not least, the coordinate  singularity close to the AdS boundary with the asymptotic AdS can hinder numerical preci-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We have proposed an efficient algorithm for solving EWCS based on the requirement  9  θ2  θ1  C1(θ1)  C2(θ2)  P1  P2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  z  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 5: The illustration of the numerical algorithm for EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  that the minimum cross-section is locally orthogonal to the boundaries of the entanglement  wedge [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 5 shows the illustration of the key concept for our numerical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  We consider EWCS of the infinite strip along the y-direction in a homogeneous background  ds2 = gttdt2 + gzzdz2 + gxxdx2 + gyydy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (12)  The minimum surfaces of the connected configuration can be represented as C1(θ1) and  C2(θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The minimum surfaces intersect with the cross-section at points p1 and p2, and the  area of this local minimum surface (the red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 5) is,  A =  �  Cp1,p2  �  gxxgyyx′(z)2 + gzzgyydz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (13)  Variating (13), we obtain the EOM determining the local minimum surface,  x′(z)3  � gxxg′  yy  2gyygzz  + g′  xx  2gzz  �  + x′(z)  �g′  xx  gxx  + g′  yy  2gyy  − g′  zz  2gzz  �  + x′′(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (14)  Remind that the global minimum cross-section is locally orthogonal to the entanglement  wedge, which implies that  � ∂  ∂z  , ∂  ∂θ1  �  p1  = 0,  � ∂  ∂z  , ∂  ∂θ2  �  p2  = 0  (15)  where ⟨·, ·⟩ represents the vector product with metric gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We can normalize the orthogonal  relation,  Q1(θ1, θ2) ≡  ⟨ ∂  ∂z,  ∂  ∂θ1⟩  �  ⟨ ∂  ∂z, ∂  ∂z⟩⟨ ∂  ∂θ1,  ∂  ∂θ1⟩  ������  p1  = 0,  Q2(θ1, θ2) ≡  ⟨ ∂  ∂z,  ∂  ∂θ1⟩  �  ⟨ ∂  ∂z, ∂  ∂z⟩⟨ ∂  ∂θ2,  ∂  ∂θ2⟩  ������  p2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (16)  10  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='52  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='54  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='56  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='58  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  T  Tc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='7  SE  q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5, m2=3/4  l=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  l=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='20  l=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='40  l=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='60  l=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='80  l=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  T  Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  SE  q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2, m2=3/4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 6: The holographic entanglement entropy SE vs temperature T/Tc with various strip width  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The critical point is indicated by the black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The stable and metastable states are  depicted by solid and transparent lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Left plot: The second-order phase transition  at Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='01791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot: The first-order phase transition at Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='003382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Finding the cross-section located at the minimum surface at (θ1, θ2) where (16) is satisfied,  we obtain the minimum cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' To this end, we adopt the Newton-Raphson method  to locate the endpoints satisfying the local perpendicular conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Based on the above  techniques, we can study the relationship between the holographic p-wave superconductor  and the EWCS [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  THE COMPUTATION OF THE HOLOGRAPHIC QUANTUM INFORMA-  TION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The holographic entanglement entropy and mutual information  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  6, we show the relationship between the HEE and temperature T/Tc during  second-order and first-order phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  For q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5 and m2 = 3/4, where the  second-order phase transition occurs, the HEE increases with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' For  q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2 and m2 = 3/4, where the first-order phase transition occurs, the HEE jumps  abruptly when crossing the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' To understand this behavior, we can examine the  relationship between HEE and thermodynamic entropy, as when the configuration is large or  the temperature is high enough, the minimum surface will approach the horizon of the black  brane and HEE will primarily be determined by thermodynamic entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Therefore, we will  next analyze the thermodynamic entropy behavior of the black brane to better understand  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  T  Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  s  q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5, m2=3/4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  ln(δ(1-T/Tc))  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  ln(δs)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  T  Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08  s  q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2, m2=3/4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 7: The entropy density s vs temperature T/Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The black dashed line represents the entropy  density of the normal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The entropy density of the superconducting phase is represented by  the purple line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Left plot: The second-order phase transition of holographic p-wave superconductor  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The inset plot depicts the logarithm between s and 1− T  Tc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot: The first-order phase  transition of holographic p-wave superconductor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The critical temperature is indicated by  the red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  the behavior of HEE [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The entropy density is given by,  ˜s = 2πA  κ2  = 2π  �  V1(z)V2(z)  κ2  ˆV ,  (17)  where A is the area of the horizon and ˆV =  �  dxdy is the corresponding area of the region  in the dual field theory [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Dividing the entropy by the area ˆV and µ2, we have the  dimensionless entropy density s =  κ2˜s  2π ˆV µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The plot of the entropy density near the critical  point can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The above phenomena show that both HEE and entropy  density can detect the critical behavior of the holographic p-wave superconducting phase  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Similar phenomena of the HEE in the superconducting phase transition can see  in [16, 17, 26, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  MI is one of the mixed-state entanglement measures that can extract the total correlation  of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Since MI is directly defined by HEE (see (10)), it also can diagnose the  phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Moreover, a disentangling phase transition occurs when MI is zero and  entanglement exists only when MI is greater than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  8 illustrates the behavior  of the disentangling phase transition for various configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' However, in certain cases,  MI is determined by the thermodynamic entropy [25, 26, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Therefore, it is necessary to  investigate other mixed-state entanglement measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  12  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  T  Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='82  bc  a=2  a=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  a=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  a=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='20  a=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='30  a=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='40  a=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='57  cc  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 8: The critical configuration for disentangling phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The critical temperature is  indicated by the black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The solid and translucent lines represent stable and metastable  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Left plot: When b is above the bc, the disentangling phase transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot:  When c is below cc, the disentangling phase transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The minimum entanglement wedge cross section  We begin by examining the EWCS during a second-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 9 shows  that EWCS can diagnose the critical behavior of holographic p-wave superconducting phase  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' At the critical point of a second-order phase transition, EWCS is continuous, but  its first derivative is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In the superconducting phase, EWCS always decreases  with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  However, we find that the EWCS in the normal phase is  configuration-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In large configurations, it behaves similarly to the HEE, showing  a monotonically increasing trend with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In contrast, for small configurations,  the EWCS of the normal phase exhibits a monotonically decreasing trend with temperature,  opposite to the behavior of the HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Next, we investigate the behavior of the EWCS during a first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 10 illustrates EWCS behavior during this phase transition, with the inset plot showing  the derivative of EWCS with respect to temperature (∂TEw) versus temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The  inset plot illustrates that in normal phase the EWCS decreases with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Unlike the HEE, the EWCS of the superconducting phase always decreases with tempera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' When the temperature falls below a critical point, EWCS abruptly jumps from the  normal phase to the superconducting phase, this sudden change in EWCS suggests that it  can capture the first-order phase transition, similar to the HEE and MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  In addition to diagnosing the critical points, it is also important to investigate the scaling  13  c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='1T/Tc  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='20  Ew  a=2, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='00  ∂TEw  c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='80  c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='1T/Tc  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='38  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='40  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='42  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='44  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='46  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='48  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  Ew  a=1, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  ∂TEw  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 9: The EWCS Ew vs the temperature T/Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The inset graph depicts ∂T Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The black  dashed line depicts the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Left plot: The ∂T Ew of the normal phase is greater  than zero when a = 2 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot: The ∂T Ew of the normal phase is less than zero  when a = 1 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='80  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='84  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='88  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='92  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='96  c=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='3  T  Tc  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='120  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='125  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='130  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='135  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='140  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='145  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='150  Ew  a=2, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='500  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='504  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='508  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='512  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='516  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='520  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  T  Tc  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='05  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='15  Ew  a=2,c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 10: The EWCS Ew versus the temperature T/Tc in the first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The  dashed black line represents the critical temperature when Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='003382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The translucent line  represents the metastable state, whereas the solid line represents the stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Left plot: We  set a = 2 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5 with varying c values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot: We set a = 2 and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8 while varying b  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  behavior of the holographic quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Next, we analyze the critical behavior  of the quantum information-related quantities during the p-wave superconductivity phase  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  14  l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='50  l=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  l=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  9  8  7  6  5  ln(δ(1-  T  Tc  ))  9  8  7  6  5  ln(δSE)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='6  ln(δ(1-  T  Tc  )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='02  slope  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='75  c=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  c=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='25  c=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='0  ln(δ(1-  T  Tc  )  9  8  7  6  5  ln(δEw)  a=2, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='6  ln(δ(1-  T  Tc  ))  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04  ln(δEw)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 11: The scaling behavior of HEE and EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The inset plot shows the slope of the holographic  quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Left plot: ln(δSE) versus ln(δ(1− T  Tc)) with different width of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot:  we set a = 2 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5 and ln(δEw) versus ln(δ(1 − T  Tc )) with different values of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  THE SCALING BEHAVIOR OF THE QUANTUM INFORMATION  As the critical point marks the bifurcation point between the normal and superconducting  phases, to study the critical behavior, we compare the quantum information quantities of  the normal phase to those of the superconducting phase by subtracting the former from the  latter,  δSE = Scond  E  − Snormal  E  ,  δEw = Econd  w  − Enormal  w  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (18)  We propose the following critical behaviors for the HEE and EWCS,  δS ∼  �  1 − T  Tc  �αHEE  ,  δEw ∼  �  1 − T  Tc  �αEWCS  ,  (19)  where αHEE and αEWCS are the critical exponent of the HEE and EWCS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We  plot the critical scaling behavior in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 11, from which we find that both EWCS and HEE  exhibit excellent scaling behavior near the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' More importantly, they both have  the same critical exponent,  αHEE ≈ αEWCS ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (20)  It is important to note that the vector field ρµ is always zero at temperatures higher than  the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' At temperatures slightly below the critical point, the condensate  vacuum expectation value of ⟨Jx⟩ is small and can be analyzed using perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  We can expand the vector field ρµ and the metric function near the critical point as [46–48],  15  ρx = ϵρ(1) + ϵ3ρ(3) + ϵ5ρ(5) + · · · ,  U = 1 + ϵ2U (1) + ϵ4U (4) + · · · ,  V1 = 1 + ϵ2V (1)  1  + ϵ4V (4)  1  + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (21)  From (21), we can deduce that the critical exponent of the metric function U, V1, is twice  that of the condensate ⟨Jx⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' This can be understood by noting that holographic quantum  information is represented by geometric objects that depend only on the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Their  critical exponent can be written as,  δ(SE) ∼ δ(Ew) ∼ δ(⟨Jx⟩)2 ∼  �  1 − T  Tc  �2αc  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (22)  Therefore, the theoretical critical exponent of holographic quantum information should be  twice that of the condensate ⟨Jx⟩,  αHEE = αEWCS = 2αc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (23)  Although EWCS and HEE have the same critical exponent in the critical region, they do  not tend to the scaling law at the same rate in the critical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' To better investigate this  phenomenon near the critical point, we define the quasi-critical exponent (QCE) as  α ≡  d ln(δS)  d ln  �  1 − T  Tc  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (24)  QCE is a function of ln  �  1 − T  Tc  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Apparently, the QCE α behavior along ln  �  1 − T  Tc  �  can  measure the extent to which a wide range of S can converge to the scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  We show the QCE of HEE and EWCS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' From the left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 12 we find  that the width l has an impact on the scaling behavior of HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' As the width l increases, the  scaling behavior of HEE is closer to the theoretical scaling behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' As the temperature  moves away from the critical point or the width l decreases, however, the scaling behavior  of the HEE begins to deviate from the theoretical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  The QCE of EWCS is depicted in the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Comparing the left plot and  the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 12, EWCS converges to the theoretical scaling law over a broader range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  As the separation b decreases, the scaling behavior of EWCS becomes close to the theoretical  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' This behavior suggests that EWCS, as a measure for mixed-state entanglement, can  more accurately describe the scaling behavior during superconductivity phase transitions  than HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  16  l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  l=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  l=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='50  l=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  7  6  5  4  3  ln(δ(1-  T  Tc  ))  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='00  QCE(SE)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='0  ln(δ(1-T/Tc))  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='995  QCE(SE)  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='25  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='40  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='70  7  6  5  4  3  ln(δ(1-  T  Tc  ))  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='00  QCE(Ew)  a=2, c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0  1-T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='985  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='990  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='995  QCE(Ew)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 12: The QCE of the HEE and EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The red dashed line represents the twice QCE of the  condensate ⟨Jx⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Left plot: The QCE of the HEE near the critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot: The QCE  of the EWCS near the critical points, when we fix the a = 2 and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  THE GROWTH RATE OF THE HOLOGRAPHIC QUANTUM INFORMA-  TION  Several important inequalities involving the EWCS have been proposed in the literature  [19, 49, 50], such as the inequality Ew(ρAC) ≥ 1  2I(A, c), which states that the EWCS cannot  be smaller than half of the MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' These inequalities are crucial in the study of mixed-state  entanglement measures, particularly in testing the validity of holographic duals of certain  quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  In this paper, we find a new inequality behavior of EWCS and  MI related to the superconductivity phase transition: near the phase transition point, the  relative growth rate of MI along the temperature axis is always greater than that of EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  When the temperature drops below the critical temperature, the EWCS and the MI of  the superconducting phases are always larger than those of the normal phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' To take a  closer look at the relationship between the EWCS and the MI, we define the relative values  of the MI and the EWCS,  ˜Ew = Ew,cond  Ew,norm  ,  ˜I = Icond  Inorm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (25)  With this definition, ˜Ew and ˜I are fixed at 1 at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 13, we depict  the relationship between the ˜I and ˜Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Contrary to the inequality [19, 49, 50], the relative  MI is always larger than the relative values of EWCS in the critical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' To describe this  relationship quantitatively, we examine the fact that,  δ(Q) ≃ A(Q)  �  1 − T  Tc  �α  ,  (26)  17  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='20  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='47  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='73  b=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  1-  T  Tc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08  Relative Value  a=2, c=2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='14  1-T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5  Slope  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='4, c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8, c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='6  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1, c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='9  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='5, c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10  1-  T  Tc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0002  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0004  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0006  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0008  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0012  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0014  Relative Value  b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='14  1-T/Tc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='015  Slope  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 13: The relative value of MI and EWCS near the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The dashed lines represent  ˜I and the solid lines represent ˜Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The inset plot is the slope of ˜I and ˜Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Left plot: We fix the  subsystems a = c = 2 and change the separation b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Right plot: We fix the separation b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1 and  change the subsystem a and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  where Q stands for any physical quantity possessing critical behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' From (26) we find  that,  ˜Ew = 1 + A( ˜Ew)  �  1 − T  Tc  �α  ,  ˜I = 1 + A(˜I)  �  1 − T  Tc  �α  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (27)  Accordingly, it can be seen that A actually measures the increasing phenomenon of holo-  graphic quantum information in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 12, and hence we call A the growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We work  out A( ˜Ew) and A(˜I) for several different configurations and list them in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' From these  numerical results we conclude a new inequality between the EWCS and MI growth rates  near the critical point,  A(˜I) > A( ˜Ew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  (28)  The growth rate of MI is always greater than that of EWCS near the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Further-  more, the difference between the growth rates of EWCS and MI increases as the subsystem  separation b increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Near the critical point, the entanglement of the system changes  rapidly, and MI is more sensitive to these changes than EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' This behavior may be due  to the fact that MI captures the total correlation of the system, which is more than what  is captured by EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We have also tested this inequality in other thermal phase transi-  tion models, such as the holographic s-wave superconductor model, and suggest that this  inequality (28) may be universal in thermal phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  18  TABLE I: The growth rate A( ˜Ew) and A(˜I) at different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Configuration  A( ˜Ew)  A(˜I)  a = c = 2, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0426  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='0741  a = c = 2, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1162  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2947  a = c = 2, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2094  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='9855  a = c = 2, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='3212  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8712  a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='8, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='11959  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  DISCUSSION  In this study, we investigate mixed-state entanglement measures, including HEE, MI and  EWCS, in a holographic p-wave superconductor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The model exhibits both second  and first-order phase transitions when varying system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We find that HEE and  EWCS can accurately diagnose the critical behavior of these phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Additionally,  we observe that the behavior of HEE is related to thermodynamic entropy as the subsystem  configuration increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' However, as a mixed-state entanglement measure, EWCS exhibits  the opposite behavior from HEE in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Specifically, HEE always  increases with temperature, whereas EWCS in the superconducting state decreases with  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In the case of first-order phase transitions, the holographic quantum infor-  mation experiences sudden changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' However, the EWCS behavior in the normal phase is  dependent on the subsystem configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' This behavior demonstrates that EWCS can not  only detect phase transitions but also capture more information than HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  In addition to diagnosing phase transitions, we also examine the scaling behaviors of the  condensate and the holographic quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Through analyzing the scaling behav-  19  ior of various holographic quantum information measures, we find that HEE and EWCS not  only detect the critical point but also exhibit scaling behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We show both numerically  and analytically that the critical exponent of holographic quantum information is twice that  of the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Furthermore, we observe that compared to HEE, EWCS provides a more  sensitive characterization of the scaling behavior, making it more suitable as a measure for  mixed-state entanglement in superconductivity phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Additionally, we propose  a novel inequality for EWCS and MI in phase transitions and provide numerical evidence  for this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The relative growth rate of MI is always larger than that of EWCS near the  critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Next, we point out several directions worth further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' The investigation of  topological and quantum phase transitions is an important area of research in condensed  matter theory [51–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' In particular, the relationship between HEE and quantum phase tran-  sitions has been studied under holographic framework in previous works [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Further  research into the mixed-state entanglement in quantum phase transitions and topological  quantum phase transitions is therefore desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Additionally, it would be interesting to  test the inequality (28) in other thermal phase transition models, such as the d-wave su-  perconductivity model and the massive gravity model [26, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We are working on these  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Acknowledgments  Peng Liu would like to thank Yun-Ha Zha for her kind encouragement during this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  Zhe Yang appreciates Feng-Ying Deng’s support and warm words of encouragement during  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' We are also very grateful to Chong-Ye Chen, Mu-Jing Li, and Wei Xiong for their  helpful discussion and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' This work is supported by the Natural Science Founda-  tion of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' 11905083, 12005077, 11805083, the Science and Technology  Planning Project of Guangzhou (202201010655) and Guangdong Basic and Applied Basic  Research Foundation (2021A1515012374).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='8, 081602  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='081602 [arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='05505 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content=' Ling, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Wu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Zhou, “Holographic Metal-Insulator Transition in Higher  Derivative Gravity,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content=' B 766 (2017), 41-48 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+page_content='07866 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
+page_content='  24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFRT4oBgHgl3EQffDdP/content/2301.13574v1.pdf'}
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+Dynamic gravitational excitation of structural resonances in the hertz regime using
+two rotating bars
+Tobias Brack,∗ Jonas Fankhauser,∗ Bernhard Zybach,∗ Stephan Kaufmann, Francesco
+Palmegiano, Jean-Claude Tomasina, Stefan Blunier, Donat Scheiwiller, and J¨urg Dual†
+Institute for Mechanical Systems, ETH Z¨urich, Tannenstrasse 3, Zurich, 8092, Switzerland.
+Fadoua Balabdaoui
+Seminar of Statistics, ETH Z¨urich, R¨amistrasse 101, Zurich, 8092, Switzerland.
+(Dated: December 21, 2022)
+With the planning of new ambitious gravitational wave (GW) observatories, fully controlled lab-
+oratory experiments on dynamic gravitation become more and more important.
+Such new ex-
+periments can provide new insights in potential dynamic effects such as gravitational shielding or
+energy flow and might contribute to bringing light into the mystery still surrounding gravity. Here
+we present a laboratory-based transmitter-detector experiment using two rotating bars as transmit-
+ter and a 42 Hz, high-Q bending beam resonator as detector. Using a highly precise phase control
+to synchronize the rotating bars, a dynamic gravitational field emerges that excites the bending
+motion with amplitudes up to 100 nm/s or 370 pm, which is a factor of 500 above the thermal noise.
+The two-transmitter design enables the investigation of different setup configurations. The detector
+movement is measured optically, using three commercial interferometers. Acoustical, mechanical,
+and electrical isolation, a temperature-stable environment, and lock-in detection are central ele-
+ments of the setup. The moving load response of the detector is numerically calculated based on
+Newton’s law of gravitation via discrete volume integration, showing excellent agreement between
+measurement and theory both in amplitude and phase. The near field gravitational energy transfer
+is 1025 times higher than what is expected from GW analysis.
+I.
+INTRODUCTION
+With increasing research efforts in gravitational waves
+and yet unexplained differences in the measurement of G
+by different research groups, laboratory-based, dynami-
+cal gravitational experiments become more and more im-
+portant. However, experiments with frequencies > 0.1 Hz
+are very rare when it comes to a full quantitative compar-
+ison between theory and experiment, such as the grav-
+itationally induced vibration amplitude and phase of a
+detector system, the distance behavior, or energy flow.
+This is illustrated in Table III showing an overview of
+the few dynamic gravitational experiments reported dur-
+ing the last 50 years.
+To fill this gap, we presented in a previous work the
+dynamical gravitational coupling between two parallel
+beams vibrating in their first bending resonance at 42 Hz
+[1]. This setup allowed for a quantitative investigation of
+the distance behavior and the estimation of the gravita-
+tional constant G. Although this setup allowed for im-
+portant advances in dynamic gravitation measurements,
+it features some drawbacks, such as the need for match-
+ing transmitter and detector resonance frequency or the
+generation of very low detector amplitudes.
+Therefore, motivated by the pioneering works of Sinsky
+and Weber [2, 3], Astone [4, 5], and the Hirakawa group
+[6–8], we introduce here a novel setup which combines
+∗ These authors contributed equally to this work.
+† dual@imes.mavt.ethz.ch
+the experimental and analytical advantages of our initial
+setup with the benefits of a rotational excitation and its
+elaborated theory. As illustrated in Table III, all recent
+works use one single transmitter. In contrast, here we
+use two slender, 0.5 m long tungsten bars arranged sym-
+metrically to a bending beam resonant detector.
+This
+double transmitter arrangement allows for various con-
+figurations of the excitation, that is, rotation direction
+and frequency, within the same experimental setup.
+Previous dynamic experiments encountered a relatively
+large measurement uncertainty for the estimation of G.
+This was mainly caused by electrical or mechanical, non-
+gravitational crosstalk and the fact that the measurement
+accuracy reaches its limit with the resulting small dis-
+placement amplitudes. In the setup presented here, some
+of these issues can be addressed using a precisely defined
+rigid body transmitter movement and considerably larger
+detector amplitudes. These advantages, however, come
+at the expense of high requirements on the balancing of
+the rotors and their motion control, in particular their
+phase accuracy and phase jitter. To demonstrate the po-
+tential of this new setup, we use the same detector beam
+as in [1] and the same measurement procedure. Thus we
+obtain the detector resonance amplitude and phase at
+different distances, as well as an estimation of the gravi-
+tational constant G.
+While the motion of the rotating bars is trivial to model,
+its interaction with the detector beam is categorized as
+a nonstandard moving load problem [9] with varying
+speed and shape of the moving force pulse.
+Its solu-
+tion is greatly simplified by the periodicity of the loading
+and the restriction to the steady state. Considering the
+arXiv:2301.01644v1  [physics.ins-det]  4 Jan 2023
+
+2
+Gravitational force/length (nN/m)
+0.17
+0.44
+0.72
+1
+1.27
+1.54
+2
+Suspension
+LDV 3
+LDV 2
+LDV 1
+Gravitational
+force field
+Transmitter
+Detector
+Light barrier
+1
+d0
+φ2
+φ1
+x
+y
+x0
+FIG. 1.
+Illustration of the measurement principle.
+Sketch of the measurement setup using two rotating bars (or-
+ange) and one detector beam (blue), suspended in the nodal
+points of the first bending mode.
+The rotating bars cre-
+ate a dynamic gravitational force field on the detector that
+is illustrated by the colored arrows (numerical solution for
+d0 = 300 mm, ϕ1,2 = 98◦). The vibration of the detector is
+measured optically by three laser Doppler vibrometers (LDV).
+The colorbar illustrates the gravitational force density of the
+dynamic force field in the xy-plane in nN/m.
+length scales of the experiment, the detector is placed
+in the near-field of the transmitter. Since the excitation
+is not necessarily purely quadrupole, Newton’s law has
+been directly applied to all relevant mass elements. We
+compute the results using a highly accurate 3D finite el-
+ement model and a simplified 1D model for comparison.
+This article presents a theoretical and experimental de-
+scription of the experiment and shows its feasibility and
+enormous potential as well as technical pitfalls, without
+providing highly accurate G measurements yet. We show
+that the double transmitter approach provides a large
+variation of dynamic gravitational fields, and that the
+setup is able to accurately measure the resulting detec-
+tor movement in amplitude and phase.
+II.
+THEORY OF DYNAMIC GRAVITATIONAL
+FORCE FIELDS GENERATED BY ROTATING
+BAR
+The setup presented in this article consists of a trans-
+mitter and detector system, cf. Fig 1a. The transmit-
+ter system is composed of two identical, slender bars of
+square cross section that can rotate around the z-axis in
+their center of mass. The bars are arranged symmetri-
+cally to the middle of the detector beam at a distance
+d0, that is, the y-distance between the central axis of the
+detector and the centers of rotation. The detector is de-
+signed as a resonator, with the first lateral bending mode
+to be the relevant vibration mode. We selected a slender
+beam of rectangular cross section and twice the length of
+the transmitter bars. Due to the rectangular shape, it is
+ensured that horizontal and vertical bending do not have
+the same resonance frequency.
+When the bars rotate, a dynamic gravitational force field
+develops due to the periodically changing distance be-
+tween detector and transmitter mass elements. Hence,
+given that a suitable rotation frequency is selected, the
+dynamic gravitational force field periodically excites the
+first bending mode of the detector beam, which then de-
+velops a measurable deflection.
+To investigate the force field acting on the detector, New-
+ton’s law of gravitation is applied to each pair of trans-
+mitter/detector mass elements. The transmitter rotation
+is maintained at a constant frequency with very high ac-
+curacy using active rotation control (cf.
+supp.
+mate-
+rial). The much smaller gravitational forces acting on the
+transmitter bars do not affect the rotation. Newton’s law
+of gravitation yields a time dependent, three-dimensional
+gravitational body force on the detector generated by
+each bar. For a first analytical analysis of the gravita-
+tional effects a 1D approach is sought, where the body
+forces in y-direction are approximated by two line forces
+f (k)
+y
+(xd) acting on the central axis of the slender detector
+beam, where xd denotes the x coordinate of the detector
+beam center line and k the number of the rotating bar,
+cf. Fig. 1. The detector’s bending movement can then
+be calculated using an Euler-Bernoulli beam model and
+a modal approach [1, 10], in which the distributed force
+is reduced to the effective modal excitation force Fy,b of
+the first bending mode via
+Fy,b =
+� ld
+0
+�
+f (1)
+y (xd) + f (2)
+y (xd)
+�
+Ub,1(xd)dxd ,
+(1)
+where Ub,1is the mode shape function of the first bending
+mode, normalized so that Ub,1(0) = 1.
+To assess the frequency components of the excitation,
+a Fourier series is computed. Due to the nonlinear law
+of gravitation, higher harmonics must occur, which is il-
+lustrated by Fig. 2a, where the spectral components at
+multiples of the rotation frequency Ω are shown qualita-
+tively (scaled to the DC value). It can be observed that
+the force field contains frequency components at even
+multiples of the rotation frequency only with decreasing
+contribution. Hence the excitation frequency Ω must be
+chosen 1/2 or 1/4 of the detector beam’s resonance fre-
+quency ω0 to achieve maximum detector excitation.
+The use of higher harmonics as excitation and the
+possibility of different rotation configurations (one/two
+bars, same/opposite direction, pos./neg. direction) en-
+ables different setup configurations that will be discussed
+briefly. Due to the rotation, the force distribution at the
+detector beam varies in time and space and can be cate-
+gorized as a nonstandard periodic moving load problem
+[9]. Considering one rotating bar only, the variation of
+the force distribution can be illustrated by a transverse
+pulse with varying speed and shape moving along the
+
+3
+0
+2
+4
+6
+8
+n
+0
+0.2
+0.4
+0.6
+0.8
+1
+Fy,b/Fy,b(n = 0)
+a
+300
+350
+400
+450
+distance d0 (mm)
+0
+20
+40
+60
+80
+100
+|Fy,b| scaled (%)
+b
+n = 2
+n = 4
+300
+350
+400
+450
+distance d0 (mm)
+-40
+-30
+-20
+-10
+0
+10
+20
+30
+40
+Fy,b - 
+Fy,b
+CW-CW (deg)
+c
+n = 2
+n = 4
+FIG. 2. Properties of the gravitationally induced modal force Fy,b acting on the detector beam, produced by two
+identical rotating bars. Analytical results, 1D approximation. a, Scaled amplitude spectrum at multiples n of the rotation
+frequency, Ω, exemplarily for opposite rotation direction and d0 = 300 mm. b/c, Second (solid line) and fourth (dashed line)
+harmonic as a function of transmitter/detector distance, d0, for various setup configurations. b, Amplitude scaled to the highest
+value at n = 2, opposite rotation, d0 = 300 mm. Note: some lines are hidden due to equal results for same (yellow, purple) and
+opposite (red, blue) rotation direction. c, Phase relative to the phase of same rotation direction (yellow, purple).
+x-axis of the detector.
+Considering two bars rotating
+in opposite directions, the situation becomes symmetric
+and the net force in x-direction becomes zero. While the
+strength, that is, the amplitude of the force field, de-
+creases with increasing transmitter/detector distance d0,
+the behavior of the phase between transmitter rotation
+and detector vibration is less intuitive. Interpreting the
+phase as a measure for the angle of maximal excitation, it
+becomes obvious that said angle must not necessarily be
+at ϕk = 90◦ due to the mode shape of the first bending
+mode and that it will change with distance d0.
+When both bars rotate in the same direction, the maxi-
+mum of the two force fields do not occur at the same time,
+therefore the force fields don’t superpose symmetrically,
+resulting in an approx. 8% smaller amplitude and a non-
+zero net force in x-direction. For the same reason, the
+angle of maximal excitation occurs exactly at ϕk = 90◦,
+which eliminates the distance dependency of the phase.
+A phasor representation of the superposition of the modal
+forces is presented in the supplementary material.
+Figs. 2b/c show the result of an analytical calculation of
+Fy,b as a function of distance, illustrating the aforemen-
+tioned effects. As it was expected from the spectrum in
+Fig. 2a, an excitation with Ω = ω0/4 creates an ampli-
+tude of about 40% lower compared to Ω = ω0/2.
+To numerically assess the detector movement caused by
+the gravitational field, the modal force is applied as ex-
+citation force to the well-known equation of motion of a
+free–free Euler–Bernoulli beam. It can be shown that the
+complex detector bending velocity at resonance, vb,0 can
+be described via
+vb,0 = GQd
+ω0
+Γ ≈ GQd
+ω0da
+0
+γ ,
+(2)
+where G is the gravitational constant, Qd the Q factor
+of the detector beam and ω0 its first bending resonance
+frequency. Γ is a complex numerical coefficient which de-
+pends on the distance d0, the detector beam and rotat-
+ing bar’s dimensions and their relative position x0, the
+rotating bar’s masses and mass distributions, the ratio
+n = ω0/Ω, and the rotation configuration. It is calcu-
+lated numerically for any given configuration. To avoid
+errors due to the 1D approximation, the calculation of
+Γ is based on a full 3D model (cf. supp. material), for
+which Eq. 2 holds as well.
+The modulus of Γ can be approximated to follow a power
+law distance behavior Γ ≈ γ/da
+0.
+It has to be noted
+that the power law coefficient a considerably differs from
+the prediction a = 5 of the dynamic gravitational field
+around a rotating bar detected by a resonant antenna
+(quadrupole-quadrupole interaction) [11, 12]. The val-
+ues of a for the setup, length scales, and configurations
+presented in this article are summarized in Table II.
+The gravitational constant G can be estimated from
+Eq. 2, using the measured detector amplitude vb,0, the de-
+tector’s vibration properties (Qd, ω0), and the numerical
+coefficient Γ. Since this will result in a complex number,
+only the modulus is taken as estimation for G. The an-
+gle gives information about the phase difference between
+measurement and theory, cf. Fig. 7.
+III.
+EXPERIMENT DESIGN AND
+MEASUREMENT PROCEDURE
+Figure 3a shows the experimental setup as it was
+already introduced and used in [1], now with the ro-
+tating bars located in the transmitter chamber.
+The
+transmitter bars are made of tungsten both with
+length of 499.86(1) mm and a quadratic cross section
+of 10.06(1) mm edge length, each driven by a 50W
+maxon EC-i 40 brushless motor to rotate with fre-
+quency Ω.
+The distance between the rotation cen-
+ters is x0 = 800.0(3) mm.
+The bars’ mass has been
+measured equal to 970.8(1) g and 971.8(1) g, respec-
+tively. The detector is made of titanium with dimensions
+1000.00(1) mm × 16.97(1) mm × 8.49(1) mm and a Q fac-
+tor of Qd = 44819(535) at a detector chamber pressure of
+
+4
+Transmitter
+Chamber
+Detector
+Chamber
+LDV 3
+LDV 2
+LDV 1
+d0
+Lock-In
+Amplifier3
+Lock-In
+Amplifier 2
+Lock-In
+Amplifier 1
+Motor
+Controller 2
+Motor
+Controller 1
+PC
+Frequency 
+Divider
+Ref. Signal
+Generator
+Rotation
+Controller
+M1
+M2
+LB2
+LB1
+Distance
+Controller
+ω
+Ω
+x0
+Transmitter
+chamber
+Detector
+chamber
+Laser
+stage
+LDV 3
+LDV 2
+LDV 1
+x
+z
+y
+Light Barrier
+2
+1
+a
+b
+x
+y
+z
+d0
+x0
+c
+measurement parameters
+measurement data
+various
+FIG. 3. Illustration of the measurement setup. a, Photo of the setup: The rotating transmitter bars (red drawing)
+are located inside the transmitter chamber. The chamber itself is hanging from springs attached to a carrier bar movable in
+y-direction. Inside the detector chamber the detector beam (blue) is hanging on rubber wires attached at the nodal points of
+the first bending mode. The detector chamber is fixed to an anti-vibration table, isolating the beam from the environment, and
+minimizing transmission of non-gravitational forces. The detector movement is measured using three laser Doppler vibrometers
+(LDV), positioned on a separate stage, likewise isolated via springs. The distance d0 is varied by moving the transmitter
+chamber. The whole setup is located in an underground laboratory, providing excellent temperature stability and minimal
+seismic noise. b, Separate view of the rotating bars mount with the light barriers used for the rotation control. c, Block
+diagram of the operation scheme: orange lines illustrate control signals from the PC that set the distance d0, excitation
+frequency Ω and the frequency ratio ω/Ω. Green lines mark measurement signals.
+about 0.03 mbar. The detector is hanging on two EPDM
+rubber strings glued to the beam at the nodal points of
+the first bending mode, which minimizes both external
+damping and force transmission, and enables the use of
+a free-free beam model. A mass of 7.1 g at the center
+of each rubber wire provides additional decoupling. The
+beam’s movement is measured optically using three Poly-
+tec OFV-5000/505 laser Doppler interferometers (LDV)
+placed on a separate, spring-suspended platform.
+The
+output signals of the laser interferometers are fed into
+individual digital lock-in amplifiers (Zurich Instruments
+MFLI) via a 12 dB attenuator.
+The lock-in amplifiers
+use eight cascaded low-pass filters with time constant of
+31 s to extract the velocity amplitude and phase at the
+measurement frequency ω. The reference is provided by
+a high precision signal generator (SRS FS740 with Ru-
+bidium time base, frequency error < 10 pHz, phase accu-
+racy < 1 ns). From the movement at three different posi-
+tions one can distinguish between two rigid body motions
+(translation in y and rotation around z) and the lateral
+bending motion, assuming a known bending mode shape
+Ub,1.
+Both detector and transmitter system have been placed
+in separate aluminum vacuum chambers to avoid acous-
+tic coupling effects or any excitation other than gravi-
+tation. Using two separate Edwards nXDS 10i vacuum
+scroll pumps running continuously, the pressure inside
+the detector and transmitter chambers was kept con-
+stant at 0.027(3) mbar and 0.103(5) mbar, respectively.
+No disturbing influence of the vibrations produced by
+the pumps was identified. The chamber containing the
+detector was fixed to a vibration isolation table using an
+80 mm thick aluminum base plate of 70 kg mass.
+The
+transmitter chamber is hanging on steel springs (Duro-
+vis 20/8/5) from a movable bar attached to a solid frame
+with high damping. This way, the distance d0 can be var-
+ied between 0.3 m and 0.6 m. Accelerometers mounted on
+both chambers give information about remaining move-
+ment of the chambers.
+To investigate the gravitational coupling and to compare
+it with the numerical solution, the frequency response of
+the detector is measured around the first bending reso-
+nance ω0. We use a 24-point frequency sweep with step
+times of 75 min to account for the large time constant
+of the detector beam (τ ≈ 5.5 min) and the lock-in am-
+plifiers (99% settling time approx. 9 min). By averaging
+the last 16 min of each step, a signal-to-noise ratio (SNR)
+ratio of up to 500 can be achieved. After fitting the fre-
+quency response of a single-degree-of freedom (SDOF)
+oscillator [1], the following parameters are obtained: am-
+plitude and phase at resonance, resonance frequency, the
+detector’s Q factor, and a complex offset constant.
+
+5
+300
+350
+400
+450
+distance d0 (mm)
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+|vb,0| (nm/s)
++ = !/2
+3D Theory
+3D Theory 95% confidence band
+Measurement data, scaled to Q = 45000
+300
+350
+400
+450
+distance d0 (mm)
+0
+5
+10
+15
+20
+25
+30
+35
+40
+|vb,0| (nm/s)
++ = !/4
+3D Theory
+3D Theory 95% confidence band
+Measurement data, scaled to Q = 45000
+300
+350
+400
+450
+distance d0 (mm)
+-40
+-30
+-20
+-10
+0
+10
+20
+30
+40
+vb,0 - 
+vb,0
+CW-CW (deg)
+300
+350
+400
+450
+distance d0 (mm)
+-40
+-30
+-20
+-10
+0
+10
+20
+30
+40
+vb,0 - 
+vb,0
+CW-CW (deg)
+FIG. 4. Measurement results of gravitationally induced detector beam motion. Compilation of results at different
+setup configurations and comparison with the 3D theory solution. The data points represent the detector motion at resonance,
+obtained from a SDOF fit of a 24-point frequency response measurement around the detector’s resonance. The measurement
+data point error bars comprise uncertainties that affect vb,0 in Eq. 2, that is, the SDOF fit, laser vibrometer and lock-
+in amplifier. The 95% confidence band of the theoretic result is derived from the standard uncertainties of the remaining
+parameters, G , ω0 , Qd , Γ. Left column, excitation with second harmonic Ω = ω/2. Right column, excitation with fourth
+harmonic Ω = ω/4. Upper row, Detector beam bending amplitude |vb,0| as a function of the beam distance d0. A power-law
+behavior can be identified, the exponents are given in Table II. The values lie mostly within the confidence band (shaded area)
+of the theoretical prediction. Lower row, Phase difference between rotation and detector response, relative to the phase for
+equal rotation direction (yellow/purple line).
+The low bandwidth of the detector and the need for syn-
+chronization of both beams imposes extremely high re-
+quirements on the resolution and stability of the rotation
+frequencies. Since this cannot be realized using the mo-
+tor’s built-in encoder, a custom rotation control and syn-
+chronization based on two light barriers (LB) have been
+implemented, cf. Fig. 3b/c. The rotation control pro-
+vides a stable rotation with a RMS phase jitter < 0.03◦.
+Details on the rotation control can be found in the sup-
+plementary material.
+Due to the long-time measurement, it must be ensured
+that resonance frequency and Q factor of the detec-
+tor beam remain stable during the measurement period.
+Therefore, the whole setup has been placed in an un-
+derground laboratory in the Swiss Alps where a very
+stable temperature can be guaranteed, resulting in an
+average temperature span of 0.004 ◦C per measurement
+point (75 min). Remaining temperature variations have
+been subsequently compensated based on the linear tem-
+perature dependency of the frequency.
+Further details
+can be found in [1].
+Despite rotating with a fraction of the measurement fre-
+quency ω, minimal unbalances of the bars, motor friction,
+material inhomogeneities etc. can generate small vibra-
+tions of the transmitter chamber with amplitudes up to
+2 µm/s (8 nm) at ω. These disturbances can propagate
+to the detector chamber most likely via acoustical trans-
+mission and structure borne sound.
+Therefore, as the
+decoupling of the detector is not perfect, unwanted, non-
+gravitationally induced detector vibration can occur. Ac-
+celerometers mounted on both chambers reveal that the
+use of Ω = ω/2 produces detector chamber velocity am-
+plitudes of approx. 2 nm/s at the frequency of measure-
+ment, while the detector chamber moves with amplitudes
+less than 0.5 nm/s at ω when rotating with Ω = ω/4, cf.
+Fig. 6.
+A.
+Examination of gravitationally induced detector
+vibration
+Over a period of two months, 28 measurement runs
+were conducted at different beam distances and setup
+configurations. Each measurement corresponds to a 24-
+
+6
+22-Nov-21
+28-Nov-21
+29-Nov-21
+30-Nov-21
+01-Dec-21
+07-Dec-21
+08-Dec-21
+09-Dec-21
+10-Dec-21
+11-Dec-21
+12-Dec-21
+14-Dec-21
+15-Dec-21
+20-Dec-21
+22-Dec-21
+24-Dec-21
+25-Dec-21
+27-Dec-21
+29-Dec-21
+30-Dec-21
+01-Jan-22
+03-Jan-22
+05-Jan-22
+11-Jan-22
+04-Feb-22
+06-Feb-22
+28-Feb-22
+02-Mar-22
+6.4073
+6.4741
+6.5408
+6.6076
+6.6743
+6.7410
+6.8078
+6.8745
+6.9413
+ G (10-11 m3kg-1s-2)
+Single measurement (SM) result (x: + = !/2, o: + = !/4)
+SM standard uncertainty
+Weighted mean G* of all SM results
+95% confidence band of weighted mean (combined)
+CODATA 2018
+296
+325
+350
+370
+335
+345
+360
+390
+410
+360
+320
+296
+296
+300
+310
+320
+330
+340
+350
+360
+370
+380
+380
+300
+300
+300
+330
+330
+Beam distance d0 (mm)
+-4
+-3
+-2
+-1
+0 
+1 
+2 
+3 
+4 
+Relative deviation from CODATA (%)
+FIG. 5. Measurement result of the gravitational constant. Over a period of about 2 months, 28 single measurement
+runs were conducted at different beam distances d0 and setup configurations. The rotation configuration is color coded, while
+the marker represent excitation with the second (x) or fourth (o) harmonic. Each measurement yields the resonance amplitude
+and phase of the detector, whereas only the amplitude has been used to estimate the gravitational constant using the analytical
+3D model, cf. Eq. 2. The figure illustrates the single and averaged results for G (left y axis) and the deviation from the
+CODATA 2018 value (right y axis). Lower x axis, measurement date; upper x axis, beam distance d0. An inverse-variance
+weighted mean (black dashed line) yields an overall estimate of G∗ = 6.68157 × 10−11 m3kg−1s−2, with a combined standard
+uncertainty of about 1.46%. The estimation is about 0.1% higher than the CODATA 2018 value (red dash–dot line). The
+plotted 95% confidence band (black dotted lines) represents the extended combined measurement uncertainty (k = 1.96) based
+on statistical and systematic uncertainties (Table I).
+point frequency sweep of 36 h duration, where resonance
+amplitude and phase, as well as the resonance frequency
+and Q-factor, have been extracted by fitting the fre-
+quency response function of a SDOF oscillator. To detect
+measurements that were affected by unstable conditions,
+the following quality criteria have been applied: Detec-
+tor chamber pressure span < 0.01 mbar, detector cham-
+ber temperature span < 0.1◦C, SDOF fit coefficient of
+determination R2 > 99%.
+The results of the measured complex bending velocity at
+resonance vb,0 are summarized in Fig. 4. Additionally,
+the results of the 3D simulation are displayed. Since the
+simulation uses a fixed detector Q factor and resonance
+frequency, the measured amplitudes are scaled to these
+values using the linear dependency of the velocity on Q
+and ω−1
+0 , cf. Eq. 2. As shown in the first row of Fig. 4,
+the measured amplitudes match the numerical prediction
+very well for all setup configurations. Considering the
+uncertainty of the theoretical prediction that comprises
+statistical and systematic uncertainties as summarized
+in Table I, the measured amplitudes mostly lie within
+the 95% tolerance band of the theory/simulation. In the
+second row of Fig. 4, experimental and theoretical re-
+sults of the phase of the detector vibration are depicted.
+Since the phase for configurations using the same rota-
+tion direction (yellow lines) should be independent on the
+distance, it has been used as reference. The experimen-
+tal phase values show the predicted number range and
+trend, but do not match the theory exactly. Besides the
+measurement system that might introduce some uncer-
+tainties, mechanical crosstalk, which is more pronounced
+in the phase, is suspected to be the main reason for the
+observed deviation. A small movement of the detector
+chamber was measured, as shown in Fig. 6, where the
+average velocity amplitudes of both detector and trans-
+mitter chamber are displayed. However, the mechanism
+and magnitude of the force transmission from detector
+chamber to detector beam are not yet fully understood.
+Therefore, a rough estimate of a 1% contribution to the
+systematic uncertainties due to mechanical crosstalk has
+been added to the error budget.
+Finally, G was estimated from each measurement re-
+sult using Eq. 2 (modulus), where the detector’s Q fac-
+tor has been incorporated individually for each mea-
+surement. In Fig. 5, the single results are depicted as
+mean value and standard deviation from statistical er-
+rors (colored patches). Combining the single measure-
+ments of G by means of inverse-variance weighting yields
+G∗ = 6.68157×10−11 m3kg−1s−2 with a relative standard
+deviation of 0.02% (black dashed line in Fig. 5).
+The
+overall 95% confidence band (black dotted line in Fig. 5)
+represents the extended combined measurement uncer-
+
+7
+Standard uncertainty
+∆G/G (%)
+Systematic errors
+Laser vibrometer amplitude
+0.67 %
+0.39 %
+Laser angular misalignment
+0.01 %
+0.01 %
+Lock-in amplifier
+0.01 %
+0.00 %
+Transmitter/detector distance, d0
+0.17 %
+0.98 %
+Mechanical crosstalk
+1.00 %
+1.00 %
+Model parameter amplitude, |γ|
+0.14 %
+0.14 %
+Model parameter phase, ∠(γ)
+0.24 deg
+-
+Transmitter/detector dimensions
+0.01 mm
+-∗
+Transmitter rotation centre x/z offset
+0.20 mm
+-∗
+Transmitter bar eccentricity
+1.00 µm
+-∗
+Transmitter/detector angular misalignment
+0.03 ◦
+-∗
+Transmitter/detector mass
+100.00 mg
+-∗
+Transmitter bar angle difference
+0.10 ◦
+-∗
+Statistical errors
+Detector bending amplitude at resonance, vb,0 0.10 %
+0.10 %
+Detector beam resonance frequency, ω0
+< 0.01 %
+< 0.01 %
+Detector beam phase shift at resonance
+0.11 deg
+-
+Detector beam Q factor, Qd
+0.18 %
+0.18 %
+Transmitter/detector distance, d0
+< 0.01 %
+< 0.01 %
+Statistical errors of all measurements
+0.02 %
+Combined error
+Single measurement
+1.46 %
+All measurements
+1.46 %
+TABLE I. One-sigma error budget used for the assessment of the combined measurement uncertainty of G estimated from
+one single measurement and from all 28 measurements. For parameters where different values apply, e.g. distance errors, the
+highest value of the uncertainty is reported. Note that the phase uncertainties do not contribute to ∆G.
+∗Included in uncertainty of model parameter γ.
+CCW-CW CW-CCW CCW-CCW CW-CW single bar*
+Ω = ω/2
+4.35
+4.35
+4.65
+4.65
+4.35
+Ω = ω/4
+5.83
+5.83
+5.78
+5.78
+5.83
+TABLE II. Value of the power law exponent a as introduced in Eq. 2 for distances d0 between 300 and 600 mm. Coefficients
+obtained from a power law fit based on 3D numerical results. The fit’s average coefficient of determination is R2 = 99.97% (34
+values). CW = clockwise; CCW = counterclockwise.
+∗Equal values for all single bar variations.
+tainty (k = 1.96) based on the statistical and systematic
+uncertainties as summarized in Table I. Finally, G is esti-
+mated with G∗ = 6.68(10)×10−11 m3kg−1s−2. We would
+like to note that although the mean value we obtain is
+only about 0.1% higher than the CODATA 2018 value
+[13], the single values obtained from the measurements
+have a standard deviation which is 1.32% of the mean.
+This can be mostly attributed to the uncertainty of the
+measurement chain which is relatively high at this stage
+of the project (95% confidence band = ±2.85%). Figure 7
+illustrates the phase difference between measurement and
+theory, increasing with larger distances, supporting the
+assumption of remaining mechanical, non-gravitational
+coupling.
+Based on a power balance analysis, the near field gravita-
+tional energy flow between transmitter and detector can
+be computed [1]. At steady state, incoming energy at the
+detector is dissipated according to the Q factor of the de-
+tector. If all this energy is attributed to gravitation, this
+yields a gravitational power of maximal 3.75×10−18 W at
+d0 = 296 mm, Ω = ω0/2, CW-CCW rotation. This is
+about 1025 times higher than the expected power of grav-
+itational waves radiated from two equivalent quadrupole
+gravitational wave generators [14].
+IV.
+DISCUSSION AND OUTLOOK
+In the last couple of years, fully characterized dy-
+namic gravitation experiments turned out to be a promis-
+ing approach to better understand gravitational inter-
+actions [15].
+They open up the path to an investiga-
+tion of dynamic gravitational effects in a frequency range
+> 1 Hz, covering for example the gravitational wave high-
+frequency band [16].
+The combination of two similar rotating bars as transmit-
+ter system and a bending beam as detector system proves
+to be a considerable improvement to previous experi-
+
+8
+ments, cf. Table III. Besides considerably higher ampli-
+tudes of one order of magnitude, the double-transmitter
+setup enables the investigation of numerous setup config-
+urations, based on rotation direction combinations and
+the use of different harmonics as excitation. Therefore,
+gravitational and remaining non-gravitational coupling
+can be better distinguished, making the results more re-
+liable. The extremely precise rotation control, necessary
+for the double excitation, has the additional merit of en-
+abling a precise phase measurement between rotation and
+detector response, which can be very helpful to further
+understand dynamic, non-gravitational coupling.
+The
+work presented here establishes a new experiment, how-
+ever, it does not yet claim to be highly precise. Nonethe-
+less, the observed discrepancy between theory and mea-
+surement is considerably less than 1%.
+To further increase the measurement quality and reliabil-
+ity, future work focuses on improvements that comprise a
+more precise distance control, reduced temperature sensi-
+tivity, characterization of mass distributions, passive and
+active crosstalk cancellation, and improved model accu-
+racy. The latter requires narrower tolerances and a bet-
+ter understanding of the material structure and behavior
+of both the rotating bars (material homogeneity, ultra-
+precise mass measurement, etc.) and detector (temper-
+ature behavior, influences on structural damping, etc.).
+Since the setup allows for the use of even larger transmit-
+ter bars the signal level can be increased by about one
+order of magnitude. Although higher amplitudes reduce
+problems associated with the optical vibration measure-
+ment of such small amplitudes [1, 17–19], an individual
+calibration of the interferometric measurement system in
+the nm/s range is still necessary to considerably reduce
+the measurement uncertainty.
+Alternatively, a custom
+made demodulation of the laser output might enable to
+directly trace back the detector displacement to the wave-
+length of light.
+We believe that these improvements can bring the ex-
+periment way beyond a proof-of-concept state towards a
+highly precise measurement that might become the new
+standard dynamic gravitational experiment. It will help
+to reveal new insights in dynamic gravitation, such as
+frequency dependency or amplitude/phase effects due to
+objects between transmitter and detector (gravitational
+shielding). The results and findings of this and future re-
+lated works can also help to advance the research and
+application of Newtonian calibrators that are used in
+gravitational-wave detectors [5, 6, 20–23].
+
+9
+Transmitter properties
+Receiver properties
+Measurement parameters
+Evaluation
+Work
+Design
+Mode
+Mass
+(kg)
+Design
+Mode
+Sensor
+principle
+Freq.
+(Hz)
+Q
+Mass
+(kg)
+TX/RX
+distance
+(m)
+Number of
+force field
+harmonics
+Number of
+excitation
+configurations
+(investigated)
+Phase
+shift
+G
+Sinsky68 [3]
+Aluminum
+cylinder
+Longitudinal
+resonance
+136
+Aluminum
+cylinder
+Longitudinal
+resonance
+Piezoelectric
+1660
+≈ 1E5 1500 1.72. . . 1.92
+1
+1 (1)
+No
+No
+Hirakawa80 [6]
+Steel bar
+Rotation
+44
+Mass quadrupole
+antenna
+Structural
+resonance
+Electrostatic
+60.5
+4100
+1400
+2.1. . . 4.2
+1
+2 (1)
+No
+No
+Ogawa82 [7]
+Steel bar
+Rotation
+401
+Mass quadrupole
+antenna
+Structural
+resonance
+Electrostatic
+60.8
+5300
+1400
+2.6. . . 10.6
+1
+2 (1)
+No
+No
+Kuroda85 [8]
+Aluminum disk
+lead-filled holes
+Rotation
+≈ 1
+Torsional
+antenna
+Torsional
+resonance
+Electrostatic
+61
+96
+≈ 1E4 0.85
+15
+0.1. . . 0.3
+1
+2 (1)
+No
+Yes
+Astone98 [5]
+Aluminum
+constant stress
+bar
+Rotation
+14
+Cryogenic
+aluminum
+cylinder [24]
+Longitudinal
+resonance
+Capacitive
+910
+930
+1.1E6
+5.6E6 2270
+1.9. . . 3.5
+1
+2 (1)
+Yes
+No
+Ross21 [20]
+Aluminum disk
+void/tungsten-
+filled holes
+Rotation
+1
+Test mass
+Displacement Optical (GW
+detector [25]) 8. . . 30
+-
+39.7
+1.18
+2
+2 (1)
+No
+Yes
+Brack22 [1]
+Tungsten beam
+Bending
+resonance
+3.9
+Titanium
+beam
+Bending
+resonance
+Optical
+42.6
+3.5E4
+0.6
+0.06. . . 0.12
+1
+2 (1)
+Yes
+Yes
+This work
+Two tungsten bars
+Rotation
+2×1
+Titanium
+beam
+Bending
+resonance
+Optical
+42.6
+4.5E4
+0.6
+0.3. . . 0.42
+2
+8 (4)
+Yes
+Yes
+TABLE III. Overview of past transmitter/detector macro-scale experiments that investigate dynamic gravitational forces. Numbers are rounded.
+
+10
+ACKNOWLEDGMENTS
+We gratefully acknowledge the support of ETH Zurich,
+maxon motor ag, ZC Ziegler Consultants AG, and Zurich
+Instruments AG.
+Appendix A: Superposition of force fields
+The situation of the superposition of the force fields
+generated by the two identical rotating bars has been de-
+scribed only qualitatively in the main article. To mathe-
+matically confirm this description, it is useful to look at
+the Fourier component at ω of the individual 1D force
+components
+F (k)
+y,b =
+� ld
+0
+f (k)
+y
+Ub,1(xd)dxd ≈ ckeiωt ,
+(A1)
+that are summed up to build the total modal force Fy,b,
+cf. Eq. (1). The variable k denotes the number of the
+rotating bar. The Fourier series yields the complex am-
+plitude at the detector’s resonance frequency ω, given by
+the coefficient ck. Assuming identical bars, perfect sym-
+metry with respect to the detector beam and a perfectly
+symmetric first bending mode shape Ub,1, the amplitudes
+|ck| of both complex forces must be equal. The angle ∠ck,
+however, differs in sign depending on the rotation direc-
+tion. It can be shown that, in case of opposite direction,
+c1 equals c2, hence the total force has twice the amplitude
+and the same phase as the components ck. Considering
+the bars rotating in the same direction c1 = c∗
+2, where
+∗ denotes the complex conjugate. Consequently, c1 + c2
+is a real value which means that no phase shift occurs.
+However, the amplitude gets smaller since the contribu-
+tion of the imaginary part vanishes. Therefore, a phase
+angle of ∠ck = ±23◦ yields an amplitude reduction of
+8%, as observed both experimentally and theoretically,
+cf. Figs. 2 and 4. Figure 8 illustrates the superposition
+of the forces qualitatively.
+The situation of both bars rotating in the same direction
+is thus very convenient to test both the numerical and
+experimental results. Numerical errors can be detected
+easily as a phase shift ̸= 0, while experimentally mea-
+sured phase shift variations indicate either crosstalk or a
+deviation from the ideal situation of two identical, per-
+fectly synchronized, symmetrically oriented bars. Like-
+wise, differences between the bars can be tested by using
+one rotating bar only.
+Appendix B: Numerical model
+The one dimensional Euler-Bernoulli approach as-
+sumes a line distribution of the mass of the detector
+beam. For the setup presented here this approximation
+gives reasonable results only if the rotating bars are far
+away from the detector beam (d0 > 0.5 m). For improved
+accuracy, a numerical, 3D finite element (FE) simulation
+of the transmitter/detector setup is necessary to accu-
+rately predict the gravitationally induced motion of the
+detector beam and to compute the coefficient Γ in Eq. 2
+for all distances d0.
+The detector beam is modeled as a freely suspended, ho-
+mogeneous, linear-elastic beam, attached to two linear
+springs in the nodes of its first bending mode. Due to
+the lock-in measurement technique, the experimentally
+measured response of the detector is available at the ex-
+citation frequency ω only. Hence, it suffices to solve for
+the steady state response of the FE model. The damping
+of the system is modeled using modal damping [10].
+Since the force field generated by the rotating bars, il-
+lustrated as a transverse force pulse moving along the
+x-axis of the detector beam, represents a moving load
+problem, it is possible that higher bending modes of the
+detector are excited as well [9]. This, in turn, could lead
+to an additional contribution to the motion measured at
+the frequency of the first bending mode, which would re-
+quire an adjustment of Eq. 2, which is based on a SDOF
+approximation of the detector.
+Therefore, the contribution of higher order modes was
+investigated numerically: Since the modal damping al-
+lows to assign experimentally measured Q factors to the
+first three in-plane bending modes, that is, Q1 = 45000,
+Q2 = 381, Q3 = 5150, as well as for the rigid body modes
+of the beam, Qrb = 2900, a comparison was performed
+between a simulation using specific modal Q factors Qi
+and a simulation where all modes have the same damp-
+ing.
+The results revealed that the influence of higher
+modes is negligible for the setup presented in this article,
+thus the first bending mode can be assumed to be fully
+decoupled from the other modes. Consequently, Eq. 2
+represents a valid approximation of the structural reso-
+nance.
+The coefficient Γ has been calculated individually for all
+different setup configurations and 37 discrete distance
+values d0 between 290 mm and 600 mm using COMSOL
+Multiphysics 5.6 and MATLAB R2020b.
+If necessary,
+amplitude values at the specific distances are interpolated
+using a power law fit with the coefficients from Table II,
+while the phase is interpolated linearly.
+The detector beam is modeled as a linear-elastic rectan-
+gular beam with Young’s modulus E = 107.4 MPa and
+Poisson’s ratio of ν = 0.37, using the solid mechanics
+module. The Young’s modulus is determined through a
+fit to the first resonance frequency of the beam. All other
+properties of the beam are the same as described in [1].
+The beam is attached to two linear springs in the nodes
+of its first bending mode. These springs have a nonzero
+stiffness in y-direction only and their stiffness is set such
+that it matches the experimentally determined resonance
+frequency of the beam’s translational pendulum motion
+in y-direction of f = 858.3 mHz.
+The computation of the detector beam velocity is a three-
+step process: In a first step, the resonance frequency
+ω0 of the beam is computed in the absence of external
+
+11
+fields. This resonance frequency is later used to run the
+frequency space simulation exactly at the resonance fre-
+quency of the beam. In a second step, the gravitational
+force field is computed.
+The gravitation force on the
+beam is given by the integral of Newton’s force law over
+the volume of each rotating bar. Since the force is peri-
+odic in time, it can be written as a Fourier series, where
+the Fourier coefficients are computed in every node of the
+FE model. In the last step, the FE model of the beam is
+solved in frequency space at the resonance frequency ω0.
+Depending on the selection of the Fourier coefficient, the
+solution corresponds to different rotation speeds ω of the
+bars relative to the resonance frequencies.
+A convergence study has been performed to ensure the
+numerical accuracy of the simulation. The gravitational
+force of the rotating bars has been computed by discretiz-
+ing the bars into 396 mass elements.
+For the ensuing
+discrete Fourier transform of the force, the rotation pe-
+riod of the bars was evaluated at 64 points per rota-
+tion. Finally, the FE model is solved for approximately
+35000 tetrahedral elements using quadratic interpolation
+(DOF ≈ 170000). Taking all three discretization steps
+into account, we have found the numerical error of the
+simulation to be far below the measurement errors in the
+experiment and can thus be neglected in the error bud-
+get.
+Appendix C: Gravitational waves generated by
+rotating bars
+The setup of a rotating bar is a classical textbook ex-
+ample of a laboratory quadrupole generator of gravita-
+tional waves [14, 26], in which the power of a slender bar
+radiated as gravitational waves can be approximately cal-
+culated via
+LGW ≈ 32
+5
+G
+c5 Ω6I2
+z ,
+(C1)
+where Iz denotes the rotational inertia of the bar rotating
+around the z axis with frequency Ω and the speed of
+light c. For one of the rotating bars presented in this
+setup rotating with Ω = ω0/2, this power calculates to
+LGW ≈ 4.2 × 10−43 W.
+In contrast, the power transmitted from two oppositely
+rotating bars to a detector in a distance d0 = 296 mm
+resonating at ω0 has been estimated to 3.75×10−18 W,
+based on the measured resonance amplitude and Q factor
+of the detector [1].
+Hence it can be concluded that the generated movement
+of the detector beam is not attributed to gravitational
+waves, emitted by the rotating beams but by the dynamic
+gravitational (near) field.
+Appendix D: Rotational system - mechanics and
+control
+The rotation system is composed of two independent,
+identical rotary units, of which one system is exemplarily
+described in this section.
+The rotating bar is placed on a turntable, where it can be
+adjusted and fixated using a clamping cover with align-
+ment pins. The bar exhibits a precisely manufactured
+indentation in its center of rotation for additional fix-
+ation. The rotary unit is mounted with two preloaded
+spindle bearings (HQW Precision GmbH SV7902) that
+are clamped to a bearing bracket mounted to a massive
+aluminum base plate of 15 mm thickness. Attached to
+the shaft is a maxon EC-i 40 brushless 50 W electric mo-
+tor with ENC 16 EASY encoder, connected to the base
+plate as well.
+The motor is internally controlled by a maxon EPOS4
+50/5 controller. To have full control over the rotation,
+a master controller based on an Arduino MEGA 2560 is
+additionally used that communicates with the EPOS sys-
+tem via CAN bus. Both the master and EPOS controller
+and the power supply are placed outside of the vacuum
+chamber.
+For the gravitational excitation it is of utmost impor-
+tance that the rotation of both bars is synchronized to a
+reference signal as precisely as possible. For this task, the
+built-in controller and rotation sensors are not suitable,
+due to their limited resolution and non-synchronized
+clock rates. Therefore, two independent light barriers are
+embedded in the base plate (cf. Fig. 3c). The light bar-
+riers are built from a high-speed PIN photodiode (SFH
+2701, 730 nm), a 500 MHz LTC6268 operational amplifier
+and a 280 MHz LTC6752 comparator. A TLC555 timer
+is used as line driver, producing a delay of the pulse of
+244 ns. The emitted light is reflected from a small alu-
+minum patch (0.035 g) glued to the rotating bar. To en-
+sure mechanical balancing, the same patch is also glued
+to the opposite side of the bar.
+To establish a rotation with a frequency Ω being a frac-
+tion of the frequency of a sinusoidal reference signal ω,
+that is, the frequency of measurement provided by a fre-
+quency generator, a comparator converts the reference
+signal to a rectangular signal of frequency Ω, which can
+then be fed into a digital, flip-flop-based frequency di-
+vider. Finally, a PID controller adjusts the frequency of
+rotation by synchronizing the edges of both the rectan-
+gular signal and the pulse of the light barrier with a time
+resolution of 62.5 ns.
+Appendix E: Measurement uncertainty
+Assuming uncorrelated input quantities, the combined
+standard uncertainty associated with estimating G can
+be calculated from Eq. 2 and Table II using a first-order
+Taylor approximation for each measurement individually.
+The uncertainties associated with estimating Qd and ω0
+
+12
+result directly from the SDOF fit of the measurement.
+Initially, the distance d0 was manually adjusted, where
+a systematic error of 0.5 mm was assumed. The auto-
+matic positioning system itself works very precisely with
+an error of 1 µm. As mentioned in [1], the uncertainty
+of the measured velocity vb,0 is yet unclear for the ex-
+tremely small amplitudes relevant in the measurements
+presented here. Therefore, the uncertainties reported in
+the data sheets of the laser vibrometers have been used
+for a first assessment. Since the detector bending ampli-
+tude is calculated from a linear combination of three laser
+measurement signals, the contribution to the combined
+uncertainty reduces by a factor 0.58. Further, an angu-
+lar misalignment of the laser beam with respect to the
+detector/transmitter surface of max. 1◦ contributes to
+the uncertainty. Errors that can be attributed to a non-
+gravitational detector excitation, most likely due to me-
+chanical crosstalk, are estimated with a 1.00 % system-
+atic error, based on an evaluation of the detector cham-
+ber acceleration. To estimate the variance of the model
+parameter γ, a quasi-Monte Carlo method has been ap-
+plied, using 2000 quasi-random sequences drawn from the
+probability distributions specified for the input parame-
+ters (Sobol method) [27, 28]. The used input parameters
+are assumed to be normally distributed with the follow-
+ing standard uncertainties: detector and transmitter di-
+mensions, 0.01 mm; transmitter masses, measured with
+a Mettler-Toledo XP6002S scale with 100.00 mg com-
+bined uncertainty; an uncertainty of x, y and z position
+of the rotation centers of each 0.20 mm; an eccentricity
+of ±1.00 µm of the rotating bars, a constant angle offset
+between the transmitter bars of ±0.10 ◦ and an angu-
+lar misalignment of 0.03 ◦ between the detector’s central
+axis and the transmitter axis given by the centers of ro-
+tation. The Monte Carlo simulation has been performed
+for each setup variation and distance used in the exper-
+iments. Therefore, an individual combined uncertainty
+results for each measurement. In Table II the maximum
+values are exemplarily reported, resulting in a combined
+standard error of max. 1.46% for an individual measure-
+ment.
+In this work, the distance d0 has the highest contribu-
+tion to the uncertainty of G, since the distance uncer-
+tainty scales with a factor of up to 5.83, cf. Table II.
+To achieve an uncertainty ∆G/G ≪ 1%, future activ-
+ities must therefore focus primarily on the distance d0,
+crosstalk elimination, the velocity measurement, and the
+Q factor determination. Further, influences that are not
+yet included such as material inhomogeneities, form tol-
+erances, numerical errors etc. must be considered as well.
+Appendix F: Dynamic influence of phase jitter
+To achieve a rotation of both bars exactly at a fraction
+of the resonance frequency, a control loop is necessary
+that enables the synchronization of the bars’ rotation fre-
+quencies to an external reference signal. However, a cer-
+tain disturbance of said rotation, quantified by a phase
+jitter Jp(t), might influence the resonance excitation of
+the detector. Therefore, the influence of a certain phase
+variation on the detector shall be discussed briefly.
+To estimate the errors associated with said disturbance,
+one can reduce the bending motion to a single-degree-of-
+freedom oscillator with an external, harmonic excitation.
+This is a valid approach since the Q factor of the first
+bending mode is very large (Qd ≈ 45000) and resonance
+frequencies are well separated. Considering excitation at
+resonance ω0, we can introduce the jitter as phase mod-
+ulation to the excitation, that is,
+Fn = F0 cos (ω0t + Jp(t)) .
+(F1)
+The response x(t) of the oscillator is described by its
+amplitude and phase x(t) = Aeiω0t+iφ. The effects of the
+phase noise Jp(t) on the output phase can be analyzed
+using the transfer function [29, 30]
+∆¯φ(ω) =
+ω0
+2Qdiω + ω0
+¯Jp(ω)
+(F2)
+where ∆¯φ denotes the absolute variation of the detec-
+tor’s phase and amplitude around the value at resonance
+in the frequency domain (indicated by the overbar). The
+response of the amplitude, however, cannot be easily de-
+scribed by the transfer function of a linear time-invariant
+system. It can be shown, however, that the amplitude ap-
+proximately shows a similar low-pass behavior [31].
+As a consequence, the detector acts as a low-pass filter to
+disturbances of the excitation phase. Disturbances trans-
+mit in the phase via a first order low-pass filter with a
+cut-off frequency of ωc = ω0/(2Qd) ≈ 0.5 mHz. Numeri-
+cal simulations with MATLAB Simulink show a relative
+amplitude deviation < 1 × 10−4 for a white phase noise
+corresponding to 0.03◦ RMS jitter. In summary, phase
+variations of the excitation force have a negligible effect
+on the detector’s behavior, since they are minimized by
+the inertia of the transmitter bars, the motor control sys-
+tem and the high-Q detector itself.
+
+13
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+vy Detector Chamber (nm/s)
++ = !/2
+mean
+std
+noise level ca. 0.26 nm/s
++ = !/4
+mean
+std
+noise level ca. 0.26 nm/s
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+vz Detector Chamber (nm/s)
+mean
+std
+noise level ca. 0.26 nm/s
+mean
+std
+noise level ca. 0.26 nm/s
+28-Nov-21
+29-Nov-21
+30-Nov-21
+01-Dec-21
+07-Dec-21
+08-Dec-21
+09-Dec-21
+10-Dec-21
+11-Dec-21
+12-Dec-21
+14-Dec-21
+15-Dec-21
+05-Jan-22
+04-Feb-22
+06-Feb-22
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+vy Transmitterchamber (7m/s)
+mean
+std
+noise level ca. 0.03 7m/s
+22-Nov-21
+20-Dec-21
+22-Dec-21
+24-Dec-21
+25-Dec-21
+27-Dec-21
+29-Dec-21
+30-Dec-21
+01-Jan-22
+03-Jan-22
+11-Jan-22
+28-Feb-22
+02-Mar-22
+mean
+std
+noise level ca. 0.03 7m/s
+FIG. 6. Detector and transmitter chamber movement. Average velocity amplitude at the measurement frequency ω
+measured with lock-in amplifiers (8th order lowpass, 31 s time constant, 16 min averaging) First two rows, Velocity amplitudes
+of the detector chamber both in y and z direction, measured with a Kinemetrics EPI ES-T FBA triaxial accelerometer.
+Third row, Velocity amplitude in y direction of the transmitter chamber, measured with a Bruel&Kjær 4535-B-001 triaxial
+accelerometer. The dashed lines represent the noise level of the measurement. Left column, Excitation with Ω = ω/2. Right
+column, Excitation with Ω = ω/4.
+
+14
+280
+300
+320
+340
+360
+380
+400
+420
+Beam distance d0 (mm)
+-8
+-6
+-4
+-2
+0
+2
+4
+ G (deg)
+Single measurement (SM) result (x: + = !/2, o: + = !/4)
+SM 95% uncertainty
+FIG. 7. Phase difference between measurement and theory as a function of transmitter/detector distance, d0. From
+Eq. 2, the phase difference between theory and measurement can be determined as the angle of the resulting complex value
+of G. Both the theoretical and experimental phase have been expressed relative to a reference phase, that is, the phase at
+same rotation direction (yellow colour coded results). The experimental reference phase has been determined by an average of
+the corresponding results, separately for each harmonic. The y-extent of the error patches denote statistical uncertainties of
+the phase, the x-range has no numerical meaning but is for illustrative purposes only. The illustration indicates an increasing
+phase difference for increasing distance, d0, both for the second (x) and fourth (o) harmonic. A certain, however yet unknown,
+systematic error seems to be present, which is most likely due to remaining crosstalk (mechanical, acoustical). For the setup
+configuration CW-CCW (red color), the error seems to be a bit higher.
+c1,CCW
+c2,CW
+c2,CW
+c1,CW
+real
+imaginary
+real
+imaginary
+real
+ imaginary
+ imaginary
+0
+0
+0
+0
+c1,CCW
+c2,CCW
+c1,CW
+c2,CCW
+0
+0
+0
+0
+real
+Φ
+Φ
+Φ
+Φ
+Φ
+Φ
+FIG. 8. Superposition of modal forces. Qualitative illustration of the modal forces of each rotating bar (black arrows)
+in the complex plane for different setup configurations. The total modal force (colored arrows) originates from a complex
+superposition of the individual modal forces. Phase angle of φ = ±23◦, as for Ω = ω/2, d0 = 300 mm.
+
+15
+Parameter
+Unit
+Transmitter 1 Transmitter 2
+Detector
+Value SU
+Value SU
+Value
+SU
+Mass
+g
+970.8
+0.1
+971.8
+0.1
+647.7
+0.1
+Length
+mm
+499.86 0.01
+499.86 0.01
+1000.00
+0.01
+Width
+mm
+10.06
+0.01
+10.06
+0.01
+8.49
+0.01
+Height
+mm
+10.06
+0.01
+10.07
+0.01
+16.97
+0.01
+Cross sectional area*
+mm2
+101.2
+0.14
+101.3
+0.14
+144.1
+0.19
+Second moment of inertia
+with respect to z-axis*
+10−10m4
+8.65
+0.03
+Resonance frequency @ 11.4◦C Hz
+42.650759 0.000015
+Mass per unit length*
+kg/m
+0.6477
+0.0001
+Bending stiffness*
+Pa m4
+92.924
+0.014
+Young’s modulus*
+GPa
+107.4
+0.4
+Transmitter distance x0
+mm
+800
+0.3
+TABLE IV. Detector and transmitter model parameters and corresponding standard uncertainties (SU). ∗Derived parameters.
+
+16
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diff --git a/v9FJT4oBgHgl3EQffixw/content/tmp_files/2301.11557v1.pdf.txt b/v9FJT4oBgHgl3EQffixw/content/tmp_files/2301.11557v1.pdf.txt
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+A Ray-tracing and Deep Learning Fusion
+Super-resolution Modeling Method for Wireless
+Mobile Channel
+Zhao Zhang∗, Danping He∗†, Xiping Wang∗, Ke Guan∗‡, Zhangdui Zhong∗§, Jianwu Dou¶∥
+∗ State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, 100044 Beijing, China
+† Beijing Engineering Research Center of High-speed Railway Broadband Mobile Communications, 100044 Beijing, China
+‡ Frontiers Science Center for Smart High-speed Railway System, 100044 Beijing, China
+§ Key Laboratory of Railway Industry of Broadband Mobile Information Communications, 100044 Beijing, China
+¶ State Key Laboratory of Mobile Network and Mobile Multimedia Technology, 518055 Shenzhen, Guangdong, China
+∥ ZTE Corporation, 518055 Shenzhen, Guangdong, China
+*Corresponding author: Danping He, E-mail: hedanping@bjtu.edu.cn
+Abstract—Mobile channel modeling has always been the core
+part for design, deployment and optimization of communication
+system, especially in 5G and beyond era. Deterministic channel
+modeling could precisely achieve mobile channel description,
+however with defects of equipment and time consuming. In this
+paper, we proposed a novel super resolution (SR) model for
+cluster characteristics prediction. The model is based on deep
+neural networks with residual connection. A series of simulations
+at 3.5 GHz are conducted by a three-dimensional ray trac-
+ing (RT) simulator in diverse scenarios. Cluster characteristics
+are extracted and corresponding data sets are constructed to
+train the model. Experiments demonstrate that the proposed
+SR approach could achieve better power and cluster location
+prediction performance than traditional interpolation method
+and the root mean square error (RMSE) drops by 51% and
+78% relatively. Channel impulse response (CIR) is reconstructed
+based on cluster characteristics, which could match well with the
+multi-path component (MPC). The proposed method can be used
+to efficiently and accurately generate big data of mobile channel,
+which significantly reduces the computation time of RT-only.
+Index Terms—ray tracing, deep learning, super resolution,
+mobile channel modeling, cluster prediction.
+I. INTRODUCTION
+The global internet of vehicles market is projected to grow
+from $ 95.62 billion in 2021 to $ 369.60 billion in 2028
+in forecast period [1]. New communication services such
+as intra-vehicle, Vehicle-to-Everything (V2X) communication
+and Industrial Internet of Things (IIoT) have put forward more
+stringent requirements for mobile communication systems [2]
+[3]. It has been the consensus that the 5G and B5G will realize
+a high data-rate and ultra-reliable low-latency communication
+albeit with soaring demand. However, current 5G could hardly
+provide a guarantee in harsh electromagnetic and time-varying
+environments. To achieve B5G superior features for mobile
+communication in diverse scenarios, mobile channel modeling
+is imperatively needed to better understand channel character-
+istics, plan and optimize communication systems.
+Channel modeling is the process of modeling the sig-
+nal propagation mechanism to obtain an accurate channel
+description. Generally, geometry-based stochastic modeling
+(GBSM) and ray-tracing (RT) based deterministic modeling
+are two main modeling approaches for mobile channel. GBSM
+can theoretically generate channel impulse responses (CIRs)
+through assumed scattering geometry to analyze performance,
+but practically the CIRs are evaluated numerically via mea-
+surement and calculation. On the other hand, RT can generate
+highly accurate channel characteristics for specific scenario
+with the defect of computational complexity [4]–[6]. Thereby,
+a fast and reliable channel characteristic generation method
+based on coarse-grained RT is desperately needed to accelerate
+the modeling process.
+Mobile channel modeling not only refers to time-varying
+and non-stationary channels but also involves the influence of
+variant multi-path components and small-scale fading, which
+is more dominant. In this regard, cluster-based approaches are
+increasingly adopted in recent research especially since GBSM
+was proposed. In [7], a clustering and tracking algorithm was
+proposed and analyzed for vehicle-to-infrastructure channel.
+To recognize and track the clusters in time-varying channels,
+a clustering and tracking algorithm based on power-angle-
+spectrum is proposed and investigated [8]. In [9], a semi-
+deterministic channel modeling method is presented based on
+RT and cluster. Many classical radio channel models are also
+based on the concept of cluster, e.g. COST 2100, 3GPP Spatial
+Channel Model and WINNER.
+Deep learning (DL) based channel modeling methods are
+getting popular in recent years because of its excellent infor-
+mation integration and inferring ability. Numerous research
+including our previous work [10] are dedicated on large scale
+channel characteristics prediction, e.g. pathloss, delay spread
+and number of clusters. Multi-layer perceptron artificial neural
+network is presented for path loss prediction in [11]. The
+author in [12] proposes a procedure of predicting channel
+characteristics based on convolutional neural network (CNN)
+for multi-dimensional millimeter wave channel characteristics
+prediction. However, small scale channel characteristics (e.g.
+arXiv:2301.11557v1  [eess.SP]  27 Jan 2023
+
+Fig. 1. Framework of the study.
+power, angle and location of each cluster) related research
+based on DL are inadequate.
+In this paper, we propose a deep learning and ray tracing
+fusion super-resolution (SR) method for cluster characteristics,
+the power and location of each cluster more specifically.
+Overview of our work is shown in Fig. 1. We first use ray-
+tracing simulator to generate ray level parameters, from which
+cluster level parameters are extracted utilizing the proposed
+clustering method. Subsequently, high resolution data and low
+resolution data are divided to train the deep learning SR
+model. Evaluation and comparison are implemented in cluster
+prediction and CIR reconstruction aspects. Specifically, we
+make the following contributions:
+• Massive RT simulation by self develop CloudRT [13]
+was conducted in four restored scenarios via SketchUp
+and 3D electronic map and multi-dimensional ray level
+characteristics data are generated.
+• A novel object-based clustering method is presented
+for better rays division. Special filtering, tracking and
+segmenting method are utilized to extract crucial char-
+acteristics, unify cluster dataset and maintain continuity.
+• Multi-layer deep learning model (MLL) is proposed to
+predict cluster characteristics more accurately. Residual
+connection and pre-upsampling techniques are integrated
+for eliminating the vanishing gradients problem and
+achieving better performance. Ablation study and gen-
+eralization test demonstrated the necessity of adopted
+crucial techniques and adaptability of proposed model.
+The remainder of this paper is organized as follows.
+Section II introduces the simulation configuration and data
+construction method. The proposed SR and baseline model
+are explained in Section III. Experiments and evaluations are
+implemented in section IV. Finally, conclusions are drawn in
+Section V.
+II. SIMULATION AND DATA CONSTRUCTION
+A. Scenario Modeling
+The overall simulation is based on two kinds of scenarios.
+As shown in Fig. 2 and Fig. 3, a dense urban scenario is
+based on 3D electronic map and three street scenarios are
+manually modeled via SketchUp. Note that all four scenarios
+are completely sourced from real environment, which are Cen-
+tral business district, Malianwa street and Jianting viaduct and
+Xinxi road in Beijing, respectively. The dense urban scenario
+mainly consists of different types of buildings and terrain (e.g.
+regular buildings, parallel buildings, dry land, green land). The
+Fig. 2. Simulations in dense urban scenario.
+Fig. 3. Simulations in street scenarios.
+TABLE I
+SIMULATION CONFIGURATION
+Parameter
+Value
+Carrier frequency [GHz]
+3.55
+System bandwidth [MHz]
+100
+Tx transmit power [dBm]
+0.1
+Tx location
+5-10 m above ground
+Rx location
+2 m above ground
+Tx & Rx attenna
+Omni-directional vertical polarization
+street scenario also contains many fine scatterers, covering
+from cars to pedestrians and trees. As depicted, the transmitter
+(Tx) and route of receivers (Rx) are presented clearly. A total
+of 7 Rx routes in different color are simulated separately.
+It is noteworthy that Tx location is elaborately sited twice
+so that both LOS and NLOS scenes are simulated for route
+1 ∼ 4. Only LOS scene is simulated for route 5 ∼ 7. Through
+simulation in various and close-to-real scenarios, the diversity
+of data is guaranteed in the case of limited data.
+B. Simulation Settings
+Based on aforementioned scenario and electromagnetic
+(EM) parameters provided by ITU-R P.1238-7, Self developed
+CloudRT platform is utilized to obtain channel characteristics
+data. Rx is located 2 meters above ground along the road at
+1 m intervals. Tx is placed on the side of road, near the end
+of receiver’s trajectory. The propagation mechanisms includes
+light of sight, scattering, reflection, penetration and diffraction.
+The detailed simulation configuration is shown in TABLE I.
+C. Clustering
+For a single Tx-Rx link, hundreds to thousands of rays can
+be traced, most of which share the similar delay, angle and
+power. Therefore, it is an accepted practice to equate similar
+rays as clusters and study channel characteristics on cluster
+level. The overall data process workflow is shown in Fig.
+4. Clustering is the process of cluster data generation based
+on rays. There are enormous approaches for clustering (e.g.
+power-angle-spectrum based clustering, power-delay profile
+
+Data set construction
+Training
+Evaluation
+Ray-tracing
+HR & LR
+Cluster
+SR model
+simulator
+data
+prediction
+Ray level
+Cluster level
+Baseline
+CIR
+model
+reconstruction
+parameters
+parametersRoute 1~4
+(a) 3D view of dense urban scenario
+(b) Aerial View of Rx routeRoute 5
+Route 6
+Route 7
+(a) Xinxi street
+(b) Malianwa road
+(c) Jianting viaductFig. 4. Workflow of data process.
+based clustering). Based on RT, the propagation of each ray
+can be accurately acquired, including 3D coordinates, object
+and microfacet identity of reflection and scatter points with
+scene. Microfacet-based and object-based clustering method
+are hence considered and compared. In this work, rays that
+hit on the same object are grouped into a cluster, after which
+cluster characteristics including power and center locations
+are extracted for each snapshot. The author in [8] proposed
+an algorithm for identifying and tracking multipath clusters,
+which also introduces the conception of power-weighted clus-
+ter centers, intra-cluster angle expansion and cluster shape.
+Similarly, we adopt the power-weighted cluster center and
+coherent superposition cluster power approach for filtering.
+The equation is as follows:
+C(j) =
+�K
+k=1 C(rk)P(rk)
+�K
+k=1 P(rk)
+, rk ∈ j
+(1)
+P(j) =
+K
+�
+k=1
+P(rk), rk ∈ j
+(2)
+C, P denotes the 3D coordinate of reflection or scatter
+point and power. j, rk denotes jth cluster and kth ray in
+this cluster. Due to different cluster between snapshots, cluster
+tracking is implemented between snapshots for continuity.
+Every 17 adjacent snapshots are then segmented into a sample,
+exactly enough to implement SR test at scale 16. Within each
+sample, the number of clusters is compromised to a certain
+value to form a regular data structure. Data structure is shown
+in Fig. 4, the data we refer here is the cluster power and
+intersections with scene. In total, 2024 samples were generated
+and combined to construct the dataset. After that, the input data
+are processed by down-sampling by certain SR scale factors.
+III. METHODOLOGY
+A. Problem Definition
+Super resolution means generate high resolution (HR) data
+ˆIHR from low resolution (LR) data ILR. The objective of SR
+is to minimize the gap between ˆIHR and ground truth IHR
+while obtaining the best model parameters θ, which is shown
+in (3) and (4) respectively.
+ˆIHR = F(ILR, θ)
+(3)
+θ = arg min
+θ
+L(IHR, ˆIHR)
+(4)
+As shown in Fig. 5(a), given a certain Rx track and SR scale
+factor δ, M, N denote the known LR snapshots and unknown
+Fig. 5. SR problem and clustering result.
+HR snapshots of receiver. δ−1 snapshots between two adjacent
+M are to be predicted. There could be dozens of clusters for
+a single snapshot, as in Fig. 5(b). The SR model intents to
+capture and predict the variation of cluster characteristics (e.g.
+the movement of cluster center present in black dotted line).
+B. Baseline model
+The baseline model is linear interpolation. Considering Rx
+track can be seen as straight line in small ranges, the linear
+interpolation method is as follows:
+fLI(Ni,j) = ||M1Ni||
+||M1M2||f(M2,j) + ||NiM2||
+||M1M2||f(M1,j)
+∀i, j ∈ I, J
+(5)
+where, I,J describe the snapshots to be predicted and cluster
+of snapshot Si(S = M, N). f represents cluster characteristics
+which can be 3D coordinate and power of cluster center, as
+in (1) and (2).
+C. Residual Network based Multi-layer Learning Model
+Deep neural networks have achieved great success and
+high-quality reconstruction for image super-resolution. So the
+network needs to be designed very deep for a better mapping
+and inference between LR and HR data. As illustrated in
+Fig. 6, the multi-layer learning model is composed of three
+ensemble linear blocks (ELB). Six hidden layers with suddenly
+increasing and gradually declining dimension changes are
+designed in each ELB, thus iterative up-and-down change in
+feature dimension is forming to filter irrelevant information in
+input data. However, the vanishing gradients issue will become
+more apparent as the model deepens. Two techniques are
+utilized to eliminate this problem. First, residual connection,
+which has exhibited superior performance in computer vision
+problems, is added between each ELB to ease training process
+and accelerate convergence. Second, a pre-upsampling opera-
+tion using baseline method is implemented, which outperforms
+transposed convolution layer demonstrated by previous experi-
+ment. Specifically, let E denote the transform in ELB, the final
+cluster characteristics fMLL could be written as in (6). The
+ELB number is set to be 3 for a balance of performance and
+training complexity. In addition, proposed model framework
+
+Cluster
+Clustering
+Filtering
+Tracking
+Segmenting
+data set
+Cluster 1
+Clustercenter
+Cluster 2
+Extract cluster
+Tracking
+characteristics
+Pow
+tion
+Rays
+locati
+Snapshot
+Tx
+Tx
+Filter low-power
+MiNiN2NiM2
+Rx
+Rx
+cluster
+Sample
+SegmentingI:snapshotstobepredicted
+J:clusters of snapshot S
+I'x
+Rx
+M1
+N,
+N2
+Ni
+M2
+Rx
+SR scale
+(a) Illustration of SR problem
+(b) Cluster in a snapshoFig. 6.
+The overview of proposed residual network based multi-layer deep
+learning model.
+can be regarded as a general channel characteristics generation
+architecture that also performs well in our previous work [10].
+fMLL(Ni,j) = E3(fLI(Ni,j)) + E2(fLI(Ni,j)) + E(fLI(Ni,j))
+(6)
+D. Loss Functions and Evaluation Metrics
+For SR task, only predicted snapshots (Ni) need to be
+evaluated for characteristics differences so the loss function
+LMLL in training could be written as follows:
+LMLL(ˆIHR, IHR) =
+I
+�
+i=0
+J
+�
+j=0
+C,P
+�
+f
+( ˙fMLL(Ni,j) − ˙f(Ni,j))2
+(7)
+According to previous experiment, L2 loss reduces char-
+acteristics error to lower level compared with L1 Loss. The
+prediction errors for different characteristics of each cluster
+in each snapshot will be added up successively for back
+propagation and parameter update. Instead of birth and death
+prediction, we intend to train the model to better understand
+the evolution of clusters and correlation between snapshots.
+However, cluster birth and death occasionally arise among
+consecutive snapshots. Therefore, f is transformed to ˙f by
+multiplying the weighted matrix with value 10−2 for these
+inconsecutive clusters and 1.0 for normal clusters to achieve
+better training and evaluation. Weight values are evaluated
+from 10−1 to 10−5, and 10−2 is the optimal. Absolute mean
+error (AME), mean absolute error (MAE) and root mean
+square error (RMSE) are basic evaluation metrics in this work.
+IV. EXPERIMENT AND EVALUATION
+A. Training and Implementation Details
+In this study, training, validation and test experiments are
+conducted by PyTorch 1.9.0 on a core server with 1 NVIDIA
+RTX 3090 GPU, Intel Core i9-9900K CPU and 32 GB DDR4
+RAM. To be noted, we elaborately divide the overall data as
+training, validation and test parts. The ratio of training set to
+validation set plus test set is about 5:1. Specifically, simulation
+results in dense urban scenarios route 1 ∼ 4 are divided into
+training set and validation set. Results in route 5 ∼ 7 are test
+set for generalization test. The model is trained for 80 epochs
+TABLE II
+SUPER RESOLUTION PERFORMANCE OF BASELINE AND PROPOSED MODEL
+Absolute mean error (AME)
+LOS
+NLOS
+scale
+method
+AME of
+power
+[dB]
+AME of
+location
+[m]
+AME of
+power
+[dB]
+AME of
+location
+[m]
+2
+Baseline
+1.57
+2.09
+0.73
+1.52
+Proposed
+0.80
+0.12
+0.87
+0.44
+4
+Baseline
+1.66
+2.18
+0.81
+1.65
+Proposed
+0.14
+0.25
+0.16
+0.45
+8
+Baseline
+1.90
+2.49
+0.87
+1.80
+Proposed
+0.56
+0.31
+0.60
+0.30
+16
+Baseline
+2.13
+2.74
+0.96
+1.84
+Proposed
+0.08
+0.93
+0.22
+0.82
+Root mean squared error (RMSE)
+LOS
+NLOS
+scale
+method
+RMSE of
+power
+[dB]
+RMSE of
+location
+[m]
+RMSE of
+power
+[dB]
+RMSE of
+location
+[m]
+2
+Baseline
+9.71
+13.20
+6.99
+8.83
+Proposed
+4.81
+2.33
+4.63
+2.06
+4
+Baseline
+10.32
+13.25
+7.63
+8.24
+Proposed
+5.04
+2.56
+4.89
+2.19
+8
+Baseline
+11.08
+13.63
+7.85
+7.52
+Proposed
+5.37
+2.92
+5.23
+2.25
+16
+Baseline
+11.79
+14.03
+8.26
+7.31
+Proposed
+5.70
+3.82
+5.51
+4.07
+before validation and test. The learning rate is set as 10−5.
+Adam optimizer is used for gradient descent. Experiments
+were carried out at SR scale factor 2, 4, 8 and 16.
+B. Performance of Proposed Model
+The best prediction results achieved by proposed model are
+illustrated in TABLE II, which exhibits the AME and RMSE
+of cluster power and location of cluster center. The training
+and validation of model are conducted in dense urban LOS
+and NLOS scenarios respectively. Prediction error is generally
+larger in LOS due to large quantity and severe variation of
+cluster. The evaluation metrics, AME and RMSE, are far
+smaller than baseline model, with error drops by 49∼94%
+in LOS scene. It is noted worthy that the power prediction
+performance of proposed model deteriorates slightly in less
+harsh NLOS environment. We will further investigate this part
+in future research.
+C. Channel Impulse Response Reconstruction
+To better evaluate the SR performance for cluster character-
+istics, we regenerate the CIR based on predicted 3D positions
+and power of clusters. The simulated CIR indicated by the red
+asterisk is generated directly by RT. As can be seen in Fig.
+7, restored CIR could match most MPCs at different scales.
+Different from interpolation method, MLL model generates
+precise cluster characteristics that could restore CIR consistent
+with simulated at larger SR scales.
+D. Ablation Study and Generalization Test
+Ablation study was implemented to investigate the effective-
+ness of specific parts and designs in proposed model. Multiple
+
+Ensemblelayerblock
+Leaky
+Pre-up
+ReLU
+sampling
+17
+Linear
+Residual
+512
+256
+128
+64
+32
+layer
+connection
+power&
+location
+snapshot
+Linear
+Linear
++
+Linear
++
+Block
+Block
+Block
+batchsizeFig. 7. Restored CIR at different scales.
+TABLE III
+CUMULATIVE SUPER RESOLUTION ERROR DECLINE OF CLUSTER POWER
+ELB
+Max hidden
+dimension
+32
+64
+128
+256
+512 (ours)
+1024
+MAE
+0
++1.9%
+-9.1%
+-20.9%
+-26.4%
+-24.3%
+RES
+Residual connection
+w/o
+w
+MAE
+0
+-11.9%
+TABLE IV
+SUPER RESOLUTION PERFORMANCE (RMSE) IN GENERALIZATION TEST
+Jianting viaduct
+Xinxi street
+Malianwa road
+scale
+RMSE of
+power
+[dB]
+RMSE of
+location
+[m]
+RMSE of
+power
+[dB]
+RMSE of
+location
+[m]
+RMSE of
+power
+[dB]
+RMSE of
+location
+[m]
+2
+3.76
+2.63
+3.49
+2.70
+5.40
+5.59
+4
+3.97
+2.82
+3.94
+2.53
+6.19
+5.41
+8
+4.29
+3.94
+3.88
+3.94
+5.93
+5.30
+linear layers with different hidden dimensions are integrated
+in ELB. By gradually increasing the layer number and its
+hidden dimension, the model could extract and learn a deeper
+variation of clusters. As demonstrated in TABLE III, the max
+hidden dimension in ELB is set to 512, which obtains the
+best performance. Residual connection is also indispensable
+to speed up convergence process and reduce errors, achieving
+more than 10% performance improvement. Generalization test
+is also implemented at Jianting viaduct, Xinxi street and
+Malianwa road, as in TABLE IV. Compared with the LOS
+results in dense urban scenarios, result is better in first two
+scenes and worse in Malianwa road. Without apparent model
+overfitting, it can be applied to other scenarios for channel
+modeling.
+V. CONCLUSION
+In this paper, an efficient SR approach for cluster charac-
+teristics based on ray tracing and deep learning is proposed.
+Object-based clustering method is conducted to generate clus-
+ter characteristics. A multi-layer deep learning model is then
+proposed for cluster characteristic prediction. Based on LR
+data, MLL achieves fairly good performance both in LOS
+and NLOS area. Best result for RMSE of cluster power and
+location reduces to 3.49 dB and 2.06 m. The generalization
+experiments demonstrate that proposed model could be used
+to other scenarios without a significant drop in performance.
+Ablation study is also implemented to verify the important role
+of each module in proposed model. Additionally, CIRs are
+regenerated utilizing the predicted cluster center and power,
+accurately matching MPC at different scales. In the future, we
+will continue to study and analyze the channel characteristics
+super-resolution issue in depth, looking forward to finding
+better rules and strategies to achieve higher-quality and real-
+time CIR reconstruction for mobile channel modeling.
+ACKNOWLEDGMENT
+This work is supported by National Key R&D Program
+of China under Grant 2020YFB1806604, NSFC under Grant
+62271043, the Ministry of Education of China under Grant
+8091B032123, ZTE Corporation and the State Key Laboratory
+of Mobile Network and Mobile Multimedia Technology.
+REFERENCES
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+impact analysis and regional forecast,” [Online], Available:https://www.
+fortunebusinessinsights.com/internet-of-vehicles-market-105345.html.
+[2] J. Tan, X. Sha, B. Dai, and T. Lu, “Analysis of industrial internet of
+things and digital twins,” ZTE Communications, vol. 19, no. 2, pp. 53–
+60, 2021.
+[3] X. Cheng, D. Duan, L. Yang, and N. Zheng, “Cooperative intelligence
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+50, 2019.
+[4] T. Qingtao, M. Ziang, G. Ke, L. Dan, and X. Huan, “Research on
+wireless signal coverage in urban tunnels based on high-performance ray
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+[5] Z. Youping and G. Jiaqi, “An improved sbr ray-tracing channel simu-
+lation method,” Journal of Beijing Jiaotong University, vol. 45, no. 5,
+2021.
+[6] X. Lin, B. Ai, D. He, K. Guan, and Z. Zhong, “Calibration of ray tracer
+and channel analysis for high-speed railway viaduct scenario,” Chinese
+journal of radio science, vol. 33, no. 1, pp. 41–47, 2018.
+[7] L. Zhu, D. He, B. Ai, K. Guan, S. Dang, J. Kim, H. Chung, and
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+munications and Information Networks, vol. 5, no. 3, pp. 265–281, 2020.
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+loss prediction based on multi-layer perceptron artificial neural network,”
+Chinese journal of radio science, vol. 36, no. 3, pp. 396–404, 2021.
+[12] L. Bai, C.-X. Wang, J. Huang, Q. Xu, Y. Yang, G. Goussetis, J. Sun, and
+W. Zhang, “Predicting wireless mmwave massive mimo channel charac-
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+and Mobile Computing, vol. 2018, 2018.
+[13] D. He, B. Ai, K. Guan, L. Wang, Z. Zhong, and T. K¨urner, “The
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+platform for 5G and beyond wireless communications: A tutorial,” IEEE
+Communications Surveys & Tutorials, vol. 21, no. 1, pp. 10–27, 2018.
+
+-60
+-60
+Restored results
+Restored results
+*RT results
+*RT results
+-80
+-80
+[dBm]
+dBm]
+-100
+-100
+2
+2
+h
+h
+-120
+-120
+-140
+-140
+200
+400
+600
+200
+400
+0
+0
+600
+Delay [ns]
+Delay [ns]
+(a) Restored CIR based on
+(b) Restored CIR based on
+predicted cluster at scale 2
+predicted cluster at scale 4
+-60
+-60
+Restored results
+Restored results
+* RT results
+* RT results
+-80
+-80
+dBm]
+[dBm]
+-100
+,-100
+米
+2
+2
+h
+h
+-120
+-120
+-140
+-140
+0
+200
+400
+600
+0
+200
+400
+600
+Delay [ns]
+Delay [ns]
+(c) Restored CIR based on
+(d) Restored CIR based on
+predicted cluster at scale 8
+predicted cluster at scale 16
\ No newline at end of file
diff --git a/v9FJT4oBgHgl3EQffixw/content/tmp_files/load_file.txt b/v9FJT4oBgHgl3EQffixw/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='A Ray-tracing and Deep Learning Fusion  Super-resolution Modeling Method for Wireless  Mobile Channel  Zhao Zhang∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
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+page_content=' Xiping Wang∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Ke Guan∗‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Zhangdui Zhong∗§,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Jianwu Dou¶∥  ∗ State Key Laboratory of Rail Traffic Control and Safety,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Beijing Jiaotong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 100044 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' China  † Beijing Engineering Research Center of High-speed Railway Broadband Mobile Communications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 100044 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' China  ‡ Frontiers Science Center for Smart High-speed Railway System,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 100044 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' China  § Key Laboratory of Railway Industry of Broadband Mobile Information Communications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 100044 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' China  ¶ State Key Laboratory of Mobile Network and Mobile Multimedia Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 518055 Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Guangdong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' China  ∥ ZTE Corporation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 518055 Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Guangdong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' China  Corresponding author: Danping He,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' E-mail: hedanping@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='cn  Abstract—Mobile channel modeling has always been the core  part for design, deployment and optimization of communication  system, especially in 5G and beyond era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Deterministic channel  modeling could precisely achieve mobile channel description,  however with defects of equipment and time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' In this  paper, we proposed a novel super resolution (SR) model for  cluster characteristics prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The model is based on deep  neural networks with residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' A series of simulations  at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='5 GHz are conducted by a three-dimensional ray trac-  ing (RT) simulator in diverse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Cluster characteristics  are extracted and corresponding data sets are constructed to  train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Experiments demonstrate that the proposed  SR approach could achieve better power and cluster location  prediction performance than traditional interpolation method  and the root mean square error (RMSE) drops by 51% and  78% relatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Channel impulse response (CIR) is reconstructed  based on cluster characteristics, which could match well with the  multi-path component (MPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The proposed method can be used  to efficiently and accurately generate big data of mobile channel,  which significantly reduces the computation time of RT-only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Index Terms—ray tracing, deep learning, super resolution,  mobile channel modeling, cluster prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' INTRODUCTION  The global internet of vehicles market is projected to grow  from $ 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='62 billion in 2021 to $ 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='60 billion in 2028  in forecast period [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' New communication services such  as intra-vehicle, Vehicle-to-Everything (V2X) communication  and Industrial Internet of Things (IIoT) have put forward more  stringent requirements for mobile communication systems [2]  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' It has been the consensus that the 5G and B5G will realize  a high data-rate and ultra-reliable low-latency communication  albeit with soaring demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' However, current 5G could hardly  provide a guarantee in harsh electromagnetic and time-varying  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' To achieve B5G superior features for mobile  communication in diverse scenarios, mobile channel modeling  is imperatively needed to better understand channel character-  istics, plan and optimize communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Channel modeling is the process of modeling the sig-  nal propagation mechanism to obtain an accurate channel  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Generally, geometry-based stochastic modeling  (GBSM) and ray-tracing (RT) based deterministic modeling  are two main modeling approaches for mobile channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' GBSM  can theoretically generate channel impulse responses (CIRs)  through assumed scattering geometry to analyze performance,  but practically the CIRs are evaluated numerically via mea-  surement and calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' On the other hand, RT can generate  highly accurate channel characteristics for specific scenario  with the defect of computational complexity [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Thereby,  a fast and reliable channel characteristic generation method  based on coarse-grained RT is desperately needed to accelerate  the modeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Mobile channel modeling not only refers to time-varying  and non-stationary channels but also involves the influence of  variant multi-path components and small-scale fading, which  is more dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' In this regard, cluster-based approaches are  increasingly adopted in recent research especially since GBSM  was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' In [7], a clustering and tracking algorithm was  proposed and analyzed for vehicle-to-infrastructure channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  To recognize and track the clusters in time-varying channels,  a clustering and tracking algorithm based on power-angle-  spectrum is proposed and investigated [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' In [9], a semi-  deterministic channel modeling method is presented based on  RT and cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Many classical radio channel models are also  based on the concept of cluster, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' COST 2100, 3GPP Spatial  Channel Model and WINNER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Deep learning (DL) based channel modeling methods are  getting popular in recent years because of its excellent infor-  mation integration and inferring ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Numerous research  including our previous work [10] are dedicated on large scale  channel characteristics prediction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' pathloss, delay spread  and number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Multi-layer perceptron artificial neural  network is presented for path loss prediction in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The  author in [12] proposes a procedure of predicting channel  characteristics based on convolutional neural network (CNN)  for multi-dimensional millimeter wave channel characteristics  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' However, small scale channel characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='11557v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='SP]  27 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Framework of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  power, angle and location of each cluster) related research  based on DL are inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  In this paper, we propose a deep learning and ray tracing  fusion super-resolution (SR) method for cluster characteristics,  the power and location of each cluster more specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Overview of our work is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' We first use ray-  tracing simulator to generate ray level parameters, from which  cluster level parameters are extracted utilizing the proposed  clustering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Subsequently, high resolution data and low  resolution data are divided to train the deep learning SR  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Evaluation and comparison are implemented in cluster  prediction and CIR reconstruction aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Specifically, we  make the following contributions:  Massive RT simulation by self develop CloudRT [13]  was conducted in four restored scenarios via SketchUp  and 3D electronic map and multi-dimensional ray level  characteristics data are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  A novel object-based clustering method is presented  for better rays division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Special filtering, tracking and  segmenting method are utilized to extract crucial char-  acteristics, unify cluster dataset and maintain continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Multi-layer deep learning model (MLL) is proposed to  predict cluster characteristics more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Residual  connection and pre-upsampling techniques are integrated  for eliminating the vanishing gradients problem and  achieving better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Ablation study and gen-  eralization test demonstrated the necessity of adopted  crucial techniques and adaptability of proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Section II introduces the simulation configuration and data  construction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The proposed SR and baseline model  are explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Experiments and evaluations are  implemented in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Finally, conclusions are drawn in  Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' SIMULATION AND DATA CONSTRUCTION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Scenario Modeling  The overall simulation is based on two kinds of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 3, a dense urban scenario is  based on 3D electronic map and three street scenarios are  manually modeled via SketchUp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Note that all four scenarios  are completely sourced from real environment, which are Cen-  tral business district, Malianwa street and Jianting viaduct and  Xinxi road in Beijing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The dense urban scenario  mainly consists of different types of buildings and terrain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  regular buildings, parallel buildings, dry land, green land).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Simulations in dense urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Simulations in street scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  TABLE I  SIMULATION CONFIGURATION  Parameter  Value  Carrier frequency [GHz]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='55  System bandwidth [MHz]  100  Tx transmit power [dBm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='1  Tx location  5-10 m above ground  Rx location  2 m above ground  Tx & Rx attenna  Omni-directional vertical polarization  street scenario also contains many fine scatterers, covering  from cars to pedestrians and trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' As depicted, the transmitter  (Tx) and route of receivers (Rx) are presented clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' A total  of 7 Rx routes in different color are simulated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  It is noteworthy that Tx location is elaborately sited twice  so that both LOS and NLOS scenes are simulated for route  1 ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Only LOS scene is simulated for route 5 ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Through  simulation in various and close-to-real scenarios, the diversity  of data is guaranteed in the case of limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Simulation Settings  Based on aforementioned scenario and electromagnetic  (EM) parameters provided by ITU-R P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='1238-7, Self developed  CloudRT platform is utilized to obtain channel characteristics  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Rx is located 2 meters above ground along the road at  1 m intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Tx is placed on the side of road, near the end  of receiver’s trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The propagation mechanisms includes  light of sight, scattering, reflection, penetration and diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  The detailed simulation configuration is shown in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Clustering  For a single Tx-Rx link, hundreds to thousands of rays can  be traced, most of which share the similar delay, angle and  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Therefore, it is an accepted practice to equate similar  rays as clusters and study channel characteristics on cluster  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The overall data process workflow is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Clustering is the process of cluster data generation based  on rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' There are enormous approaches for clustering (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  power-angle-spectrum based clustering, power-delay profile  Data set construction  Training  Evaluation  Ray-tracing  HR & LR  Cluster  SR model  simulator  data  prediction  Ray level  Cluster level  Baseline  CIR  model  reconstruction  parameters  parametersRoute 1~4  (a) 3D view of dense urban scenario  (b) Aerial View of Rx routeRoute 5  Route 6  Route 7  (a) Xinxi street  (b) Malianwa road  (c) Jianting viaductFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Workflow of data process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  based clustering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Based on RT, the propagation of each ray  can be accurately acquired, including 3D coordinates, object  and microfacet identity of reflection and scatter points with  scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Microfacet-based and object-based clustering method  are hence considered and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' In this work, rays that  hit on the same object are grouped into a cluster, after which  cluster characteristics including power and center locations  are extracted for each snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The author in [8] proposed  an algorithm for identifying and tracking multipath clusters,  which also introduces the conception of power-weighted clus-  ter centers, intra-cluster angle expansion and cluster shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Similarly, we adopt the power-weighted cluster center and  coherent superposition cluster power approach for filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  The equation is as follows:  C(j) =  �K  k=1 C(rk)P(rk)  �K  k=1 P(rk)  , rk ∈ j  (1)  P(j) =  K  �  k=1  P(rk), rk ∈ j  (2)  C, P denotes the 3D coordinate of reflection or scatter  point and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' j, rk denotes jth cluster and kth ray in  this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Due to different cluster between snapshots, cluster  tracking is implemented between snapshots for continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Every 17 adjacent snapshots are then segmented into a sample,  exactly enough to implement SR test at scale 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Within each  sample, the number of clusters is compromised to a certain  value to form a regular data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Data structure is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 4, the data we refer here is the cluster power and  intersections with scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' In total, 2024 samples were generated  and combined to construct the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' After that, the input data  are processed by down-sampling by certain SR scale factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Problem Definition  Super resolution means generate high resolution (HR) data  ˆIHR from low resolution (LR) data ILR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The objective of SR  is to minimize the gap between ˆIHR and ground truth IHR  while obtaining the best model parameters θ, which is shown  in (3) and (4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  ˆIHR = F(ILR, θ)  (3)  θ = arg min  θ  L(IHR, ˆIHR)  (4)  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 5(a), given a certain Rx track and SR scale  factor δ, M, N denote the known LR snapshots and unknown  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' SR problem and clustering result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  HR snapshots of receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' δ−1 snapshots between two adjacent  M are to be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' There could be dozens of clusters for  a single snapshot, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The SR model intents to  capture and predict the variation of cluster characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  the movement of cluster center present in black dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Baseline model  The baseline model is linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Considering Rx  track can be seen as straight line in small ranges, the linear  interpolation method is as follows:  fLI(Ni,j) = ||M1Ni||  ||M1M2||f(M2,j) + ||NiM2||  ||M1M2||f(M1,j)  ∀i, j ∈ I, J  (5)  where, I,J describe the snapshots to be predicted and cluster  of snapshot Si(S = M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' f represents cluster characteristics  which can be 3D coordinate and power of cluster center, as  in (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Residual Network based Multi-layer Learning Model  Deep neural networks have achieved great success and  high-quality reconstruction for image super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' So the  network needs to be designed very deep for a better mapping  and inference between LR and HR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' As illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 6, the multi-layer learning model is composed of three  ensemble linear blocks (ELB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Six hidden layers with suddenly  increasing and gradually declining dimension changes are  designed in each ELB, thus iterative up-and-down change in  feature dimension is forming to filter irrelevant information in  input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' However, the vanishing gradients issue will become  more apparent as the model deepens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Two techniques are  utilized to eliminate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' First, residual connection,  which has exhibited superior performance in computer vision  problems, is added between each ELB to ease training process  and accelerate convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Second, a pre-upsampling opera-  tion using baseline method is implemented, which outperforms  transposed convolution layer demonstrated by previous experi-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Specifically, let E denote the transform in ELB, the final  cluster characteristics fMLL could be written as in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The  ELB number is set to be 3 for a balance of performance and  training complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=" In addition, proposed model framework  Cluster  Clustering  Filtering  Tracking  Segmenting  data set  Cluster 1  Clustercenter  Cluster 2  Extract cluster  Tracking  characteristics  Pow  tion  Rays  locati  Snapshot  Tx  Tx  Filter low-power  MiNiN2NiM2  Rx  Rx  cluster  Sample  SegmentingI:snapshotstobepredicted  J:clusters of snapshot S  I'x  Rx  M1  N,  N2  Ni  M2  Rx  SR scale  (a) Illustration of SR problem  (b) Cluster in a snapshoFig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  The overview of proposed residual network based multi-layer deep  learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  can be regarded as a general channel characteristics generation  architecture that also performs well in our previous work [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  fMLL(Ni,j) = E3(fLI(Ni,j)) + E2(fLI(Ni,j)) + E(fLI(Ni,j))  (6)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Loss Functions and Evaluation Metrics  For SR task, only predicted snapshots (Ni) need to be  evaluated for characteristics differences so the loss function  LMLL in training could be written as follows:  LMLL(ˆIHR, IHR) =  I  �  i=0  J  �  j=0  C,P  �  f  ( ˙fMLL(Ni,j) − ˙f(Ni,j))2  (7)  According to previous experiment, L2 loss reduces char-  acteristics error to lower level compared with L1 Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The  prediction errors for different characteristics of each cluster  in each snapshot will be added up successively for back  propagation and parameter update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Instead of birth and death  prediction, we intend to train the model to better understand  the evolution of clusters and correlation between snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  However, cluster birth and death occasionally arise among  consecutive snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Therefore, f is transformed to ˙f by  multiplying the weighted matrix with value 10−2 for these  inconsecutive clusters and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='0 for normal clusters to achieve  better training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Weight values are evaluated  from 10−1 to 10−5, and 10−2 is the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Absolute mean  error (AME), mean absolute error (MAE) and root mean  square error (RMSE) are basic evaluation metrics in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' EXPERIMENT AND EVALUATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Training and Implementation Details  In this study, training, validation and test experiments are  conducted by PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='0 on a core server with 1 NVIDIA  RTX 3090 GPU, Intel Core i9-9900K CPU and 32 GB DDR4  RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' To be noted, we elaborately divide the overall data as  training, validation and test parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The ratio of training set to  validation set plus test set is about 5:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Specifically, simulation  results in dense urban scenarios route 1 ∼ 4 are divided into  training set and validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Results in route 5 ∼ 7 are test  set for generalization test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The model is trained for 80 epochs  TABLE II  SUPER RESOLUTION PERFORMANCE OF BASELINE AND PROPOSED MODEL  Absolute mean error (AME)  LOS  NLOS  scale  method  AME of  power  [dB]  AME of  location  [m]  AME of  power  [dB]  AME of  location  [m]  2  Baseline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='57  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='73  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='52  Proposed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='44  4  Baseline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='66  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='65  Proposed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='45  8  Baseline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='90  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='87  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='80  Proposed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='30  16  Baseline  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
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+page_content='84  Proposed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='82  Root mean squared error (RMSE)  LOS  NLOS  scale  method  RMSE of  power  [dB]  RMSE of  location  [m]  RMSE of  power  [dB]  RMSE of  location  [m]  2  Baseline  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='71  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='20  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='99  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='83  Proposed  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='81  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='33  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='63  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='06  4  Baseline  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='32  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='25  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='63  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='24  Proposed  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='56  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='89  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='19  8  Baseline  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='08  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='63  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='85  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='52  Proposed  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='37  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='92  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='23  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='25  16  Baseline  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='79  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='03  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='26  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='31  Proposed  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='70  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='82  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='51  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='07  before validation and test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The learning rate is set as 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Adam optimizer is used for gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Experiments  were carried out at SR scale factor 2, 4, 8 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Performance of Proposed Model  The best prediction results achieved by proposed model are  illustrated in TABLE II, which exhibits the AME and RMSE  of cluster power and location of cluster center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The training  and validation of model are conducted in dense urban LOS  and NLOS scenarios respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Prediction error is generally  larger in LOS due to large quantity and severe variation of  cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The evaluation metrics, AME and RMSE, are far  smaller than baseline model, with error drops by 49∼94%  in LOS scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' It is noted worthy that the power prediction  performance of proposed model deteriorates slightly in less  harsh NLOS environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' We will further investigate this part  in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Channel Impulse Response Reconstruction  To better evaluate the SR performance for cluster character-  istics, we regenerate the CIR based on predicted 3D positions  and power of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The simulated CIR indicated by the red  asterisk is generated directly by RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  7, restored CIR could match most MPCs at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Different from interpolation method, MLL model generates  precise cluster characteristics that could restore CIR consistent  with simulated at larger SR scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Ablation Study and Generalization Test  Ablation study was implemented to investigate the effective-  ness of specific parts and designs in proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Multiple  Ensemblelayerblock  Leaky  Pre-up  ReLU  sampling  17  Linear  Residual  512  256  128  64  32  layer  connection  power&  location  snapshot  Linear  Linear  +  Linear  +  Block  Block  Block  batchsizeFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Restored CIR at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  TABLE III  CUMULATIVE SUPER RESOLUTION ERROR DECLINE OF CLUSTER POWER  ELB  Max hidden  dimension  32  64  128  256  512 (ours)  1024  MAE  0  +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='9%  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='1%  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='9%  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='4%  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='3%  RES  Residual connection  w/o  w  MAE  0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='9%  TABLE IV  SUPER RESOLUTION PERFORMANCE (RMSE) IN GENERALIZATION TEST  Jianting viaduct  Xinxi street  Malianwa road  scale  RMSE of  power  [dB]  RMSE of  location  [m]  RMSE of  power  [dB]  RMSE of  location  [m]  RMSE of  power  [dB]  RMSE of  location  [m]  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='76  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='63  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='49  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='70  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='40  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='59  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='97  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='82  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='94  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='53  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='41  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='29  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='94  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='88  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='94  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='93  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='30  linear layers with different hidden dimensions are integrated  in ELB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' By gradually increasing the layer number and its  hidden dimension, the model could extract and learn a deeper  variation of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' As demonstrated in TABLE III, the max  hidden dimension in ELB is set to 512, which obtains the  best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Residual connection is also indispensable  to speed up convergence process and reduce errors, achieving  more than 10% performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Generalization test  is also implemented at Jianting viaduct, Xinxi street and  Malianwa road, as in TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Compared with the LOS  results in dense urban scenarios, result is better in first two  scenes and worse in Malianwa road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Without apparent model  overfitting, it can be applied to other scenarios for channel  modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' CONCLUSION  In this paper, an efficient SR approach for cluster charac-  teristics based on ray tracing and deep learning is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Object-based clustering method is conducted to generate clus-  ter characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' A multi-layer deep learning model is then  proposed for cluster characteristic prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Based on LR  data, MLL achieves fairly good performance both in LOS  and NLOS area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Best result for RMSE of cluster power and  location reduces to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='49 dB and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='06 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' The generalization  experiments demonstrate that proposed model could be used  to other scenarios without a significant drop in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  Ablation study is also implemented to verify the important role  of each module in proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' Additionally, CIRs are  regenerated utilizing the predicted cluster center and power,  accurately matching MPC at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content=' In the future, we  will continue to study and analyze the channel characteristics  super-resolution issue in depth, looking forward to finding  better rules and strategies to achieve higher-quality and real-  time CIR reconstruction for mobile channel modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This work is supported by National Key R&D Program  of China under Grant 2020YFB1806604, NSFC under Grant  62271043, the Ministry of Education of China under Grant  8091B032123, ZTE Corporation and the State Key Laboratory  of Mobile Network and Mobile Multimedia Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='  REFERENCES  [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
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+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
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+page_content='  60  60  Restored results  Restored results  RT results  RT results  80  80  [dBm]  dBm]  100  100  2  2  h  h  120  120  140  140  200  400  600  200  400  0  0  600  Delay [ns]  Delay [ns]  (a) Restored CIR based on  (b) Restored CIR based on  predicted cluster at scale 2  predicted cluster at scale 4  60  60  Restored results  Restored results  RT results  RT results  80  80  dBm]  [dBm]  100  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
+page_content='-100  米  2  2  h  h  120  120  140  140  0  200  400  600  0  200  400  600  Delay [ns]  Delay [ns]  (c) Restored CIR based on  (d) Restored CIR based on  predicted cluster at scale 8  predicted cluster at scale 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf'}
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@@ -0,0 +1,868 @@
+> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+1
+ 
+Abstract—This letter proposes an improved CNN predictor 
+(ICNNP) for reversible data hiding (RDH) in images, which 
+consists of a feature extraction module, a pixel prediction module, 
+and a complexity prediction module. Due to predicting the 
+complexity of each pixel with the ICNNP during the embedding 
+process, the proposed method can achieve superior performance 
+than the CNN predictor-based method. Specifically, an input 
+image does be first split into two different sub-images, i.e., the “Dot” 
+image and the “Cross” image. Meanwhile, each sub-image is 
+applied to predict another one. Then, the prediction errors of 
+pixels are sorted with the predicted pixel complexities. In light of 
+this, some sorted prediction errors with less complexity are 
+selected to be efficiently used for low-distortion data embedding 
+with a traditional histogram shift scheme. Experimental results 
+demonstrate that the proposed method can achieve better 
+embedding performance than that of the CNN predictor with the 
+same histogram shifting strategy. 
+ 
+Index Terms—Convolutional neural network, multitasking, 
+reversible data hiding, histogram shifting 
+ 
+I. INTRODUCTION 
+EVERSIBLE data hiding (RDH) can losslessly recover 
+both the embedded data and the cover medium [1]. Due to 
+the trait, RDH has gradually become a hot research field in the 
+information hiding community and has been widely used in 
+several realistic scenarios [1], including medical, military, and 
+law forensics et. al. According to the domain hiding a secret 
+message, RDH can be categorized as two main branches: spatial 
+domain-based RDH [2-23] and JPEG domain-based RDH [24-
+27]. The spatial domain-based RDH generally exploits three 
+technologies, i.e., lossless compression (LC) [2-4], difference 
+expansion (DE) [5-15], and histogram shifting (HS) [16-23]. 
+While the JPEG domain-based RDH is mainly based on DCT 
+ 
+This work was supported in part by the National Key R&D Program of 
+China (Grant 2021YFE0205400), the Natural Science Foundation of Xiamen, 
+China (Grant 3502Z20227192), and the Natural Science Foundation of China 
+(Grant U20B2051, 61972168, 62072114, 62002124, 61871434). Corresponding 
+author: Zhenxing Qian. 
+Y. Qiu, X. Lin, and H. Zeng are with the College of Information Science & 
+Engineering, Huaqiao 
+University, Xiamen 361021, China. (e-mail: 
+yqqiu@hqu.edu.cn, echo.linxd@gmail.com, zeng0043@hqu.edu.cn). 
+W. Peng and Z. Qian are with the School of Computer Science, Fudan 
+University, Shanghai 200433, China. (e-mail: pengwanli@fudan.edu.cn, 
+zxqian@fudan.edu.cn). 
+Y. Qiu and W. Peng contribute equally to this work. 
+ 
+ 
+coefficients modification [24, 25] or Huffman table 
+modification [26, 27]. 
+Currently, in the RDH community, pixel prediction has 
+become a critical problem, which dramatically affects the 
+performance of RDH algorithms [14]. The traditional predictors 
+include the median edge direction (MED) predictor [6], 
+interpolation predictor [7], gradient-adjusted predictor (GAP) 
+[8], pixel-value-ordering (PVO) predictor [9, 12, 22], linear 
+predictor [10], rhombus predictor [17-20], and ridge regression 
+predictor [23], etc. Although these predictors have achieved 
+supervising improvement, there is still a notable weak point, 
+that is few neighboring pixels are used for pixel prediction [14]. 
+If more adjacent pixels are served as reference pixels, higher 
+prediction performance can be achieved. Due to its strong 
+capabilities of different receptive fields fusion and whole 
+optimization, a convolutional neural network (CNN) can be 
+established and trained to predict pixels accurately by building 
+a non-linear mapping for pixel prediction. In light of this, Luo 
+et al. [13] presented a CNN-based stereo image RDH method 
+by leveraging the correlations between the left view and the 
+right view. Hu et al. [14] proposed a CNN predictor (CNNP) 
+based RDH method, where a grayscale image was split into two 
+sub-images, and each one is predicted with another one 
+alternatively by using the CNNP. After that, Hu et al. [15] 
+divided an image into four parts, and each part was predicted 
+with the other three parts in turn by using a CNNP for a better 
+prediction performance. In addition, a better visual quality of the 
+marked image is achieved through adaptive embedding. Overall, 
+the prediction performance of CNN predictors can be better 
+than that of the traditional predictors.  
+From the above discussion, in order to improve performance, 
+the existing methods conduct pixel prediction by leveraging 
+adjacent pixels. While these methods [13-15] ignore the 
+complexity of each pixel with deep learning, which limits the 
+performance of RDH. 
+To tackle the above limitation, in this letter, we improve the 
+CNNP presented in [14] by adding a complexity prediction part 
+to predict the pixel’s complexities precisely, which is called 
+improved CNNP (ICNNP) in the rest of this letter. Specifically, 
+during data embedding, we first split a grayscale image into two 
+sub-images, where one sub-image is predicted by other one. 
+Then, we sort the prediction errors of the predicted pixels 
+according to their complexities, and the prediction errors with 
+less complexity are used for data embedding with a classical HS 
+Improved CNN Prediction Based Reversible 
+Data Hiding 
+Yingqiang Qiu, Wanli Peng, Xiaodan Lin, Huanqiang Zeng, Senior Member, IEEE, and Zhenxing 
+Qian, Member, IEEE  
+R 
+
+> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
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+strategy. Finally, experimental results show that the 
+performance of the proposed method is better than that of the 
+CNNP presented in [14]. 
+The rest of this letter is organized as follows. The proposed 
+improved RDH method is described in detail in Section II, and 
+the experimental results and discussions are provided in Section 
+III. Finally, we conclude our work in Section IV. 
+II. PROPOSED IMPROVED METHOD  
+A. Network Architecture 
+As shown in Fig. 1, according to the checkerboard context 
+model [17], the original image is split into two sub-images 
+which consist of "Dot" and "Cross" pixels, respectively. For the 
+“Dot” image, the values of the “Dot” pixels are remained, while 
+those of the “Cross” pixels are set to 0. Meanwhile, just the 
+values of the “Dot” pixels are set to 0 for the “Cross” image. 
+Based on the pixel correlation of the two sub-images, each sub-
+image is applied to predict the pixel values and complexities of 
+another sub-image.  
+The overall of the proposed ICNNP is shown in Fig. 2. The 
+architecture of the ICNNP is composed of three parts, i.e., 
+feature extraction, pixel prediction, and complexity prediction. 
+The “Cross” image I2 is fed to the network to predict the 
+values and complexities of the “Dot” pixels, where the values 
+of complexity are adjusted to [0,255] for a good visualization 
+display. The lower value, the lower complexity. The feature 
+extraction consists of some convolution layers with different 
+filter sizes (3 × 3, 5 × 5, 7 × 7, ⋯), which are parallelized and 
+appended with a 3×3 convolution layer respectively to extract 
+features from different receptive fields. A residual block is then 
+applied to further aggregate and refine the learnt features from 
+different branches. With the extracted feature, the pixel 
+prediction yields the predicted “Dot” image 𝐼�� , and the 
+complexity prediction yields the predicted complexity 𝐶�� of the 
+“Dot” image I1. “Conv” stands for the convolution unit with 
+kernel size S × S and the number of channels is output × input. 
+A LeakyReLU activation function [28] locates between each 
+two convolution layers.  
+It is worthy to note that the complexity prediction is similar 
+to the pixel prediction, i.e., instead of orthogonal adjacent pixels 
+[14, 17], more adjacent pixels are used to nonlinearly predict 
+the complexity of the pixel area, improving the performance of 
+RDH. 
+B. Training 
+In the ICNNP, the well-trained parameters of CNNP [14] are 
+loaded into the feature extraction and pixel prediction. Note that, 
+these parameters are fixed and the parameters of complexity 
+prediction are updated during the training of the ICNNP. In the 
+training, the input is the “Cross” image I2, the outputs are the 
+predicted “Dot” image 𝐼�� and the predicted complexity 𝐶�� of 
+the “Dot” image I1. Since the filter parameters of the feature 
+extraction and the pixel prediction are fixed, the target is no 
+longer the “Dot” image I1 but the referenced complexity 𝐶� of 
+I1. The definition of 𝐶� is described as follows. 
+(1) For “Cross” pixels, 𝐶��𝑖, 𝑗� is set to 0; 
+(2) For “Dot” pixels, if 𝑖 � 1 or 𝑖 � 𝑀 or 𝑗 � 1 or 𝑗 � 𝑁, 
+𝐶��𝑖, 𝑗� is set to 0; otherwise, 𝐶��𝑖, 𝑗��2 � 𝑖 � 𝑀 � 1,2 � 𝑗 �
+𝑁 � 1� is calculated as 
+𝐶��𝑖, 𝑗� �
+�
+� ∙ �∑
+∑
+�𝐼�𝑖 � 𝑘, 𝑗 � 𝑙� � 𝐼�𝑖, 𝑗���
+��
+����
+��
+����
+,     (1) 
+where,  
+�
+𝑘� � �1, 𝑘� � 2    , 𝑖 � 2                 
+𝑘� � �2, 𝑘� � 1    , 𝑖 � 𝑀 � 1        
+𝑘� � �2, 𝑘� � �2 , 2 � 𝑖 � 𝑀 � 1
+ ,               (2) 
+�
+𝑙� � �1, 𝑙� � 2    , 𝑗 � 2                 
+𝑙� � �2, 𝑙� � 1    , 𝑗 � 𝑁 � 1        
+𝑙� � �2, 𝑙� � �2 , 2 � 𝑗 � 𝑁 � 1
+ ,                (3) 
+𝑅 � �𝑘� � 𝑘� � 1� � �𝑙� � 𝑙� � 1� � 1.           (4) 
+In the proposed method, the max predicted pixel area is 5 × 5, 
+which can accurately calculate the pixel complexity.   
+As with CNNP in [14], we leveraged back-propagation [29] 
+and Adam algorithm [30] to optimize the objective function 
+defined as below: 
+Loss �
+�
+� ∑
+�𝐶�� � 𝐶��� � 𝜆‖𝜔‖�
+�
+�
+���
+,                  (5) 
+where N stands for the number of training examples, ω 
+represents all weights of the network, and λ denotes the weight 
+decay. 
+C. ICNNP	based	RDH 
+Fig. 3 depicts the data embedding architecture of the ICNNP-
+based RDH method. The adopted double embedding strategy 
+[17] with the HS technique [6] involves consecutive usage of  
+…… 
+⊕ 
+⊕ 
+Conv|3×3|32×32 
+LeakyReLU 
+Conv|3×3|1×32 
+Conv|3×3|1×32 
+Conv|3×3|32×32 
+LeakyReLU 
+Conv|3×3|32×32 
+LeakyReLU 
+Conv|3×3|32×32 
+Conv|3×3|32×1 
+LeakyReLU 
+Conv|3×3|32×32 
+Conv|5×5|32×1 
+LeakyReLU 
+Conv|3×3|32×32 
+Conv|7×7|32×1 
+LeakyReLU 
+Conv|3×3|32×32  
+ Feature extraction 
+Pixel prediction 
+Complexity prediction 
+ “Cross” image I2 
+Predicted “Dot” image 𝐼�� 
+Predicted complexity of “Dot” image 𝐶�� 
+Fig. 2. The overall architecture of the proposed ICNNP. 
+ 
+(a) Original image I      (b) “Dot” image I1      (c) “Cross” image I2 
+Fig. 1. Illustration of splitting an original image into two sub-images.  
+
+444十
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+04> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
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+the “Dot” embedding and the “Cross” embedding, and the 
+“Cross” embedding is achieved after the “Dot” embedding. 
+The original image I is firstly divided into two sub-images, 
+i.e., a “Dot” image I1 and a “Cross” image I2. Next, the predicted 
+“Dot” image 𝐼�� and the predicted complexity 𝐶��  of I1 are 
+predicted with I2 as follows: 
+�𝐼��, 𝐶��� � 𝐼𝐶𝑁𝑁𝑃�𝐼��.                        (6) 
+Then, the prediction errors of I1 are calculated as 
+ 
+𝑒��𝑖, 𝑗� � 𝐼��𝑖, 𝑗� � 𝐼���𝑖, 𝑗�, �𝑖 � 𝑗�mod2 ≡ 0.          (7) 
+According to the magnitude of predicted complexities and the 
+size of the additional data S1, we select the predicted errors with 
+less complexity and determine two threshold Tn1 (Tn1<0) and Tp1 
+(Tp1≥0) for HS-based data embedding, which is achieved as 
+𝐸��𝑖, 𝑗� � �
+2𝑒��𝑖, 𝑗� � 𝑏           , 𝑖𝑓 𝑒��𝑖, 𝑗� ∈ �𝑇��, 𝑇��� 
+𝑒��𝑖, 𝑗� � 𝑇�� � 1 , 𝑖𝑓 𝑒��𝑖, 𝑗� � 𝑇��           
+𝑒��𝑖, 𝑗� � 𝑇��        , 𝑖𝑓 𝑒��𝑖, 𝑗� � 𝑇��          
+,     (8) 
+where 𝑏 ∈ �0,1� is the embedded data including the encrypted 
+additional data and some auxiliary data [17]. Therefore, the 
+marked “Dot” image MI1 is generated as 
+𝑀𝐼��𝑖, 𝑗� � 𝐼���𝑖, 𝑗� � 𝐸��𝑖, 𝑗�.               (9) 
+ 
+During 
+the 
+“Cross” 
+embedding, 
+due 
+to 
+the 
+checkerboard pattern, the marked “Dot” image MI1 cannot be 
+fed into the network to predict the “Cross” image I2 directly. As 
+illustrated in Fig. 4, if the height/width is even, after being 
+rotated clockwise/counterclockwise by 90 degrees, the resulting 
+image 𝑀𝐼�
+� has the same pattern as the “Cross” image I2. With 
+the same ICNNP, as shown in Eqn. (10), we feed the rotated 
+marked “Dot” image 𝑀𝐼�
+� into the network, then we obtain the 
+predicted rotated “Cross” image 𝐼��
+� and its complexity 𝐶��
+�. 
+�𝐼��
+�, 𝐶��
+�� � 𝐼𝐶𝑁𝑁𝑃�𝑀𝐼�
+��.                       (10) 
+Then, 𝐼��
+� and 𝐶��
+� are rotated counterclockwise/clockwise by 90 
+degrees to get the predicted “Cross” image 𝐼�� and the predicted 
+complexity 𝐶�� of I2. Similar to the “Dot” embedding, another 
+part of additional data S2 is encrypted with the hiding key K and 
+embedded into I2 to obtain the marked “Cross” image MI2.  
+Finally, we combine the marked “Dot” image MI1 and the 
+marked “Cross” image MI2 to obtain the marked image MI. 
+Fig. 5 describes the architecture of data extraction/image 
+recovery. Data extraction and image recovery are the inverse 
+procedures of data embedding, so we operate the “Dot” 
+extraction/recovery ahead of the “Cross” extraction/recovery. 
+The marked image MI is firstly divided into two sub-images, 
+i.e., the marked “Dot” image MI1 and the marked “Cross” image 
+MI2. With the rotated marked “Dot” image 𝑀𝐼�
+�, 𝐼��
+� and 𝐶��
+� are 
+predicted by using the ICNNP as Eqn. (10). 𝐼�� and 𝐶�� are then 
+obtained by rotating 𝐼��
+� and 𝐶��
+� respectively. Next, the marked 
+prediction errors of I2 are calculated as 
+𝐸��𝑖, 𝑗� � 𝑀𝐼��𝑖, 𝑗� � 𝐼���𝑖, 𝑗�, �𝑖 � 𝑗�mod2 ≡ 1.        (11) 
+According to the sorted magnitude of 𝐶��  and the extracted 
+auxiliary data Tn2 (Tn2<0) and Tp2 (Tp2≥0) as threshold, the 
+embedded data is extracted as 
+𝑏 � 𝐸��𝑖, 𝑗� mod 2 ,  𝐸��𝑖, 𝑗� ∈ �2𝑇��, 2𝑇�� � 1�,        (12) 
+and the original prediction errors of I2 are recovered as  
+𝑒��𝑖, 𝑗� � �
+⌊𝐸��𝑖, 𝑗� 2
+⁄ ⌋            , 𝑖𝑓 𝐸��𝑖, 𝑗� ∈ �2𝑇��, 2𝑇�� � 1� 
+𝐸��𝑖, 𝑗� � 𝑇�� � 1, 𝑖𝑓 𝐸��𝑖, 𝑗� � 2𝑇�� � 1             
+𝐸��𝑖, 𝑗� � 𝑇��        , 𝑖𝑓 𝐸��𝑖, 𝑗� � 2𝑇��                     
+, 
+(13) 
+where ⌊∙⌋ is the floor function. We decrypt the extracted bits to 
+get S1 with the hiding key K, and recover the original “Cross” 
+image I2 as 
+𝐼��𝑖, 𝑗� � 𝑒��𝑖, 𝑗� � 𝐼���𝑖, 𝑗�, �𝑖 � 𝑗�mod2 ≡ 1.      (14) 
+Similarly, the embedded data S1 is extracted correctly and 
+the original “Dot” image I1 is recovered losslessly. Finally, we  
+ 
+(a) MI1                                                                    (b) 𝑀𝐼�
+�
+Fig. 4. Illustration of image rotation of marked “Dot” image.  
+
+Additional data Si & hiding key K-
+Reversible
+Marked “Dot" image MIi
+embedding
+"Dot" image Ii
+111
+12
+Original
+Image
+ICNNP
+Image
+M
+ICNNP
+Results
+Image
+Marked
+image I
+ partition
+prediction
+rotation
+prediction
+rotation
+combination
+image M
+12 *
+12
+"Cross" image I2
+Data
+Additional data S, & hiding key K
+embedding
+Marked “Cross" image Ml2
+Fig. 3. The flowchart of data embedding by using the ICNNP based RDH method
+Hiding key K
+Data extraction
+ Additional data Si
+Marked ‘Dot" image Mi
+& image recovery
+Dot" image I
+111
+C1
+Marked
+Image
+Image
+MI'
+ICNNP
+Results
+ICNNP
+Image
+Original
+image MI
+partition
+rotation
+prediction
+rotation
+prediction
+combination
+image I
++21
+12
+Marked “Cross" image Ml
+"Cross" image I2
+Data extraction
+Hiding key K
+& image recovery
+Additional data S2
+Fig. 5. The flowchart of data extraction and image recovery by using the ICNNP based RDH method.:
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+  
+  
+  
+ 
+(a) Lena               (b) Yacht                (c) Peppers            (d) Barbara 
+Fig. 6. Four cover images. 
+ 
+(a)                                                        (b) 
+ 
+(c)                                                          (d) 
+Fig. 7. Performance comparison of CNNP in [14] and the proposed ICNNP for 
+RDH in four benchmark images. 
+ 
+combine the recovered “Cross” image I2 and “Dot” image I1 to 
+obtain the original image I. 
+III. EXPERIMENTAL RESULTS 
+ To evaluate the efficiency of the proposed ICNNP, the 
+parameters of its complexity prediction model are trained by 
+using 1,000 images randomly selected from BOWS-2 [31]. The 
+proposed ICNNP is trained on an Intel Core i10 CPU (3.6 GHz) 
+with 16 GB RAM and NVIDIA GeForce RTX 2060. The 
+weight decay λ is set to 1×10−3, the batch size is 4 and the initial 
+learning rate is 1×10−3. In [14], the prediction accuracy of 
+CNNP is proved better than some traditional linear predictors, 
+including the BIP, MEDP, GAP, and DP, and the achieved rate-
+distortion performance is better than that of the above predictor 
+with the expansion embedding scheme, it's also better than that 
+of BIP with the HS scheme, and the performance of HS is far 
+better than that of expansion embedding with the CNNP. 
+Therefore, we justly evaluate the performance with ICNNP by 
+comparing it with that of CNNP [14] with the same HS 
+technique.  
+With four benchmark images in Fig. 6 as cover images at the 
+same embedding rate, we employ the peak signal-to-noise ratio 
+(PSNR) between the original image and the marked image as 
+the metric for objective image quality evaluation. Besides, we 
+randomly select 100 images different from the training images 
+from BOWS-2 [31] and test them with different embedding 
+capacities to evaluate the universality of ICNNP. 
+Fig. 7 shows the PSNR values of four test images (Lena, 
+Yacht, Peppers, and Barbara) with embedding capacities from 
+10,000 to 200,000 bits. From this figure, we can see that the 
+PSNR values of the RDH method with the ICNNP are larger 
+than those of the CNNP-based RDH method. Moreover, Table 
+I shows the average PSNR values of the 100 test images for 
+different embedding capacities. When the embedding capacity 
+is 10,000 bits, the average PSNR value of the proposed ICNNP 
+based RDH method is 62.37 dB, which is 1.06dB higher than 
+that of the CNNP-based RDH method. Along with the 
+embedding capacity increases from 20,000 to 150,000 bits, the 
+average PSNR values of the improved RDH method are still 
+0.81dB, 0.60dB, 0.55dB, 0.56dB, 0.55dB, 0.54dB, 0.53dB, 
+0.51dB, 0.47dB, 0.41dB, 0.37dB, 0.33dB, 0.26dB, and 0.20dB 
+higher respectively. 
+IV. CONCLUSION 
+In this letter, we propose an improved CNN predictor for 
+RDH, which extracts features from different receptive fields 
+with whole optimization and uses more neighboring pixels to 
+precisely predict the pixel value and its complexity. During data 
+embedding, a grayscale image is split into two sub-images, one 
+sub-image is applied to predict another sub-image alternately 
+by using the ICNNP. Then the pixels’ prediction errors are 
+sorted according to the predicted pixels’ complexities, and the 
+prediction errors with less complexity are selected for data 
+embedding with the classical HS strategy. The original image 
+is recovered losslessly after the embedded data is extracted 
+correctly, and the data extraction and image recovery are 
+separable. Experimental results show that the achieved 
+performance of the improved CNNP with the classical HS 
+strategy is better than that of the CNNP presented in [14] with 
+the same HS strategy. 
+0
+0.5
+1
+1.5
+2
+Payload(bits)
+105
+40
+45
+50
+55
+PSNR(dB)
+Lena
+ (CNNP)
+ (ICNNP)
+0
+0.5
+1
+1.5
+2
+Payload(bits)
+105
+45
+50
+55
+60
+PSNR(dB)
+Yacht
+ (CNNP)
+ (ICNNP)
+0
+0.5
+1
+1.5
+2
+Payload(bits)
+105
+35
+40
+45
+50
+PSNR(dB)
+Peppers
+ (CNNP)
+ (ICNNP)
+0
+0.5
+1
+1.5
+2
+Payload(bits)
+105
+35
+40
+45
+50
+55
+PSNR(dB)
+Barbara
+ (CNNP)
+ (ICNNP)
+TABLE I 
+AVERAGE PSNR (DB) OF 100 IMAGES OF THE PROPOSED ICNNP-BASED 
+METHOD AND THE CNNP-BASED METHOD [14] 
+Embedding Capacity (bits) 
+CNNP 
+ICNNP
+10000
+61.31 
+62.37 
+20000
+58.01 
+58.82
+30000
+55.98 
+56.58 
+40000
+54.43 
+54.98 
+50000
+53.10 
+53.66 
+60000
+51.94 
+52.49 
+70000
+50.86 
+51.40 
+80000
+49.85 
+50.38 
+90000
+48.88 
+49.39 
+100000
+47.91 
+48.38 
+110000
+46.95 
+47.36 
+120000
+46.03 
+46.40 
+130000
+45.13 
+45.46 
+140000
+44.29 
+44.55
+150000 
+43.47 
+43.67 
+
+汇
+20594
+51179619
+20651
+200
+650> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+5
+REFERENCES 
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+
diff --git a/vtAzT4oBgHgl3EQfdfwn/content/tmp_files/load_file.txt b/vtAzT4oBgHgl3EQfdfwn/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf,len=577
+page_content='> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  1  \uf020  Abstract—This letter proposes an improved CNN predictor  (ICNNP) for reversible data hiding (RDH) in images, which  consists of a feature extraction module, a pixel prediction module,  and a complexity prediction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Due to predicting the  complexity of each pixel with the ICNNP during the embedding  process, the proposed method can achieve superior performance  than the CNN predictor-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Specifically, an input  image does be first split into two different sub-images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', the “Dot”  image and the “Cross” image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Meanwhile, each sub-image is  applied to predict another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Then, the prediction errors of  pixels are sorted with the predicted pixel complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' In light of  this, some sorted prediction errors with less complexity are  selected to be efficiently used for low-distortion data embedding  with a traditional histogram shift scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Experimental results  demonstrate that the proposed method can achieve better  embedding performance than that of the CNN predictor with the  same histogram shifting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Index Terms—Convolutional neural network, multitasking,  reversible data hiding, histogram shifting  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' INTRODUCTION  EVERSIBLE data hiding (RDH) can losslessly recover  both the embedded data and the cover medium [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Due to  the trait, RDH has gradually become a hot research field in the  information hiding community and has been widely used in  several realistic scenarios [1], including medical, military, and  law forensics et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' According to the domain hiding a secret  message, RDH can be categorized as two main branches: spatial  domain-based RDH [2-23] and JPEG domain-based RDH [24-  27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The spatial domain-based RDH generally exploits three  technologies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', lossless compression (LC) [2-4], difference  expansion (DE) [5-15], and histogram shifting (HS) [16-23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  While the JPEG domain-based RDH is mainly based on DCT  This work was supported in part by the National Key R&D Program of  China (Grant 2021YFE0205400), the Natural Science Foundation of Xiamen,  China (Grant 3502Z20227192), and the Natural Science Foundation of China  (Grant U20B2051, 61972168, 62072114, 62002124, 61871434).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Corresponding  author: Zhenxing Qian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Qiu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Lin, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Zeng are with the College of Information Science &  Engineering, Huaqiao  University, Xiamen 361021, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (e-mail:  yqqiu@hqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='cn, echo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='linxd@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='com, zeng0043@hqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Peng and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Qian are with the School of Computer Science, Fudan  University, Shanghai 200433, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (e-mail: pengwanli@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='cn,  zxqian@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Qiu and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Peng contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  coefficients modification [24, 25] or Huffman table  modification [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Currently, in the RDH community, pixel prediction has  become a critical problem, which dramatically affects the  performance of RDH algorithms [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The traditional predictors  include the median edge direction (MED) predictor [6],  interpolation predictor [7], gradient-adjusted predictor (GAP)  [8], pixel-value-ordering (PVO) predictor [9, 12, 22], linear  predictor [10], rhombus predictor [17-20], and ridge regression  predictor [23], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Although these predictors have achieved  supervising improvement, there is still a notable weak point,  that is few neighboring pixels are used for pixel prediction [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  If more adjacent pixels are served as reference pixels, higher  prediction performance can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Due to its strong  capabilities of different receptive fields fusion and whole  optimization, a convolutional neural network (CNN) can be  established and trained to predict pixels accurately by building  a non-linear mapping for pixel prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' In light of this, Luo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' [13] presented a CNN-based stereo image RDH method  by leveraging the correlations between the left view and the  right view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' [14] proposed a CNN predictor (CNNP)  based RDH method, where a grayscale image was split into two  sub-images, and each one is predicted with another one  alternatively by using the CNNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' After that, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' [15]  divided an image into four parts, and each part was predicted  with the other three parts in turn by using a CNNP for a better  prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' In addition, a better visual quality of the  marked image is achieved through adaptive embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Overall,  the prediction performance of CNN predictors can be better  than that of the traditional predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  From the above discussion, in order to improve performance,  the existing methods conduct pixel prediction by leveraging  adjacent pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' While these methods [13-15] ignore the  complexity of each pixel with deep learning, which limits the  performance of RDH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  To tackle the above limitation, in this letter, we improve the  CNNP presented in [14] by adding a complexity prediction part  to predict the pixel’s complexities precisely, which is called  improved CNNP (ICNNP) in the rest of this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Specifically,  during data embedding, we first split a grayscale image into two  sub-images, where one sub-image is predicted by other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Then, we sort the prediction errors of the predicted pixels  according to their complexities, and the prediction errors with  less complexity are used for data embedding with a classical HS  Improved CNN Prediction Based Reversible  Data Hiding  Yingqiang Qiu, Wanli Peng, Xiaodan Lin, Huanqiang Zeng, Senior Member, IEEE, and Zhenxing  Qian, Member, IEEE  R  > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  2  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Finally, experimental results show that the  performance of the proposed method is better than that of the  CNNP presented in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  The rest of this letter is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The proposed  improved RDH method is described in detail in Section II, and  the experimental results and discussions are provided in Section  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Finally, we conclude our work in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' PROPOSED IMPROVED METHOD  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Network Architecture  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 1, according to the checkerboard context  model [17], the original image is split into two sub-images  which consist of "Dot" and "Cross" pixels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' For the  “Dot” image, the values of the “Dot” pixels are remained, while  those of the “Cross” pixels are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Meanwhile, just the  values of the “Dot” pixels are set to 0 for the “Cross” image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Based on the pixel correlation of the two sub-images, each sub-  image is applied to predict the pixel values and complexities of  another sub-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  The overall of the proposed ICNNP is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The  architecture of the ICNNP is composed of three parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=',  feature extraction, pixel prediction, and complexity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  The “Cross” image I2 is fed to the network to predict the  values and complexities of the “Dot” pixels, where the values  of complexity are adjusted to [0,255] for a good visualization  display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The lower value, the lower complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The feature  extraction consists of some convolution layers with different  filter sizes (3 × 3, 5 × 5, 7 × 7, ⋯), which are parallelized and  appended with a 3×3 convolution layer respectively to extract  features from different receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' A residual block is then  applied to further aggregate and refine the learnt features from  different branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' With the extracted feature, the pixel  prediction yields the predicted “Dot” image 𝐼�� , and the  complexity prediction yields the predicted complexity 𝐶�� of the  “Dot” image I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' “Conv” stands for the convolution unit with  kernel size S × S and the number of channels is output × input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  A LeakyReLU activation function [28] locates between each  two convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  It is worthy to note that the complexity prediction is similar  to the pixel prediction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', instead of orthogonal adjacent pixels  [14, 17], more adjacent pixels are used to nonlinearly predict  the complexity of the pixel area, improving the performance of  RDH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Training  In the ICNNP, the well-trained parameters of CNNP [14] are  loaded into the feature extraction and pixel prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Note that,  these parameters are fixed and the parameters of complexity  prediction are updated during the training of the ICNNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' In the  training, the input is the “Cross” image I2, the outputs are the  predicted “Dot” image 𝐼�� and the predicted complexity 𝐶�� of  the “Dot” image I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Since the filter parameters of the feature  extraction and the pixel prediction are fixed, the target is no  longer the “Dot” image I1 but the referenced complexity 𝐶� of  I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The definition of 𝐶� is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  (1) For “Cross” pixels, 𝐶��𝑖, 𝑗� is set to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  (2) For “Dot” pixels, if 𝑖 � 1 or 𝑖 � 𝑀 or 𝑗 � 1 or 𝑗 � 𝑁,  𝐶��𝑖, 𝑗� is set to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' otherwise, 𝐶��𝑖, 𝑗��2 � 𝑖 � 𝑀 � 1,2 � 𝑗 �  𝑁 � 1� is calculated as  𝐶��𝑖, 𝑗� �  �  � ∙ �∑  ∑  �𝐼�𝑖 � 𝑘, 𝑗 � 𝑙� � 𝐼�𝑖, 𝑗���  ��  ����  ��  ����  ,     (1)  where,  �  𝑘� � �1, 𝑘� � 2    , 𝑖 � 2  𝑘� � �2, 𝑘� � 1    , 𝑖 � 𝑀 � 1  𝑘� � �2, 𝑘� � �2 , 2 � 𝑖 � 𝑀 � 1  ,               (2)  �  𝑙� � �1, 𝑙� � 2    , 𝑗 � 2  𝑙� � �2, 𝑙� � 1    , 𝑗 � 𝑁 � 1  𝑙� � �2, 𝑙� � �2 , 2 � 𝑗 � 𝑁 � 1  ,                (3)  𝑅 � �𝑘� � 𝑘� � 1� � �𝑙� � 𝑙� � 1� � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (4)  In the proposed method, the max predicted pixel area is 5 × 5,  which can accurately calculate the pixel complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  As with CNNP in [14], we leveraged back-propagation [29]  and Adam algorithm [30] to optimize the objective function  defined as below:  Loss �  �  � ∑  �𝐶�� � 𝐶��� � 𝜆‖𝜔‖�  �  �  ���  ,                  (5)  where N stands for the number of training examples, ω  represents all weights of the network, and λ denotes the weight  decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' ICNNP based RDH  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 3 depicts the data embedding architecture of the ICNNP-  based RDH method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The adopted double embedding strategy  [17] with the HS technique [6] involves consecutive usage of  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  ⊕  ⊕  Conv|3×3|32×32  LeakyReLU  Conv|3×3|1×32  Conv|3×3|1×32  Conv|3×3|32×32  LeakyReLU  Conv|3×3|32×32  LeakyReLU  Conv|3×3|32×32  Conv|3×3|32×1  LeakyReLU  Conv|3×3|32×32  Conv|5×5|32×1  LeakyReLU  Conv|3×3|32×32  Conv|7×7|32×1  LeakyReLU  Conv|3×3|32×32  Feature extraction  Pixel prediction  Complexity prediction  “Cross” image I2  Predicted “Dot” image 𝐼��  Predicted complexity of “Dot” image 𝐶��  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The overall architecture of the proposed ICNNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  (a) Original image I      (b) “Dot” image I1      (c) “Cross” image I2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Illustration of splitting an original image into two sub-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  444十  x  0  0  0  0  0  +  0  x  0  x  x  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  0  +  0  x  0  X  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  X  X  X  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  x  :  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  0  x  0  x  0  x  0  x  X  X  0  0  0  X  0  0  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  X  X  X  X  x  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  x  0  X  0  X  0  x  0  x  x  0  0  0  0  x  0  x  0  x  0  x  X  X  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  X  0  x  0  X  04> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  3  the “Dot” embedding and the “Cross” embedding, and the  “Cross” embedding is achieved after the “Dot” embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  The original image I is firstly divided into two sub-images,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', a “Dot” image I1 and a “Cross” image I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Next, the predicted  “Dot” image 𝐼�� and the predicted complexity 𝐶��  of I1 are  predicted with I2 as follows:  �𝐼��, 𝐶��� � 𝐼𝐶𝑁𝑁𝑃�𝐼��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (6)  Then, the prediction errors of I1 are calculated as  𝑒��𝑖, 𝑗� � 𝐼��𝑖, 𝑗� � 𝐼���𝑖, 𝑗�, �𝑖 � 𝑗�mod2 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (7)  According to the magnitude of predicted complexities and the  size of the additional data S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' we select the predicted errors with  less complexity and determine two threshold Tn1 (Tn1<0) and Tp1  (Tp1≥0) for HS-based data embedding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' which is achieved as  𝐸��𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑗� � �  2𝑒��𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑗� � 𝑏           ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑖𝑓 𝑒��𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑗� ∈ �𝑇��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑇���  𝑒��𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑗� � 𝑇�� � 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑖𝑓 𝑒��𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑗� � 𝑇��  𝑒��𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑗� � 𝑇��        ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑖𝑓 𝑒��𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝑗� � 𝑇��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='     (8)  where 𝑏 ∈ �0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='1� is the embedded data including the encrypted  additional data and some auxiliary data [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Therefore, the  marked “Dot” image MI1 is generated as  𝑀𝐼��𝑖, 𝑗� � 𝐼���𝑖, 𝑗� � 𝐸��𝑖, 𝑗�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (9)  During  the  “Cross”  embedding,  due  to  the  checkerboard pattern, the marked “Dot” image MI1 cannot be  fed into the network to predict the “Cross” image I2 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' As  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 4, if the height/width is even, after being  rotated clockwise/counterclockwise by 90 degrees, the resulting  image 𝑀𝐼�  � has the same pattern as the “Cross” image I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' With  the same ICNNP, as shown in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (10), we feed the rotated  marked “Dot” image 𝑀𝐼�  � into the network, then we obtain the  predicted rotated “Cross” image 𝐼��  � and its complexity 𝐶��  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  �𝐼��  �, 𝐶��  �� � 𝐼𝐶𝑁𝑁𝑃�𝑀𝐼�  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (10)  Then, 𝐼��  � and 𝐶��  � are rotated counterclockwise/clockwise by 90  degrees to get the predicted “Cross” image 𝐼�� and the predicted  complexity 𝐶�� of I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Similar to the “Dot” embedding, another  part of additional data S2 is encrypted with the hiding key K and  embedded into I2 to obtain the marked “Cross” image MI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Finally, we combine the marked “Dot” image MI1 and the  marked “Cross” image MI2 to obtain the marked image MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 5 describes the architecture of data extraction/image  recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Data extraction and image recovery are the inverse  procedures of data embedding, so we operate the “Dot”  extraction/recovery ahead of the “Cross” extraction/recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  The marked image MI is firstly divided into two sub-images,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', the marked “Dot” image MI1 and the marked “Cross” image  MI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' With the rotated marked “Dot” image 𝑀𝐼�  �, 𝐼��  � and 𝐶��  � are  predicted by using the ICNNP as Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 𝐼�� and 𝐶�� are then  obtained by rotating 𝐼��  � and 𝐶��  � respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Next, the marked  prediction errors of I2 are calculated as  𝐸��𝑖, 𝑗� � 𝑀𝐼��𝑖, 𝑗� � 𝐼���𝑖, 𝑗�, �𝑖 � 𝑗�mod2 ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (11)  According to the sorted magnitude of 𝐶��  and the extracted  auxiliary data Tn2 (Tn2<0) and Tp2 (Tp2≥0) as threshold, the  embedded data is extracted as  𝑏 � 𝐸��𝑖, 𝑗� mod 2 ,  𝐸��𝑖, 𝑗� ∈ �2𝑇��, 2𝑇�� � 1�,        (12)  and the original prediction errors of I2 are recovered as  𝑒��𝑖, 𝑗� � �  ⌊𝐸��𝑖, 𝑗� 2  ⁄ ⌋            , 𝑖𝑓 𝐸��𝑖, 𝑗� ∈ �2𝑇��, 2𝑇�� � 1�  𝐸��𝑖, 𝑗� � 𝑇�� � 1, 𝑖𝑓 𝐸��𝑖, 𝑗� � 2𝑇�� � 1  𝐸��𝑖, 𝑗� � 𝑇��        , 𝑖𝑓 𝐸��𝑖, 𝑗� � 2𝑇��  ,  (13)  where ⌊∙⌋ is the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' We decrypt the extracted bits to  get S1 with the hiding key K, and recover the original “Cross”  image I2 as  𝐼��𝑖, 𝑗� � 𝑒��𝑖, 𝑗� � 𝐼���𝑖, 𝑗�, �𝑖 � 𝑗�mod2 ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' (14)  Similarly, the embedded data S1 is extracted correctly and  the original “Dot” image I1 is recovered losslessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Finally, we  (a) MI1                                                                    (b) 𝑀𝐼�  �  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Illustration of image rotation of marked “Dot” image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Additional data Si & hiding key K-  Reversible  Marked “Dot" image MIi  embedding  "Dot" image Ii  111  12  Original  Image  ICNNP  Image  M  ICNNP  Results  Image  Marked  image I  partition  prediction  rotation  prediction  rotation  combination  image M  12 *  12  "Cross" image I2  Data  Additional data S, & hiding key K  embedding  Marked “Cross" image Ml2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The flowchart of data embedding by using the ICNNP based RDH method  Hiding key K  Data extraction  Additional data Si  Marked ‘Dot" image Mi  & image recovery  Dot" image I  111  C1  Marked  Image  Image  MI\'  ICNNP  Results  ICNNP  Image  Original  image MI  partition  rotation  prediction  rotation  prediction  combination  image I  +21  12  Marked “Cross" image Ml  "Cross" image I2  Data extraction  Hiding key K  & image recovery  Additional data S2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The flowchart of data extraction and image recovery by using the ICNNP based RDH method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' :  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  0  0  :  0  0  0  0  0  0  0  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Rotate by 90 degrees  0  0  0  0  (clockwise/counterclockwise)  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  0  0  0  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  0  0> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  4  (a) Lena               (b) Yacht                (c) Peppers            (d) Barbara  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Four cover images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  (a)                                                        (b)  (c)                                                          (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Performance comparison of CNNP in [14] and the proposed ICNNP for  RDH in four benchmark images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  combine the recovered “Cross” image I2 and “Dot” image I1 to  obtain the original image I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' EXPERIMENTAL RESULTS  To evaluate the efficiency of the proposed ICNNP, the  parameters of its complexity prediction model are trained by  using 1,000 images randomly selected from BOWS-2 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The  proposed ICNNP is trained on an Intel Core i10 CPU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='6 GHz)  with 16 GB RAM and NVIDIA GeForce RTX 2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The  weight decay λ is set to 1×10−3, the batch size is 4 and the initial  learning rate is 1×10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=" In [14], the prediction accuracy of  CNNP is proved better than some traditional linear predictors,  including the BIP, MEDP, GAP, and DP, and the achieved rate-  distortion performance is better than that of the above predictor  with the expansion embedding scheme, it's also better than that  of BIP with the HS scheme, and the performance of HS is far  better than that of expansion embedding with the CNNP." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Therefore, we justly evaluate the performance with ICNNP by  comparing it with that of CNNP [14] with the same HS  technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  With four benchmark images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 6 as cover images at the  same embedding rate, we employ the peak signal-to-noise ratio  (PSNR) between the original image and the marked image as  the metric for objective image quality evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Besides, we  randomly select 100 images different from the training images  from BOWS-2 [31] and test them with different embedding  capacities to evaluate the universality of ICNNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 7 shows the PSNR values of four test images (Lena,  Yacht, Peppers, and Barbara) with embedding capacities from  10,000 to 200,000 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' From this figure, we can see that the  PSNR values of the RDH method with the ICNNP are larger  than those of the CNNP-based RDH method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Moreover, Table  I shows the average PSNR values of the 100 test images for  different embedding capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' When the embedding capacity  is 10,000 bits, the average PSNR value of the proposed ICNNP  based RDH method is 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='37 dB, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='06dB higher than  that of the CNNP-based RDH method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Along with the  embedding capacity increases from 20,000 to 150,000 bits, the  average PSNR values of the improved RDH method are still  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='81dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='60dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='55dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='56dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='55dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='54dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='53dB,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='51dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='47dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='41dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='37dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='33dB, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='26dB, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='20dB  higher respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' CONCLUSION  In this letter, we propose an improved CNN predictor for  RDH, which extracts features from different receptive fields  with whole optimization and uses more neighboring pixels to  precisely predict the pixel value and its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' During data  embedding, a grayscale image is split into two sub-images, one  sub-image is applied to predict another sub-image alternately  by using the ICNNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Then the pixels’ prediction errors are  sorted according to the predicted pixels’ complexities, and the  prediction errors with less complexity are selected for data  embedding with the classical HS strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' The original image  is recovered losslessly after the embedded data is extracted  correctly, and the data extraction and image recovery are  separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Experimental results show that the achieved  performance of the improved CNNP with the classical HS  strategy is better than that of the CNNP presented in [14] with  the same HS strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='5  2  Payload(bits)  105  40  45  50  55  PSNR(dB)  Lena  (CNNP)  (ICNNP)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='5  2  Payload(bits)  105  45  50  55  60  PSNR(dB)  Yacht  (CNNP)  (ICNNP)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='5  2  Payload(bits)  105  35  40  45  50  PSNR(dB)  Peppers  (CNNP)  (ICNNP)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='5  2  Payload(bits)  105  35  40  45  50  55  PSNR(dB)  Barbara  (CNNP)  (ICNNP)  TABLE I  AVERAGE PSNR (DB) OF 100 IMAGES OF THE PROPOSED ICNNP-BASED  METHOD AND THE CNNP-BASED METHOD [14]  Embedding Capacity (bits)  CNNP  ICNNP  10000  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='31  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='37  20000  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='01  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='82  30000  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='98  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='58  40000  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content='10  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='66  60000  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content='67  汇  20594  51179619  20651  200  650> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  5  REFERENCES  [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Wu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Ma, “Reversible data hiding:  Advances in the past two decades,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 3210–3237,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Saber, “Lossless  generalized-LSB data embedding,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Li, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Yu, “Optimal transition probability of  reversible data hiding for general distortion metrics and its applications,”  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 294–304, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Yang, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Yu, “Reversible data hiding under  inconsistent distortion metrics,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 5087–5099, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Circuits Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Video Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 890–896, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  721–730, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Zeng, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Xiong, “Reversible image  watermarking using interpolation technique,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Forensics  Secur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' 412-417, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Yang, “High-fidelity reversible data hiding  scheme based on pixel-value-ordering and prediction-error expansion,”  Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Yu, “Adaptive reversible data hiding by extending  the generalized integer transformation,” IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Cai, “An insight into pixel value ordering prediction-based  prediction-error expansion,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Jiang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Zhong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Pan, Convolutional  neural networks-based stereo image reversible data hiding method, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Hu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Xiang, “CNN Prediction Based Reversible Data Hiding,”  IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Hu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content='  [16] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Shi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Ansari, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Su, “Reversible data hiding,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Circuits Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Video Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  [17] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Nam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Suresh, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' -Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Shi, “Reversible  watermarking algorithm using sorting and prediction,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Circuits Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content='  [18] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Gui, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Yang, “Efficient reversible data hiding  based on multiple histograms modification,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Forensics  Secur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 2016–2027, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  [19] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Zhang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Guo, “Optimal Reversible Data Hiding  Scheme Based on Multiple Histograms Modification,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Circuits Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Video Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Ni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Mao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Shi, “Multiple Histograms-Based  Reversible Data Hiding: Framework and Realization,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Video Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Zhao, “High capacity reversible data hiding based on  multiple histograms modification,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Video  Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content='  [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Guo, “Location-based pvo and adaptive  pairwise modification for efficient reversible data hiding,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='  Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Ma, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' -Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content=' Shi, “High Precision Error  Prediction Algorithm Based on Ridge Regression Predictor for Reversible  Data Hiding,” IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
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+page_content='  Available at http://bows2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='ec-lille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
+page_content='fr/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfdfwn/content/2301.01420v1.pdf'}
diff --git a/vtFKT4oBgHgl3EQf4S7l/content/tmp_files/2301.11933v1.pdf.txt b/vtFKT4oBgHgl3EQf4S7l/content/tmp_files/2301.11933v1.pdf.txt
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+In our mind’s eye: Visible and invisible in quantum theory, with Schrödinger’s cat experiment  
+ 
+Arkady Plotnitsky* 
+ 
+*Literature, Theory, Cultural Studies Program; Philosophy and Literature Program, Purdue University, 
+West Lafayette, IN 47907, USA; Email: plotnits@purdue.edu 
+ 
+Abstract. This article aims to reconsider E. Schrödinger’s famous thought experiment, the cat-paradox experiment, 
+and its place in quantum foundations from a new perspective, grounded in the type of interpretation of quantum 
+phenomena and quantum mechanics, which belongs to the class of interpretations designated here as “reality without 
+realism” (RWR) interpretations. Such interpretations have not been previously brought to bear on the cat 
+experiment, including by N. Bohr, whose interpretation in its ultimate form (as he changed his interpretation a few 
+times) is an RWR interpretation, but who does not appear to have commented on the cat experiment. The 
+interpretation adopted in this article follows Bohr’s interpretation, as based on two assumptions or postulates, the 
+Heisenberg and Bohr postulates, but it adds a third postulate, the Dirac postulate. The article also introduces, in 
+conjunction with the concept of reality without realism, the concepts of visible and invisible to thought and 
+considers their role in the cat-paradox experiment. 
+ 
+Key words: the cat paradox experiment, the cut, classical objects, quantum objects, quantum phenomena, reality 
+without realism, visible to thought, invisible to thought 
+ 
+Hamlet: 
+My father — methinks I see my father. 
+Horatio: 
+Where, my lord? 
+Hamlet: 
+In my mind’s eye, Horatio. 
+--William Shakespeare, Hamlet, Act 1, Scene 2, ll. 183-185 
+ 
+To die for the invisible. This is metaphysics. 
+--Emmanuel Levinas, Totality and infinity: An essay on exteriority 
+ 
+1. Introduction 
+ 
+This article aims to reconsider E. Schrödinger’s famous thought experiment, the cat-paradox experiment 
+(hereafter “the cat experiment”), and its place in quantum foundations from a new perspective, which 
+removes any paradox from it [Schrödinger 1935]. So, admittedly, do some other views of the experiment, 
+and Schrödinger himself did not call it a paradox, rather a “ridiculous situation” [Schrödinger 1935, p. 
+157]. The present view of it, however, is grounded the type of interpretation of quantum phenomena and 
+quantum mechanics (QM) that has not, to my knowledge, been previously brought to bear on the cat 
+experiment, including by N. Bohr, whose interpretation, especially in its ultimate form, is the closest to 
+the one adopted here, but who does not appear to have commented on the cat experiment. It is worth 
+keeping in mind that, while an interpretation of QM, commonly, including Bohr’s or the present 
+interpretation, involves an interpretation of quantum phenomena, the latter have separate interpretable 
+aspects (noted whenever necessary in this article) that do not depend on, and hence could be interpreted 
+independently of, any theory predicting them. Quantum phenomena will be assumed here to be defined by 
+the fact that in considering them (technically, the data found in them, pertinent to quantum experiment), 
+the Planck constant, h, which is a classically measurable quantity, must be taken into account.1 
+ 
+1 I put aside qualifications of this definition, necessary in general but not germane for this article, because all 
+quantum phenomena and measurements considered involve h (e.g., [Plotnitsky 2021a, pp. 37-38], also [Khrennikov 
+2021]). I might only add that all quantum-mechanical equations used for actually predicting the data observed in 
+
+ 
+2 
+Bohr eventually came to see quantum phenomena as revealing “a novel feature of atomicity in the 
+laws of nature,” “disclosed” by “Planck’s discovery of the quantum of action [h], supplementing in such 
+unexpected manner the old [Democritean] doctrine of the limited divisibility of matter” [Bohr 1938, p. 
+94]. Atomicity and, thus, discreteness or discontinuity initially emerged on this Democritean model, with 
+M. Planck’s discovery of the quantum nature of radiation in 1900, which led Planck to his concept of the 
+quantum of action, h, physically defining this discontinuity, and then A. Einstein’s introduction of the 
+concept of a photon, as a particle of light, in 1906. The situation, however, gradually, especially with the 
+discovery of QM in 1925 by W. Heisenberg, revealed itself to be more complex, eventually leading Bohr 
+to his concepts of phenomenon and atomicity (essentially equivalent to that of phenomenon, but 
+highlighting some of the features of the latter concept, such as discreteness), and the interpretation of 
+quantum phenomena and QM based in these concepts. 
+This interpretation, developed in the later 1930s, became the ultimate version of Bohr’s interpretation, 
+following a decade of the development of, and some significant changes in, his views (with only a few 
+minor refinements added later). This requires one to specify to which version of his interpretation one 
+refers, which I shall do as necessary, while focusing on his ultimate interpretation, unavoidably, in the 
+present interpretation of his interpretation. Unless qualified, “Bohr’s interpretation” will refer to his 
+ultimate interpretation. (The designation “the Copenhagen interpretation” requires even more 
+qualifications as concerns whose interpretation it is, say, that of Heisenberg, Dirac, or von Neumann. 
+Accordingly, I avoid this designation altogether.) The interpretation adopted in this article follows this 
+interpretation, in particular as based on two assumptions or postulates, the Heisenberg and Bohr 
+postulates, but it adds a third postulate, the Dirac postulate. All three postulates are defined below. I 
+would like, however, to emphasize from the outset that these postulates are interpretive assumptions that 
+could, in principle, be falsified, even though, as discussed later in this article, a falsification of an 
+interpretation is not the same (and a more complex matter than) that of a theory by experimental evidence. 
+By virtue of the first two postulates, especially the Heisenberg postulate, both interpretations belong to 
+the class of interpretations of quantum phenomena and QM, or quantum field theory (QFT), designated 
+here as “reality without realism” (RWR) interpretations. This article is only concerned with QM and, 
+marginally, QFT (in high-energy regimes) in their currently standard forms, and puts aside, except in 
+passing, alternative quantum theories, such as Bohmian mechanics or spontaneous collapse theories.2 
+The Heisenberg postulate, most essentially defining RWR interpretations, was in effect introduced by 
+Bohr’s 1913 atomic theory, in considering the transitions, “quantum jumps,” between stationary states of 
+electrons, while retaining a realist view of stationary states by assuming them to be represented as orbits 
+of electrons around nuclei. The RWR understanding of quantum phenomena and QM emerged in its full 
+form in Heisenberg’s approach to quantum theory that led him to his discovery of QM, which is why I 
+use the designation “the Heisenberg postulate.” The Heisenberg postulate places the emergence of 
+quantum phenomena beyond representation or knowledge, or even conceptions, beyond the reach of 
+ 
+quantum phenomena contain h or ℏ (or something mathematically equivalent, for example, by suitable changing the 
+values of the parameters, such as time), which fact is sometimes hidden in more abstract, such as Hilbert-space, 
+versions of the formalism, unless one properly unfolds its relevant elements to make actual predictions possible. 
+2 The interpretation offered in this article was considered previously in [Plotnitsky 2021a,b; 2022a,b]. The last 
+article cited expressly adopts the three postulates in question, under the headings of Heisenberg, Bohr, and Dirac 
+discontinuity, in considering the double-slit experiment. RWR interpretations without the Bohr postulate may be 
+possible, but they will be put aside here, because both Bohr and the present interpretation adopt this postulate, along 
+with the Heisenberg postulate, which defines all RWR interpretations. It is possible and technically more rigorous to 
+see a different interpretation of a given theory as forming a different theory, because each interpretation may involve 
+concepts not be shared by other interpretations. This is the case, for example, in different versions of “the 
+Copenhagen interpretation,” not all of which are RWR interpretations, which too may be different, as are Bohr’s and 
+the present interpretation, because the present interpretation assumes the Dirac postulate. What is shared is the 
+mathematical formalism used, at least in terms of the equivalence or mutual translatability of its different versions. 
+For simplicity, however, I shall continue to speak of different interpretations of a theory itself containing a given 
+mathematical formalism, specifically, of different interpretations of QM or QFT.  
+
+ 
+3 
+thought, or in terms I shall adopt here, make this emergence invisible to thought. Realism, by contrast, is 
+defined by the assumption of the possibility of either representation or knowledge, or at least conception 
+of how the phenomena considered are possible, thus making them visible to thought. I shall speak of weak 
+RWR interpretations (or the weak form of the Heisenberg postulate) when this emergence is assumed to 
+be beyond representation or knowledge, and of strong RWR interpretation (or of the strong form of 
+Heisenberg postulate) when it is assumed to be beyond conception and thus made invisible to thought. 
+This article adopts a strong RWR interpretation, as did Bohr in the ultimate version of his interpretation. 
+Unless qualified, the term “RWR interpretation” will, hereafter, refer to strong RWR interpretations. 
+The concepts of classical physics, specifically classical mechanics, emerged as mathematized 
+refinements of our daily concepts—concepts arising from our general phenomenal experience of the 
+world. All modern physics (classical, relativistic, or quantum) only deals with suitably mathematized 
+idealizations of physical reality. This connection between physical and daily concepts has proven to be 
+difficult to use in quantum theory, even in realist interpretations, which would assume that QM or QFT 
+provides a mathematized representation of quantum objects and processes. Such a representation is no 
+longer a mathematical refinement of our general phenomenal experience of the world, although QM or 
+QFT formally adopts some (but only some) mathematics used in classical physics. RWR interpretations 
+preclude any representation, including any mathematical representation, or even conception of the 
+ultimate reality responsible for quantum phenomena. I qualify because classical physics remains an 
+essential part of quantum theory, including in RWR interpretations, if they assume, as both Bohr’s and the 
+present interpretation do, the Bohr postulate. By the Bohr postulate, quantum phenomena, defined by 
+what is observed in measuring instruments, along with the observable parts of these instruments, are 
+represented by classical physics.3 By classical physics I mean (as Bohr appears to have done, although he 
+did not always specify the term) to classical mechanics and classical electromagnetic theory, with the 
+addition of special relativity in high-energy (QFT) quantum regimes. By classical mechanics I refer to 
+Newton’s mechanics defined by its three main laws, the law of inertia, the law of the changes a force can 
+have on the motion of a body, F = ma, and the law of action and reaction between interacting bodies, as 
+equal in magnitude by opposite in direction. These three theories—classical physics, relativity, and 
+quantum theory—are sufficient for the present purposes for representing the observable part of measuring 
+instruments in quantum physics. (Gravity, governed by Newton’s forth law, is a special case of Newton’s 
+mechanics, which can be put aside for the present purposes, as can be the equivalence principle, not 
+involved in any phenomena considered in this article.) Accordingly, by a quantum instrument I 
+understand any technological device able of establishing quantum phenomena and registering quantum 
+data, the data that is represented by classical physics but that cannot be predicted by classical physics 
+(including as concerns the role of h in these data), and thus requires an alternative, quantum theory, such 
+as QM or QFT, of possibly some alternative theory. (This article, again, will put such alternatives, be they 
+actual or hypothetical, aside.) At least, such is the case, as things stand now (a qualification assumed 
+throughout this article), for it is in principle possible that one or another form of classical theory able to 
+predict this data can be developed, and there are attempt in this direction, which will be put aside here, 
+because this article is only concerned with RWR interpretations of QM or QFT and, a subject not 
+considered previously, how the cat paradox appears in this interpretation. I make no other claims here. 
+Measuring instruments, it follows, also have quantum strata through which they interact with quantum 
+objects. Eventually, Bohr adopted the term “phenomenon” to refer strictly to what is observed in 
+measuring instruments, as effects of their interaction with quantum objects [Bohr 1987, v. 2, p. 64]. The 
+Bohr postulate, thus, also reflects the transition, via measuring instruments, from the ultimate, “quantum,” 
+reality considered to the classical reality of observation, and conversely, in the initial stage (preparation) 
+of an experiment, from the classical reality of observation to the ultimate, “quantum,” reality considered. 
+In strong RWR interpretations, when referring to this ultimate reality, “quantum,” or, in the first place 
+ 
+3 RWR interpretations on the Heisenberg postulate alone (defining all RWR interpretations), without assuming the 
+Bohr postulate, are possible. Conversely, the Bohr postulate need not be limited to RWR interpretations and is found 
+in realist interpretations of quantum phenomena and QM. 
+
+ 
+4 
+“reality,” is a term to which no concept we can form can be associated. Any such association is only 
+possible and necessary at the level of observation or phenomena. At this level, that of visible to thought or 
+even available to immediate human perception, thus defining quantum phenomena or quantum events, the 
+term “quantum” has (classical) physical concepts, such as discreteness or individuality, associated with it. 
+By the Heisenberg postulate, how quantum phenomena come about cannot, in RWR interpretations, 
+be represented by QM or QFT, but only predicted by it, in general probabilistically. QM or QFT, has, in 
+these interpretations, no physical connections apart from making these predictions, to either the ultimate 
+nature of reality responsible for quantum phenomena or, because they are described by classical physics, 
+to these phenomena themselves. Hence, in these interpretations, the capacity of the mathematics of QM or 
+QFT to predict the outcomes of quantum experiments, even if only probabilistically (which is, however, 
+in accord with the experimental evidence now in place), becomes in turn beyond knowledge or even 
+conception. We know how this mathematics works (how to use it), but we do not know and perhaps 
+cannot know or even conceive of why it works. Fortunately for us, however, it does work. 
+By contrast, in classical physics or relativity (special or general), the mathematical formalism 
+(ideally) represents, make visible to thought, the physical reality responsible for the phenomena 
+considered and connects, by continuous processes, these phenomena. The latter can, moreover, be 
+identified with the physical objects considered, because the interference of measuring instruments can be 
+neglected for all practical purposes. This identification is no longer possible in considering quantum 
+phenomena, in the constitution of which the role of measuring instruments is irreducible. Nobody has 
+ever seen a moving electron or photon. It is invisible to an observation and, in the RWR view, even 
+invisible to thought, is entirely beyond the reach of thought. It is only possible to observe traces, which 
+are visible even to our immediate sense perception and consciousness (such traces may also be “clicks” 
+that we hear rather than see) of their interactions with measuring instruments. These traces make it 
+difficult and, in strong RWR interpretations, impossible to reconstitute the ultimate nature of the reality 
+responsible for them, whether one sees this reality in terms of quantum objects or assumes, as I do here, 
+that a quantum object is an idealization applicable only at the time of measurement. (This assumption is 
+the content of the Dirac postulate.) Either way, this situation entails an unavoidable discrimination 
+between quantum objects and instruments, and hence phenomena, a discrimination that, according to 
+Bohr, “may indeed be said to form a principal distinction between classical and quantum-mechanical 
+description of physical phenomena” [Bohr 1935, p. 701].4 
+While adopting this structure of observation in quantum physics, the present interpretation further 
+stratifies it by the Dirac postulate, not found in Bohr. Bohr’s argumentation might be seen as, at certain 
+points, suggesting the Dirac postulate. Bohr, however, never formulated this type of postulate or the 
+corresponding view and appears to have always assumed that the concept of a quantum object is an entity 
+that exist independently, while still being beyond representation or even conception, by the Heisenberg 
+postulate. According to the Dirac postulate, the concept of a quantum object is only applicable at the time 
+of observation, but not to anything assumed to exist independently in nature. In Bohr’s interpretation in 
+ 
+4 The difference between phenomena and objects has its genealogy, in modern times (it had earlier precursors, even 
+in ancient Greek philosophy), in I. Kant’s distinction between objects as things-in-themselves in their independent 
+existence and phenomena as representations created by our mind, which may not correspond to the objects which 
+they are aiming to represent or to which they may be representationally unrelated at all [Kant 1997]. The latter is in 
+fact the case in considering quantum phenomena vis-à-vis quantum objects because quantum phenomena represent 
+classical physical objects observed in measuring instruments. As the strong RWR view, Bohr’s or the present view 
+is more radical than that of Kant. While Kant’s things-in-themselves are assumed to be beyond knowledge, they are 
+not beyond conception, at least a hypothetical conception, even if such a conception cannot be guaranteed to be 
+correct and is only practically justified in its applications [Kant 1997, p. 115]. By contrast, in the strong RWR view 
+what is practically justified is not a possible conception of the ultimate nature of reality responsible for quantum 
+phenomena, but the impossibility of such a conception, thus in precluding this reality from being visible to thought, 
+even hypothetically. No other justification than practical is possible by virtue of the impossibility of this conception. 
+The concept of a quantum object can of course be considered from alternative, including realist, perspectives. See, 
+for example, [Jaeger 2014], which offers a rigorous argument for such a concept.  
+
+ 
+5 
+all its versions, the ultimate reality responsible for quantum phenomena was associated with quantum 
+objects, eventually as independent RWR-type entities, different from quantum phenomena, defined by the 
+irreducible role of measuring instruments, the observable parts of which are described by classical 
+physics. A quantum object is, in Bohr view, a physical object responsible for the existence of a quantum 
+phenomenon, as an effect of the interaction between this object and a measuring instrument or some 
+(classical) object existing in nature that function as an instrument for us. Nothing, either built by us or by 
+nature, can be defined as an instrument apart from us in the present and, I would argue, in Bohr’s view. 
+As RWR type entities quantum objects were assumed by Bohr to be beyond conception, and hence could 
+not be assigned any properties, including h, even at the time of observation and measurement, as all 
+physical properties were only assignable to observables parts of measuring instruments, described 
+classically. As will be seen, it is possible in a quantum experiment to consider as the object under 
+investigation an object that also contains a classical part, such as the cat in the cat experiment, but this 
+composite object must still contain a properly quantum object for the observed phenomenon to be a 
+quantum phenomenon. QM would predict such observed properties of measuring instruments and only 
+them, rather than any properties of quantum objects. The Dirac postulate introduces a triple rather the 
+double, stratification into this situation, following [Plotnitsky 2021a, 2022a,b]. The ultimate RWR reality 
+responsible for quantum phenomena is an idealization assumed to exist independently of our interactions 
+with it, and thus independently of observation. By contrast, the concept of a quantum object, elementary, 
+such as a photon or electron, or composite (possibly macroscopic) is an idealization that, while still of the 
+RWR-type, only applies at the time of an observation. An observation becomes a creation of a quantum 
+phenomenon by the interaction between the ultimate RWR-type reality, and the instrument we use, and 
+the capacity of our thought to observe the phenomena thus created, which also allows enables one to 
+apply the concept of quantum object, by the Dirac postulate only after an observation has taken place. In 
+all three cases—the (independent) ultimate RWR reality responsible for quantum phenomena, quantum 
+objects, and quantum phenomena—one only deals with idealizations created by our thought. 
+The reason for using the designation “Dirac postulate” is that, while, unlike the Heisenberg postulate 
+by Heisenberg and the Bohr postulate by Bohr (even if without using these designations as such), this 
+postulate was not considered by Dirac himself, it may be seen as having emerged from Dirac’s famous 
+equation for a relativistic electron. While originally written for an electron, Dirac’s equation 
+ 
+!𝛽𝑚𝑐! +	 ' 𝛼"𝑝"
+#
+"$%
+𝑐* 𝜓(𝑥, 𝑡) = 𝑖ℏ 𝜕𝜓(𝑥, 𝑡)
+𝜕𝑡
+ 
+ 
+ 
+(I4 is the identity matrix) 
+ 
+ 
+revealed itself to an equation for both the (free) electron and the (free) positron, including their spins, 
+which the equation contains automatically, in contrast to QM, where predicting the spin of an electron 
+needs to be handled separately, via Pauli matrices combined with Schrödinger’s equation. Dirac’s 
+equation reflected and, as it happened, led to the discovery that a different particle (in the present view, 
+defined in terms of effects observed in measuring instruments) can be registered in a single experiment: 
+the initial observation can register an electron, while the next one a positron, or a photon, or an electron-
+positron pair, with the probabilities defined by the same equation. Once one moves to still higher 
+energies, the panoply of possible outcomes becomes even greater. In QED, one only deals with electrons, 
+positrons, and photons; in QFT, depending how high the energy is, one can find any known elementary 
+particle or combination, that is, the corresponding effects will be registered. Accordingly, it is reasonable 
+to apply the concept of a quantum object (still as an RWR-type entity) exclusively at the time of 
+ 
+ai
+2 = b2 = I4
+ 
+aib + bai = 0
+aia j + a jai = 0
+
+ 
+6 
+observation. There are, however, reasons to adopt this view in low-energy (QM) quantum regimes, 
+including in order more effectively to interpret quantum conundrums, such as that of the double-slit 
+experiment, a paradigmatic and, arguably, the most famous quantum experiment [Plotnitsky 2022b].5  
+Do quantum phenomena or QM require the Dirac postulate, or the Heisenberg and Bohr postulates? It 
+would be difficult to argue such a case, and it is not my aim to do so. My only claim is the logical 
+consistency of the interpretations, such as the one adopted here, grounded in these postulates, and their 
+accord with the experimental evidence currently available. As I said, new experimental evidence can 
+change the present situation of fundamental physics, just as then new evidence changed its situation 
+around 1900, leading to Planck’s discovery of quantum theory.   
+The next section outlines the (RWR-type) interpretation adopted in this article, cast in terms of the 
+relationships between visible and invisible to thought. Section 3 discusses, by way of the bridge to the cat 
+experiment, the letter exchange between Schrödinger and Bohr (at the time Schrödinger’s work on his 
+paper containing the cat paradox) concerning the use of classical concepts in quantum measurement, thus, 
+essentially the Bohr postulate. Section 4 considers Schrödinger’s cat experiment from the (RWR) 
+perspective established by the preceding analysis. The conclusion offers philosophical reflections on the 
+role of metaphysics in physics, via the relationships between visible and invisible to thought. 
+ 
+2. Reality without realism, and visible and invisible to thought in fundamental physics 
+ 
+This section outlines the RWR view of quantum phenomena and quantum theory, a view cast in  
+terms of the relationships between visible and invisible to thought. For simplicity, I shall primarily 
+discuss QM, only briefly referring to QFT, although my argument applies to and can be further supported 
+by QFT. I speak, more generally, of “the RWR view” because it can lead to various interpretations. These 
+interpretations share the Heisenberg postulate, defining RWR interpretations, but beyond being either 
+ 
+5 One can translate the argument of this article into quantum-informational terms. In the present view, information is 
+human. Nature has no information, only we do, possibly about nature, or what we assume to be nature. All 
+information obtainable in quantum experiments is contained in the data observed in measuring instruments, 
+described classically, by the Bohr postulate. Hence, this information qua information is classical, Shannon 
+information (measured in classical bits), and as such is visible to thought and communicable unambiguously given 
+the mathematical nature of Shannon information (as opposed to other forms of information with a semantic content, 
+which may allow for ambiguity). However, this information and its organization, as manifested in quantum 
+experiments cannot be predicted or processed by classical means. The emergence of this information requires the 
+assumption of quantum objects or in the present view, by the Dirac postulate, an RWR type reality ultimately 
+responsible for quantum phenomena and the use measuring instruments capable on interacting with this reality, and 
+a theory, such as QM, different from classical theories, that is capable of handling this information. In this sense, 
+while all actual information obtained in experiments is classical, one can speak, as is common, of “quantum 
+information.” In the present view, the “quantum,” as the ultimate reality responsible for quantum phenomena, in 
+only a particular way, defined by out interaction with nature, to create and communicate classical information, 
+which cannot be predicted by classical physics (or relativity) and the structure of which cannot be generate by 
+classical objects and processes. Dealing with quantum information requires quantum information science, a vast 
+subject of its own, including as concerning its impact on quantum foundations. Quantum information theory brings 
+new features to the differences between classical and quantum information in relation to QM or QFT, because at this 
+stage quantum information theory is primarily concerned with discrete variables, physically represented by spin, 
+rather than continuous variables. While there are classical and quantum versions of continuous variables, spin is a 
+quantum variable, corresponding to a strictly quantum aspect of nature, and has no classical analog. By the same 
+token, a spin may be seen as reflecting something in nature strictly beyond our thought’s capacity to form a 
+conception of, a strictly RWR-type entity. I am indebted to G. M. D’Ariano for exchanges concerning of the visible 
+and communicable or (his preferred term) “sharable” nature of classical information. I am not claiming that he 
+subscribes to the argument of this article or all of this argument, which builds on [Plotnitsky 2021a,b, 2022a,b], as 
+concerns the RWR view and the idea of the invisibility to thought, as extending to all thought, including 
+mathematical and physical thought, rather than only our immediate (conscious) phenomenal intuition or 
+visualization. In general, this article’s argument is independent of quantum information theory. 
+
+ 
+7 
+weak or strong RWR interpretations, some of them may contain additional postulates, such as the Bohr 
+postulate or the Dirac postulate. Thus, while Bohr’s interpretation and the present interpretation both 
+assume the Bohr postulate, only the present interpretation assumes the Dirac postulate.  
+The philosophical position grounding the present interpretation implies that modern physics, as a 
+mathematical-experimental science, contains two forms of thinking, which may be designated as 
+“classical” and “quantum” in view of their respective origins in classical and quantum theory. Both 
+assume the physical reality they consider to exist independently of our existence and thinking as humans, 
+but each treats this reality differently as concerns the ultimate nature of this reality, assumed to be 
+representable and, thus, visible to thought in classical thinking, and to be beyond not only representation 
+but also conception and, thus, invisible to thought in quantum thinking. I speak of the ultimate nature of 
+the reality considered or (for the sake of economy) just the ultimate reality considered, because quantum 
+thinking assumes that classical thinking applies at some levels of the reality considered, specifically, by 
+the Bohr postulate, that of the observable parts of measuring instruments. Quantum thinking also involves 
+classical thinking. In adopting only classical thinking one assumes it to apply at all levels of physical 
+reality, without allowing for quantum thinking. These two forms of thinking are as follows: 
+(1) Classical thinking, which is essentially a realist thinking, deals with a form of reality that is visible 
+to thought, as what can be perceived, imagined, visualized, represented, known, conceptualized, and so 
+forth, and as such allows one to have  statements or images of this reality that can, at least in principle, be 
+communicable unambiguously, as is necessary for the practice of science, as constituted now;  
+(2) Quantum thinking, which is essentially RWR thinking, contains classical thinking but also 
+assumes the existence of a form of reality, as the ultimate reality considered, that is no longer available to 
+classical thinking and, as such, is invisible to thought, as what  cannot be perceived, imagined, visualized, 
+represented, known, conceptualized, and so forth, which also means that nothing about it can be 
+communicated unambiguously, if at all, apart from the claim that it is beyond the reach thought.6 
+Thus, in the case of this (RWR) form of reality, quantum thinking, divorces the term “reality” from 
+any possible concept associated to it, making it akin to a mathematical symbol, like R or X, which could 
+have been used instead of the word reality here. “Reality without realism” or RWR functions in this way 
+as well. I emphasize that quantum thinking also assumes forms of reality, such as that observed, as 
+phenomena, in quantum experiments, that handled by classical thinking by the Bohr postulate and as such 
+is visible to thought. The very existence of any RWR-type reality is inferred from certain configurations 
+(such those defining the data observed in quantum experiments) of classical reality. 
+Classical and quantum thinking are not the same as physical theories or interpretations using either 
+thinking, because such theories contain additional features. Thus, while different theories, both classical 
+physics and relativity, special or general, conform to classical thinking, which quantum theory does not at 
+least in RWR interpretations. It is true that special relativity severely limits the capacity of our immediate 
+phenomenal intuition to represent or visualize the kinematic used by the theory. These qualifications, 
+however, do not prevent this kinematic from being visible to thought and allow for a realist treatment, 
+because relativity, special or general, represents all reality considered in it in terms of suitably 
+mathematized physical concepts, just as does classical physics. In considering classical physical 
+phenomena or (they can, again, be identified with each other in classical physics or relativity) objects the 
+role of both h and c can be disregarded and is by classical mechanics; in considering relativistic 
+phenomena or (they can, again, be identified with each other) objects, only h can be disregarded; and in 
+considering quantum phenomena or (they can no longer be identified with each other) objects, h be taken 
+ 
+6 By thus relating the concepts of “visible to thought” and “unambiguously communicable,” specifically, by means 
+of language (although language is not the only form of unambiguous communication, which can be visual, for 
+example), speaking of visible and invisible to thought suggests a connection to J. S. Bell’s title Speakable and 
+Unspeakable in Quantum Mechanics [Bell 2004], a collection of his writings, primarily on QM. The philosophical 
+position adopted in this article is, however, opposite of that of Bell, which also leads Bell to his discontent with 
+Bohr’s view, including the Bohr postulate, and with QM in the first place, as discussed in [Plotnitsky 2021b, 2022b].  
+
+ 
+8 
+into account, and in high-energy relativistic (QFT) regimes, c must be taken into account as well.7 This 
+formulation is different from saying that c is assumed to be infinite in classical physics and h equal to 
+zero in classical physics and relativity. In particular, classical mechanics need not be, and in the present 
+view is not, assumed to be the limit of QM by putting h equal to zero or express in this way Bohr’s 
+correspondence principle (explained below). These are, in the present view, two different theories, 
+dealing with two different types of objects, even though both are ultimately composed of quantum objects 
+or (in the present view) the same ultimate reality: classical mechanics is not a special (limit) form of QM 
+and classical objects are not a special type of quantum objects, although quantum objects can sometimes 
+be treated classically, which claim, however, requires important qualifications explained below. In the 
+present interpretation, moreover, quantum objects are only defined at the time of observation by the Dirac 
+postulate, which does not apply to classical objects, defined independently of observation. 
+Quantum thinking emerged in quantum theory in RWR interpretations in view of the nature of 
+quantum phenomena, assumed to be the effects on the interactions between the ultimate reality 
+responsible for these phenomena and suitable measuring instruments. These phenomena, or rather 
+numerical data they contain, are predicted by quantum theory, QM or QFT, without, in RWR 
+interpretations, representing the ultimate reality responsible for them. QM or QFT does not represent 
+these effects either. They are, by the Bohr postulate, represented by classical physics, which enables them 
+to be as unambiguously communicable, as is the mathematics of QM or QFT. Along with the predictive 
+capacities of both theories, the possibility of this unambiguous communication and, in this sense (a 
+qualification discussed below), objectivity make these theories conform to “the basic principles of 
+science,” as stressed by Bohr (e.g., [Bohr 1935, p. 700; Bohr 1987, v. 2, pp. 67-68, v. 3, p. 7]). On the 
+other hand, classical theories cannot predict these effects. It follows that both types of theories are 
+necessary in fundamental physics, including quantum theory, because, by the Bohr postulate, quantum 
+phenomena are represented by classical physics, with adding special relativity in high-energy (QFT) 
+regimes, while they can only be predicted by quantum theory, which in RWR interpretations represents 
+neither these phenomena not, by the Heisenberg postulate how they come about. In fact, classical physics 
+is necessary for describing measuring instruments in relativity as well, specifically in all measurements 
+within each local reference frame, even though the instrument are subjects to the relativistic laws of 
+motion. While philosophically classical, that is, conforming to classical thinking in the sense defined here, 
+general relativity is a separate part of fundamental physics as currently constituted. It may sometime play 
+a role in dealing with quantum phenomena, keeping in mind that the emergence of quantum phenomena 
+considered thus far does not involve gravity as a such, as we do not have a quantum theory of gravity.8 
+Classical theories are grounded, in Bohr’s words, “the idea [and hence the assumption] that the 
+phenomena concerned may be observed without disturbing them appreciably,” which enables one to 
+identify these phenomena with the objects considered [Bohr 1987, v. 1, p. 53]. This assumption no longer 
+appears possible in considering quantum phenomena, empirically and hence regardless of interpretation. 
+In fact, it may not be rigorously possible even in classical physics and relativity, insofar as all phenomena 
+considered are still created by our thought, which is a product of our bodies and brains, as experimental 
+technologies created by nature [Plotnitsky 2021a, pp. vii-xxiv]. However, the assumption that “the 
+phenomena concerned may be observed without disturbing them appreciably” is workable in these 
+ 
+7 See, however, note 1 above. 
+8 It may in principle be contended that, even if necessary for the description of the observable phenomena in 
+quantum or relativistic measurements, classical physics is not a separate theory, for example, by assuming or 
+arguing that ultimately all physical objects considered are quantum. In the present view, however, classical physics 
+is a necessary separate part of fundamental physics. It may not, as such, deal with the ultimate constitution of matter, 
+but I would argue, it is unavoidable in considering many, even most, macroscopic phenomena, and, again, even in 
+dealing with fundamental, such as quantum, physics, where classical physics is unavoidable in dealing with 
+observation and measurement. Given that gravity plays no role quantum (or classical) phenomena considered in this 
+article, I put it aside, although it may ultimately bear on the set of questions posed here. 
+
+ 
+9 
+theories for all practical purposes. By contrast, in Bohr’s or the present interpretation, quantum theory, 
+QM or QFT, are defined by dealing with the combination of fours features:  
+ 
+(1) the ultimate, “quantum,” reality responsible for quantum phenomena, a reality invisible to thought 
+by the Heisenberg postulate and commonly, including by Bohr, identified with quantum objects, 
+which are, however, by the Dirac postulate, defined (still as RWR-type entities and thus invisible to 
+thought) only at the time of observation in the present interpretation;  
+(2) observational technology, commonly understood as comprised of measuring instruments;  
+(3) observed phenomena, created by the interaction between quantum objects and measuring 
+instruments, phenomena that are always visible to thought and even to our immediate phenomenal 
+perception, while the numerical data observed and, in the first place, the observable parts of 
+measuring instruments, are described by classical physics by the Bohr postulate; 
+(4) the mathematical formalism of QM (cum Born’s rule), probabilistically or statistically predicting 
+the outcomes of quantum experiments, as observed, via measuring instruments, in quantum 
+phenomena, without representing the ultimate reality responsible for them by (1).  
+  
+Measuring instruments, it follows, contain both classical, observable, strata of reality and unobservable, 
+ultimately invisible-to-thought, quantum strata of reality, which enables this interaction.   
+The reasons for my emphasis on visible and invisible, extended to the idea of visible and invisible to 
+thought (essentially thinkable and unthinkable) are both conceptual and historical. Conceptually, our 
+capacity for visualizing the world, which is also related to the neurological functioning of our brain (about 
+60% of which is dealing with vision), is crucial to our thought. This capacity has shaped classical physics, 
+as a mathematical refinement of the world we observe, but it was defeated, first by special relativity, the 
+kinematic of which is beyond it, and then, more radically, by quantum theory, which brought into physics 
+that which is invisible to thought altogether, is beyond thought. 
+Historically, this emphasis follows Bohr’s appeal to the impossibility of visualization of the ultimate 
+responsible for quantum phenomena, defined as effects of the interaction between this reality and our 
+agencies of observation. Bohr’s use of visualization and its avatars was in part shaped by the German 
+term for intuition, Anschaulichkeit, which etymologically relates to what is phenomenally visualizable. 
+Even before (albeit only by a few months) Heisenberg’s discovery of QM, based on “abandoning the 
+ordinary spacetime description,” and hence on an RWR type view, at least in its weak form [Bohr 1987, 
+v. 1, p. 48], Bohr said: 
+ 
+I am forcing myself these days with all my strength to familiarize myself with the mysticism of nature and am 
+attempting to prepare myself for all eventualities, indeed even for the assumption of a coupling of quantum 
+processes in separated atoms. However, the cost of this assumption are so great that they cannot be estimated 
+within the ordinary spacetime description. [A Letter to Heisenberg, April 18, 1925, Bohr 1972–1996, v. 5, pp. 
+79–80, p. 237] 
+ 
+In a letter to Born, a few days later, he added: 
+ 
+[Quantum experiments] preclude the possibility of a simple description of the physical occurrences [at the 
+quantum level] by means of visualizable pictures . . . [S]uch pictures are of even more limited applicability than 
+is ordinarily supposed. This is of course almost a purely negative assertion, but I feel that . . . one must have 
+recourse to symbolic analogies to an even greater extent than hitherto. Just recently I have been racking my 
+brain to dream up such analogies. [Letter to Born, 1 May 1925, Bohr 1972–1996, v. 5, p. 311] 
+ 
+The word “mysticism” will soon disappear from Bohr’s writings, replaced by an emphasis on QM as a 
+rational theory, free from any “mysticism incompatible with the true spirit of science” [Bohr 1937, p. 83, 
+Bohr 1987, v. 2, p. 63]. By referring to “the assumption of a coupling of quantum processes in separated 
+atoms,” the statement also captures the core of the dilemma later posed by the Einstein-Podolsky-Rosen 
+(EPR) experiment [Einstein et al 1935]. Bohr links this dilemma to the impossibility of visualization, 
+
+ 
+10 
+ultimately making how what is observed there, or in any quantum experiment, come about invisible to 
+thought. What makes Bohr’s statements remarkable is that they were made in 1925, 10 years before 
+EPR’s article. It is true that Einstein brought up related considerations in his exchanges with Bohr already 
+in 1927, which was, however, still two years away in 1925, and followed the invention of QM [Bohr 
+1987, v. 2, pp. 41-58].  
+There are numerous invocations of the limits and in effect the impossibility of visualization 
+throughout Bohr’s writing, with an increasing emphasis, ultimately amounting to dealing with invisible to 
+thought, even if without using this language.9 To cite some key passages, proceeding chronologically: 
+ 
+* “In atomic problems we have apparently met with such a limitation of our usual means of visualization” 
+(1925) [Bohr 1987, v. 1, p. 51]; 
+* “On the whole, it would scarcely seem justifiable, in the case of the interaction problem, to demand a 
+visualization by means of ordinary space-time pictures. In fact, all our knowledge concerning the internal 
+properties of atoms is derived from experiments on their radiation or collision reactions, such that the 
+interpretation of experimental facts ultimately depends on the abstractions of radiation in free space, and free 
+material particles. Hence, our whole space–time view of physical phenomena, as well as the definition of 
+energy and momentum, depends ultimately upon these abstractions” (1927) [Bohr 1987, v. 1, p. 77]; 
+* “The resignation with regard to the desires for visualization which gives our whole language its character, 
+to which we are compelled by the situation [in QM]” (1929) [Bohr 1987, v. 1, p. 98]; 
+* “Indeed, only a conscious resignation of our usual demands of visualization and [classical] causality] was 
+it possible to make Planck’s discovery fruitful in explaining the properties if the [chemical] elements of the 
+basis of our knowledge of the building stones of atoms” (1929) [Bohr 1987, v. 1, p. 108]; 
+* “We must only be prepared for the necessity for the necessity of ever extending abstraction from our 
+customary demands for a directly visualizable description of nature. Above all, we might expect new surprises 
+in the domain [of QED and QFT] where the quantum theory meets with the theory of relativity and where 
+unsolved difficulties still stand as hindrance to a complete fusion of the extension of our knowledge and the 
+expedients to account for natural phenomena which these theories have given us” (1929) [Bohr 1987, v. 1, p. 
+108]; 
+* The fundamental indeterminacy which we meet here [in Heisenberg’s uncertainty relations] may, as the 
+writer [Bohr] has shown, be considered as a direct expression of the absolute limitation of the applicability of 
+visualizable conception in the description of [the ultimate reality of] atomic phenomena. … The resignation as 
+regards visualization and [classical] causality, to which we are thus forced in our description of atomic 
+phenomena, might well be regarded as a frustration of the hopes which formed the [Democritean] starting-point 
+of atomic conceptions. Nevertheless, from the present standpoint of the atomic theory, we must consider this 
+very renunciation as an essential advance in our understanding” (1929) [Bohr 1987, v. 1, pp. 114-115]; 
+* “Only [the] limitation of our visualizable conception of motion, which is characteristic of quantum 
+theory, enables us to understand how electrons can make their way between the metal atoms in the wire” (1929) 
+[Bohr 1987, v. 1, pp. 118]; 
+* “We must be prepared for a more comprehensive generalization of the complementary mode of 
+description [in QFT] which will demand a still more radical renunciation of the usual claims of so-called 
+visualization” (1937) [Bohr 1937, p. 88]; 
+* “The extent to which renunciation of the visualization of atomic phenomena [technically, how they come 
+about] is imposed upon us is strikingly illustrated by the following example to which Einstein very early called 
+attention and often has reverted [in effect that of the alternative, complementarity, behavior of photon, 
+depending up the two alternative set-up of the experiment, in essence equivalently to the double slit 
+experiment” (1949) [Bohr 1987, v. 2, p. 51];  
+* “… [T]he ingenious formalism of quantum mechanics … abandons pictorial representation and aims 
+directly at the statistical account of quantum processes” (1951) [Bohr 1998, p. 152]; 
+* “Indeed, renouncing pictorial description of electronic constitution of the atomic system and only making 
+use of empirical knowledge of threshold and binding energies of molecular processes, we can within a wide 
+field of experiences treat the reaction of such systems by ordinary chemical kinetics, based on the well-
+established laws of thermodynamics” (1962) [Bohr 1987, v. 3, p. 25]. 
+ 
+9 I have considered this aspect of Bohr’s argumentation in [Plotnitsky 2012, 2016, 2021a]. 
+
+ 
+11 
+* “Certainly the issue [raised by the EPR experiment] is of a very subtle character and suited to emphasize 
+how far, in quantum theory, we are beyond the reach of pictorial visualization” [Bohr 1987, v. 2, p. 59; 
+emphasis added].  
+ 
+The statement closing my traversal is Bohr’s 1949 comment on the EPR experiment. As already Bohr’s 
+1925 statements cited above make clear, for Bohr the cost of “the assumption of a coupling of the 
+processes in separated atoms,” at stake in the EPR experiment (which deals with two spatially separated 
+quantum objects) was the impossibility of “the ordinary spacetime description” of how phenomena is 
+observed there and, as such, described in space and time. The EPR experiment and, in fact, all quantum 
+experiments “preclude the possibility of a simple description of the physical occurrences [of phenomena 
+considered] by means of visualizable pictures” [Letter to Born, 1 May 1925, Bohr 1972–1996, v. 5, p. 
+311]. Bohr, thus, assumed that such may be the case well before EPR’s article, which might be one of the 
+reasons why he thought that EPR’s experiment didn’t contain anything essentially new. He might not 
+have been entirely right on this point, given the role of entanglement and correlations brought about by 
+the EPR experiment. EPR, however, did not consider these concepts either. That of entanglement was 
+introduced by Schrödinger in response to EPR’s paper, including in the cat-paradox paper [Schrödinger 
+1935, p. 161]. Correlations became prominent even later. Be it as it may on that score, Bohr argued that, 
+although “the issue [raised by the EPR experiment] is of a very subtle character and suited to emphasize 
+how far, in quantum theory, we are beyond the reach of pictorial visualization,” EPR’s argument does not 
+demonstrate, as EPR claimed, either the incompleteness of QM or else its nonlocal nature (in the sense of 
+allowing an instantaneous action at a distance). My main point at the moment is Bohr’s view that the 
+ultimate reality responsible for quantum phenomena is invisible to thought, a reality defined here as a 
+reality without realism (RWR), while quantum phenomena, as, by the Bohr postulate, observed 
+classically, are visible to thought or in fact available to our immediate phenomenal perception. 
+It might be useful to briefly consider, as a simple representative example, which illustrate and will 
+help to guide my discussion of RWR interpretations, how predicting the polarization of a photon appears 
+in these interpretations. There are two possible outcomes of measurement (after the initial preparation): 
+for example, the horizontal state x and the vertical state 𝑦, observed classically by the Bohr postulate. In 
+RWR interpretations, one could not say, as it is said sometimes, that before it is measured, the photon is 
+(or is prepared) in a superposition of two physical states, and in the present view, moreover, the very 
+concept of a photon, while it cannot be observed as such (only the corresponding effect in measuring 
+instruments can) is only applicable at the time of observation by the Dirac postulate. The wave function 
+allowing one to predict either physical state x or y is written as |𝜓⟩ = 𝛼|𝑋⟩ + 𝛽|𝑌⟩ with probability 
+amplitudes of |𝜓⟩ associated with state vector |𝑋⟩ given by 𝛼 and |𝑌⟩ given by 𝛽. In a random 
+experiment, the probability of the photon, when its polarization will be measured, to be horizontally 
+polarized is	|𝛼|! and to be vertically polarized is |𝛽|! (by Born’s rule). (Actual predictions will involve h, 
+which does not appear in these abstract notations, but will once there are properly unfolded to make actual 
+predictions possible.) That, however, need not, and in the RWR view does not, mean that |𝜓⟩ = 𝛼|𝑋⟩ +
+𝛽|𝑌⟩ represents the photon in a superposition of two physical states, x and y, as nothing can be said, by 
+the Heisenberg postulate, concerning what happens between observations in the RWR view. Only the 
+mathematical state vectors, designated |𝑋⟩ and |𝑌⟩ (in capital letters), in the Hilbert space used, are in a 
+linear (mathematical) superposition, with given amplitudes, and not quantum objects, let alone the 
+outcomes of experiments.  
+QM, then, in Bohr’s or the present interpretation, does not represent either, by the Heisenberg 
+postulate, the physical emergence of quantum phenomena or, by the Bohr postulate, the observed 
+quantum phenomena, represented by classical physics. The only relationship between quantum 
+phenomena and QM in these interpretations is defined by the fact that QM predicts, in general 
+probabilistically, the outcomes of quantum experiments. The probabilistic (or statistical) nature of these 
+predictions is in accord with what is experimentally observed because no other predictions concerning 
+such outcomes are in general possible, as concerns kinematic or dynamical variables, such as the position 
+or the momentum, or the direction of spin. Such quantities as mass, charge, or spin are invariant. (There 
+
+ 
+12 
+are certain specific situations, such those of the EPR-type experiments, where exact predictions are 
+ideally possible, bur with important qualifications, explained below.) These predictions are, moreover, 
+only possible by using rules added to the formalism rather than being part of it, such as Born’s rule, 
+which relates (essentially, by using complex conjugation) complex quantities of the formalism to real 
+numbers corresponding to the probabilities of quantum events. Arguments to the effect that such rules are 
+inherent in the formalism have been offered, but they are not commonly accepted. The Heisenberg 
+postulate remains the grounding postulate of all RWR interpretations.  
+The concept itself of reality-without-realism is based in more general concepts of reality and 
+existence, assumed here to be primitive concepts and not given analytical definitions. By “reality” I refer 
+to that which is assumed to exist, without making any claims concerning the character of this existence or 
+reality, claims that, as explained below, define realism. The absence of such claims allows one to place 
+this character beyond representation or even conception, which defines the RWR view. I understand 
+existence as a capacity to have effects on the world. The assumption that something is real, including of 
+the RWR-type, is made, by inference, on the basis of such effects. The RWR view is grounded in the 
+assumption that observable (either immediately or via a mediation of observational instruments) effects of 
+physical reality allow for a representation these effects but not necessarily a representation (the weak 
+RWR view) or even a conception (the strong RWR view) of how these effects are possible. The latter 
+representation or conception may not be possible and is not in RWR interpretation in the case of the 
+ultimate reality responsible for quantum phenomena, making this reality invisible to thought. It follows 
+this reality or nature or matter, in the first place, are assumed to exist independently in the first place, just 
+as it would be in classical theories or relativity or other realist theories, where, however, all strata of 
+nature or matter considered are assumed to be visible to thought. The assumption of the independent 
+existence of nature or matter essentially amounts to the assumption that it has existed before we existed 
+and will continue to exist when we will no longer exist. Even this assumption, which still belongs to 
+thought, has been challenged, even to the point of denying that the ultimate nature of reality is material 
+rather than mental. Plato is the most famous ancient and Bishop Berkeley as the most famous modern 
+case of this questioning. Such views are useful in suggesting that any conception of how anything exists, 
+or even that it exists, including when assumed to be unavailable to human thought, still belongs to 
+thought. It need not follow, however, that something which such concepts represent or to which they 
+relate otherwise, possibly placing it beyond representation or even conception, does not exist. 
+Quantum phenomena would not be possible without our interaction with nature by means of 
+experimental technology and our specific (human) ways of observing phenomena and thinking about 
+them, which makes them visible to thought or even to our immediate phenomenal perception or 
+consciousness. (Not all perceptions or forms of thought are conscious.) In RWR interpretations, nature 
+has no quantum objects; and when it comes to its ultimate workings, at least those responsible for 
+quantum phenomena, nature is beyond knowledge or, in strong RWR interpretations, conception. Hence, 
+the term “workings,” “nature,” or “existence” would not ultimately apply either, any more than any other 
+terms or concepts. Modern physics gave us new, mathematical-experimental, means of dealing with the 
+world by interacting with it by means of experimental technology and mathematics (as a form of thought). 
+In the present view, however, it gave us no more than such means, even in classical physics or relativity, 
+where the assumption that the theory actually (ideally) represent nature is workable for all practical 
+purposes. In any RWR interpretation, the concept of a quantum object is an idealization created in 
+response to our interactions with nature by means experimental technology resulting in quantum 
+phenomena. The present interpretation goes further by assuming the Dirac postulate, which makes the 
+concept of quantum object an idealization (of the RWR type) only applicable at the time of observation. 
+Importantly, however, the present (strong RWR) interpretation does not a uniform or otherwise 
+unified character of the ultimate, RWR-type, reality considered in QM or QFT, a character only 
+manifesting itself differently in quantum experiments. This assumption is in conflict with strong RWR 
+interpretations, which preclude any conception of this reality and, hence, that of its unity or oneness, 
+uniform or not. While each time unknowable or even unthinkable, invisible to thought, an RWR-type 
+reality is assumed to be each time different. This is what makes each quantum phenomenon, as an effect 
+
+ 
+13 
+of this reality, individual and unrepeatable, unique, manifesting the unique, but still inconceivable, nature 
+of the reality ultimately responsible for it each time one encounters this reality through its effects. One 
+can always repeat the setup of a given measurement, because this setup can be classically controlled. Not 
+so, however, as concerns the outcome of this repeated measurements. Such outcomes are ideally the same 
+and are ideally predictable exactly in in classical or relativistic experiments dealing with individual or 
+simple systems, with probability only entering when these systems have a great mechanical complexity, 
+as in classical statistical physics or chaos theory. By contrast, these outcomes will in general (apart from 
+special cases) be different in identically prepared quantum experiments, no matter how elementary the 
+quantum object considered. As explained below, while possible, even preparing a given state, say, that of 
+a “spin-up,” as manifested in the corresponding measurement, cannot in general be done in a single 
+preparation, but only by post-selecting the required preparation. 
+ A brief outline of realist thinking may help to sharpen the nature of the RWR view. Realist thinking 
+is manifested in the corresponding theories, commonly representational in character. Such theories aim to 
+represent the reality they consider, in modern, post-Galilean, physics primarily by mathematized models, 
+suitably idealizing this reality. It is even possible to aim, including in quantum theory, for a strictly 
+mathematical representation of this reality apart from physical concepts, at least as they are ordinarily 
+understood, say, in classical physics or relativity. It is also possible to assume an independent structure 
+(defined by properties and relationships among them) of the reality considered, while admitting that it is 
+either (A) not possible to represent this architecture or (B) even to form a rigorously specified concept of 
+it, either at a given moment in history or even ever. Under (A), a theory that is merely predictive could be 
+accepted for lack of a realist alternative, usually with the hope that a future theory will do better by being 
+a representational theory. Einstein and, often following him, others held this type of view of QM. What, 
+then, grounds realism most fundamentally is the assumption that the ultimate constitution of reality 
+possesses properties and the relationships between them, or, as in (ontic) structural realism, just a 
+structure, the more elemental constituents of which are not defined in terms of properties [Ladyman 
+2016]. Such properties, relationships, or structures may either be ideally represented and hence known, or 
+be unrepresentable or unknown or even unknowable, but are still conceivable, usually with a hope that 
+they will be eventually so represented. Most realist theories are representational. In considering physics, 
+the concept of realism just outlined is often called “scientific realism.” However, this outline would apply 
+to most forms of realism in science or philosophy. It does not cover all forms of realism, which would be 
+impossible. I shall also refer, as is common, to realist theories as ontological.10 
+Thus, classical mechanics (used in dealing with individual objects and small systems, apart from 
+chaotic ones), classical statistical mechanics (used in dealing, statistically, with large classical systems), 
+chaos theory (used in dealing with classical systems that exhibit a highly nonlinear behavior), or 
+relativity, special and general, are realist theories. While classical statistical mechanics does not expressly 
+represent the overall behavior of the systems considered because their mechanical complexity prevents 
+such a representation, it assumes that the individual constituents of these systems are represented by 
+classical mechanics. In chaos theory, which, too, deals with systems consisting of large numbers of 
+atoms, one assumes a mathematical representation of the behavior of these systems. Relativity posed 
+major, even insurmountable, difficulties for our immediate spatiotemporal phenomenal intuition, because 
+the relativistic law of addition of velocities (defined by the Lorentz transformation) in special 
+relativity,	𝑠 =
+!"#
+$"(!#/')), for collinear motion (c is the speed of light in a vacuum), runs contrary to any 
+possible intuitive conception. Our phenomenal intuition cannot conceive of, visualize, this kind of 
+motion, thus, making this concept of motion no longer a mathematical refinement of a daily sense of 
+motion as the concept of motion is in classical physics. Relativity was the first physical theory that 
+ 
+10 Although the terms “realist” and “ontological” sometimes designate more diverging concepts, these terms are 
+commonly close in their meaning and will be used, as adjectives, interchangeably here. I shall adopt “realism,” as a 
+noun, as a more general term and refer by an “ontology” to the representation or conception of the reality considered 
+by a given theory. Another, relatively common, term for realist theories, sometimes with additional specificities (not 
+important for my argument here), is “ontic,” coming, as does ontological, from the ancient Greek on (Being).  
+
+ 
+14 
+defeated our ability to form a phenomenal visualization of an elementary individual physical process, 
+although the concept of (classical) field in classical electromagnetism already posed certain complexities 
+in this regard. Bohr did not miss this point: “I am glad to have the opportunity of emphasizing the great 
+significance of Einstein’s theory of relativity in recent development of physics with respect to our 
+emancipation from the demands of visualization” [Bohr 1987, v. 1, p. 115-116]. Emancipation! This is 
+not a casual word choice, rarely, if ever, found in Bohr. Special, as well as general, relativity, however, 
+still offer mathematically idealized conceptual representations of the physical reality they consider, and in 
+this respect allowed this reality still to be visible to thought. 
+Quantum physics brought this emancipation to a more radical level, that of the invisible to thought, 
+mathematically reflected in the Hilbert space (or analogous) formalism over ℂ. Event this mathematics 
+itself poses difficulties of seeing this formalism as representing anything physical in space and time, 
+represented in all physical theories thus far as concepts over ℝ, to which, the formalism relates, with the 
+help of Born’s rule which converts complex quantities into real one, by means of probabilistic predictions 
+of the outcome of measurement. As Bohr noted in 1927 (before the Hilbert space version of the formalism 
+was introduced, while the role of ℂ in the formalism was already in place), “the symbolic [rather than 
+representational] character of Schrödinger’s method appears not only from the circumstance that its 
+simplicity, similarly to that of the matrix theory [of Heisenberg], depends essentially upon the use of 
+imaginary arithmetic quantities” [Bohr 1987, v.1, p. 76]. 
+All theories just mentioned, apart from QM, are based in the assumption, defining all 
+“epistemologically classical theories,” as they may be called (which designation would apply to relativity 
+as well), that one can observe the phenomena considered without disturbing them. As a result, these 
+phenomena can be identified with the corresponding physical objects and their independent behavior and, 
+ideally, represent this behavior and predict it, in the case of individual or simple systems, ideally exactly, 
+by using this representation. This is no longer possible in dealing with quantum phenomena, regardless of 
+interpretation, and hence also in realist interpretations of QM, or alternative theories, such as Bohmian 
+mechanics, of quantum phenomena. On the other hand, this situation opens the possibility of RWR 
+interpretations of QM or QFT. The irreducible role of measuring instruments in the constitution of 
+quantum phenomena grounded Bohr’s interpretation, in all its versions. As noted from the outset, Bohr 
+adjusted, sometimes significantly, his interpretation.  
+As Bohr argued in the Como lecture, which presented his first interpretation of QM (but the argument 
+was retained in all versions of his interpretation, including the ultimate, RWR, one), in classical physics 
+and relativity “our … description of physical phenomena [is] based of the idea that the phenomena 
+concerned may be observed without disturbing them appreciably” [Bohr 1987, v. 1, p. 53; emphasis 
+added]. By contrast, “any observation of atomic phenomena will involve an interaction [of the object 
+under investigation] with the agency of observation not to be neglected” [Bohr 1987, v. 1, p. 54; emphasis 
+added]. One should keep in mind the subtle nature of this contrast: the interaction between the object 
+under investigation and the agency of observation gives rise to a quantum phenomenon rather than 
+disturbs it. Bohr became weary of the language of “disturbing of phenomena by observation” [Bohr 1987, 
+v. 2, p. 64].11 Bohr grounded his interpretation (in all its versions) in this role and, in the ultimate version 
+of his interpretation, in the strong RWR concept of reality, as applied to quantum objects, placed beyond 
+conception and thus made invisible to thought. The behavior of the observable parts of measuring 
+ 
+11 Relativity represented a step in this direction, insofar as, in contrast to Newtonian mechanics, space and time were 
+no longer seen as preexisting (absolute) entities then measured by instruments, such as rods and clocks, but were 
+instead defined by the latter in each local reference frame. Still the interference of observational instruments into the 
+behavior of the objects considered could be disregarded, thus allowing, as in classical physics, identification of these 
+objects with the observed phenomena for all practical purposes. Hence, the objects under investigation can be 
+considered independently of their interactions with measuring instruments. Quantum phenomena preclude this type 
+of idealization, to the point of, in RWR interpretations, making the ultimate nature of the reality responsible for 
+quantum phenomena invisible to thought. Bohr often reflected on these affinities, as well as differences, between 
+relativity and quantum theory (e.g., [Bohr 1987, v. 1, p. 115, v. 2, p. 41, 1935, pp. 701-702]). 
+
+ 
+15 
+instruments and, with them, quantum phenomena were idealized as representable by means of classical 
+physics, by the Bohr postulate. Eventually, Bohr adopted the term “phenomenon” to refer strictly to what 
+is observed in measuring instruments, as effects of their interaction with quantum objects: 
+ 
+I advocated the application of the word phenomenon exclusively to refer to the observations obtained under 
+specified circumstances, including an account of the whole experimental arrangement. In such terminology, the 
+observational problem is free of any special intricacy since, in actual experiments, all observations are 
+expressed by unambiguous statements referring, for instance, to the registration of the point at which an electron 
+arrives at a photographic plate. Moreover, speaking in such a way is just suited to emphasize that the 
+appropriate physical interpretation of the symbolic quantum-mechanical formalism amounts only to predictions, 
+of determinate or statistical character, pertaining to individual phenomena appearing under conditions defined 
+by classical physical concepts [describing the observable parts of measuring instruments]. [Bohr 1987, v. 2, p. 
+64; emphasis added] 
+ 
+As defined by “the observations [already] obtained under specified circumstances,” phenomena refer to 
+events that have already occurred and not to possible future events, such as those predicted by QM. This 
+is the case even if these predictions are ideally exact or deterministic, which they can be in certain 
+circumstances, such as those of EPR type experiments. The reason that such a prediction cannot define a 
+quantum phenomenon is that a prediction for variable Q (for example, that related to a coordinate, q) 
+cannot, in general, be assumed to be confirmable by a future measurement, in the way they can be in 
+classical physics or relativity. One can always perform a complementary measurement, that of p (the 
+momentum), which will leave any value predicted by using Q undetermined by the uncertainty relations, 
+which in principle preclude associating a physical reality corresponding to a coordinate q when one 
+measures p [Plotnitsky 2021a, pp. 210-212]. As earlier, I use capital vs. small letters to differentiate, as is 
+necessary, Hilbert-space elements, here operators, like Q and P, associated with predicting the values of 
+measured quantities, like q and p, observed on measuring instruments. Hence, one can never speak of 
+both variables unambiguously, even if they are associated with measuring instruments, while any 
+references, even that to a single property of a quantum object considered independently is ambiguous. In 
+classical physics, this difficulty does not arise because one can, at least in principle, always define both 
+variables simultaneously and unambiguously speak of the reality associated with both variables and 
+assign them to the object itself. By contrast, in any quantum experiment we always deal with a system 
+containing an object and an instrument. Thus, in considering quantum phenomena, on the one hand, there 
+is always a discrimination between an object and an instrument, and, on the other, the impossibility of 
+physically separating them. This impossibility compelled Bohr to speak of “the essential ambiguity 
+involved in a reference to physical attributes of objects when dealing with phenomena where no sharp 
+distinction can be made between the behavior of the objects themselves and their interaction with the 
+measuring instruments,” as opposed to a reference to what is observed which, as classical by the Bohr 
+postulate, can be unambiguous and communicated as such [Bohr 1987, v. 2, p. 61].  
+There are several reasons for adding the Dirac postulate to the Heisenberg and Bohr postulates, 
+defining Bohr’s ultimate interpretation, in QM and, as noted, even more so in QFT, as considered in 
+[Plotnitsky 2021a, pp. 273-306, 2021b, 2022b], beginning with the fact that no properties can be assigned 
+to a quantum object apart from observations in Bohr’s interpretation. In fact, in any strong RWR 
+interpretation, nothing at all can be said or even thought about what happens between observations. Need 
+one, then, still speak of quantum objects between observation? Also, as explained in detail in Section 5 
+(although, as just indicated, this point is implied by Bohr’s concept of a phenomenon), in quantum 
+physics in each experimental arrangement defining an observation one must, regardless of interpretation, 
+discriminate “between those parts of the physical system considered which are to be treated as measuring 
+instruments and those which constitute the objects under investigation” [Bohr 1935, p. 701]. The 
+difference between them is, however, not uniquely defined. This is related to the arbitrariness of the “cut,” 
+considered in Section 5. For the moment, it follows that it is how one sets up an experiment that defines 
+what is “the object under investigation” in this experiment, which invites assuming the concept of a 
+quantum object to be applicable only at the time of observation. The situation has additional complexity 
+
+ 
+16 
+because, while still treatable by means of QM, “the object under investigation” in a quantum experiment 
+may not be strictly a quantum object: it may be partly classical, for example, containing the cat of the cat 
+experiment, which is why Bohr speaks here of “objects under investigation” rather than quantum objects. 
+This object, however, can only be only partly classical because it must contain, as its part, a properly 
+quantum object, such as an electron or a photon, or some composite quantum object, to observe quantum 
+effects. I shall return to this aspect of the situation, which bears importantly on the cat experiment, in 
+Section 5, merely noting here that, a classical part, such as the cat, of such a combined object is always 
+the same object and hence does not obey the Dirac postulate, which only applies to quantum objects. In 
+accordance with Bohr’s concept of a phenomenon, whatever is the object of investigation in a quantum 
+experiment, it cannot be considered independently of its interaction with the measuring instrument, thus, 
+making any quantum experiment and hence quantum theory involve both a combination and a separation 
+of an object and an instrument. Because, however, the object under investigation in a quantum experiment 
+must contain a properly quantum object, this is also a combination of what is invisible to thought and 
+cannot be communicated unambiguously, and what is visible to thought, via observational instruments, 
+and can be communicated unambiguously. Hence, as noted, the Bohr postulate also manifests the 
+transition, via observation, from the ultimate, “quantum,” reality to the classical level of observation, and 
+conversely, in the initial preparation of an experiment, from the classical level of observation to the 
+ultimate, “quantum,” reality. According to Bohr: 
+    
+The essential lesson of the analysis of measurements in quantum theory is thus the emphasis on the necessity, in 
+the account of the phenomena, of taking the whole experimental arrangement into consideration, in complete 
+conformity with the fact that all unambiguous interpretation of the quantum mechanical formalism involves the 
+fixation of the external conditions, defining the initial state of the atomic system concerned and the character of 
+the possible predictions as regards subsequent observable properties of that system. Any measurement in 
+quantum theory can in fact only refer either to a fixation of the initial state or to the test of such predictions, and 
+it is first the combination of measurements of both kinds which constitutes a well-defined phenomenon. [Bohr 
+1938, p. 101]  
+ 
+One begins an experiment by classically preparing an observational instrument and registering, at time 
+tprep, the data obtained by the interaction between this instrument and a quantum object, thus setting up the 
+workings of the ultimate reality considered, placed beyond representation or even conception, by the 
+Heisenberg postulate. This is the classical to the quantum, the visible to thought to the invisible to 
+thought, conversion. Then by setting up a new observational device, one makes a new observation at time 
+tobserv registering an outcome of the experiment, possibly as predicted by QM, in which case the 
+observational instrument needs to be prepared accordingly, for, as explained, one can always perform a 
+different type of measurement at this moment in time. This is the quantum to the classical, the invisible to 
+thought to the visible to thought, conversion.  
+If, however, in assuming, as I do here, the Dirac postulate, a quantum object is only an idealization 
+defined by an observation, rather than of something that exist independently (vis-à-vis the ultimate reality 
+responsible for quantum phenomena assumed to exist independently), could one still speak of the same 
+quantum object, say, the same electron, in two successive observations, with the second confirming the 
+prediction based on the first of the formalism of QM? The case can be given a strictly RWR 
+interpretation, insofar as all these properties are, physically, those of measuring devices, impacted by 
+quantum objects, rather than of these objects themselves, placed beyond representation or conception. 
+Rigorously speaking, if the concept of a quantum object is only applicable at the time of observation, then 
+a prediction based on a given measurement and the new measurement based on this prediction could only 
+concern a new quantum object, and not an object that one measured earlier in making a prediction. 
+Accordingly, one deals with two different quantum objects, two different electrons, for example. To 
+consider them as the same electron is, however, a permissible idealization in low-energy QM, or low-
+energy QFT, regimes. By contrast, speaking of the same electron in successive measurements in high-
+energy (QFT) regimes is meaningless, because these measurements can register quantum objects of 
+
+ 
+17 
+different types, say, in the case quantum electrodynamics (QED) an electron in the initial and a positron 
+or photon in the next measurement [Plotnitsky 2021a, pp. 279-292, 2021b]. QFT supports adding the 
+Dirac postulate to the Heisenberg and Bohr postulate, in RWR interpretations, but, as I argue here, there 
+are reasons also to do so in low-energy (QM) regimes, including, it may be shown, the complexities 
+involved in the double-slit and related experiments [Plotnitsky 2022b]. 
+On the other hand, there is no difficulty in speaking of the same classical object, even if it is part of 
+the object of investigation by quantum means, such QM, in a quantum experiment, which, again, require a 
+properly quantum object, such as emitted particle in the cat experiment, to be a quantum experiment. At 
+all stages of the cat experiment, we deal with the same cat, dead or alive. The state of a classical object 
+can change in time, just as the state of a measuring instruments does when impacted by a quantum object. 
+If one tosses a coin, its state will change throughout its trajectory before it falls with either head or tail 
+side up. Of course, this “sameness” is an idealization, possible and necessary in classical physics or 
+relativity, or in dealing with classical objects, including measuring instruments, in quantum physics. As 
+Heraclitus famously said, one cannot step in the same river twice because neither the river nor the one 
+who steps into it is the same. Such concepts, however, do not apply to quantum objects, such as electrons, 
+which, while they can change their location, momentum, or energy, are considered as strictly 
+indistinguishable from each other in terms of any invariant characteristics, such as mass, changes, or spin. 
+In RWR interpretations, these quantities are still only observable as effects of the interactions between 
+quantum objects and measuring instruments, because no concepts apply to quantum objects, whether one 
+defines them as existing independently, as in Bohr, only at the time of measurement, as here. 
+Two key concepts defining classical physics and relativity, (classical) “measurement” and (classical) 
+“causality,” become no longer applicable in quantum theory in RWR interpretations. The term 
+“measurement” is a remnant of classical physics and the history that shaped it, beginning with ancient 
+Greek thinking and the rise of geometry, geo-metry, there. In Bohr’s and the present view, a quantum 
+measurement does not measure or, in the first place, is not an observation of any property of the ultimate 
+constitution of the reality responsible for quantum phenomena, a property that this reality would be 
+assumed to possess before or even during the act of observation. The concept of observation requires a 
+redefinition as well. An act of observation in quantum physics establishes, creates, quantum phenomena 
+by an interaction between the instrument and the quantum object. This act is a unique event of creation.12  
+This view also gives a new meaning to and gives a central significance to the category of event, as 
+defining a new physical situation each time, akin to important events that transform the situation in life or 
+culture, including politics, except that in quantum physics every event of observation radically transforms 
+the situation and redefines the possibly future vis-à-vis the preceding events, no longer meaningful for 
+predictions concerning the future from this point on. As a result, quantum theory becomes a theory of 
+transition probabilities between events, thus defined by experimental technology and our decisions 
+concerning which experiment to performed. Then what is so observed as the data or information can be 
+measured classically, just as one measures what is observed in classical physics. There, however, what is 
+observed or measured could be associated with the object considered. In quantum physics, there is a 
+difference between observations, which construct phenomena, and measurements, which classically 
+measure physical properties of the phenomena thus constructed. In speaking of “quantum measurement,” 
+I refer to this whole process. It follows that measuring instruments must contain both the visible (even to 
+our immediate phenomenal perception) observable classical stratus and the quantum stratum, which 
+ 
+12 While this formulation echoes J. A. Wheeler’s invocation, inspired by Bohr, of “an elementary act of creation,” 
+the present view and, I would argue that of Bohr, may be different from Wheeler’s view of a “participatory 
+universe” [Wheeler 1983, pp. 189, 194; Wheeler 1981]. As existing independently of us, as it is assumed to be in the 
+present and Bohr’s views, (the reality of) the universe is not participatory; only independent phenomena, as created 
+by us, and the world we experience are participatory. It is possible that by “a participatory universe,” Wheeler refers 
+to the world of our experience, including of quantum phenomena as our acts of creation. Wheeler, however, does not 
+qualify his view in this way, and in any event, he never advances, and does not appear to adopt, the idea of reality 
+without realism, which underlies the present and, I argue, Bohr’s view, even though Bohr does not use the term. 
+
+ 
+18 
+enables their interactions with quantum objects. This interaction is quantum and cannot be observed and, 
+in RWR interpretations, be visible to thought. It is, in Bohr’s language, “irreversibly amplified” to the 
+classical level of observable effects, such as a spot left on a silver screen [Bohr 1987, v. 2, p. 73].13 
+The nature of causality in QM changes as well, as classical causality is no longer possible in RWR 
+interpretations. By “classical causality” I refer to the claim that the state, X, of a physical system is 
+determined, in accordance with a law, at all future moments of time once its state, A, is determined at a 
+given moment of time, and state A is determined by the same law by any of the system’s previous states. 
+This assumption implies a concept of reality, which defines this law, thus making this concept of 
+causality ontological or realist. There are several reasons for my choice of “classical causality,” rather 
+than just causality, used more commonly for this type of concepts. The main one is that it is possible to 
+introduce alternative, probabilistic, concepts of causality, applicable in QM, including in RWR 
+interpretations, where classical causality does not apply (e.g., [Plotnitsky 2021a, pp. 207-218]). Some, 
+beginning with P. S. Laplace, have used “determinism” to designate classical causality. I define 
+“determinism” as an epistemological category referring to the possibility of predicting the outcomes of 
+classically causal processes ideally exactly. In classical mechanics, when dealing with individual or small 
+systems, both concepts become equivalent. On the other hand, classical statistical mechanics or chaos 
+theory are classically causal but not deterministic in view of the complexity of the systems considered, 
+which limit us to probabilistic or statistical predictions concerning their behavior. 
+In quantum phenomena, deterministic predictions are not possible even in considering the most 
+elementary quantum systems. This is because the repetition of identically prepared quantum experiments 
+in general leads to “different recordings” of the observed data (associated with the kinematic and 
+dynamical variables), and unlike in classical physics, this difference cannot be diminished beyond the 
+limit, defined by h, by improving the capacity of our measuring instruments [Bohr 1987, v. 2, p. 73]. 
+“Recordings” refers to both those of the initial measurement, enabling a prediction, and those of the 
+second measurement, which would verify this prediction, a combination that, as explained above, 
+generally defines an experiment in physics, but takes a new meaning in quantum physics. These 
+recordings will be different either one repeats the whole procedure in the same set of experimental 
+arrangements or if one builds a copy of the apparatus and sets it up in the same way, as we do to 
+separately verify the outcomes of experiments. Either repetition is always possible because the 
+preparations of the instruments could be controlled classically. On the other hand, their interaction with 
+quantum objects (or in the present view, the ultimate reality responsible for quantum phenomena and, at 
+the time of measurement, quantum objects) cannot be controlled, which compelled Bohr to speak of “the 
+finite and uncontrollable interaction between the object and the measuring instruments in the field of 
+quantum theory” [Bohr 1935, p. 700]. The respective probabilities of the first and the second 
+measurements are independent of each other. The most crucial, however, is the difference in the outcomes 
+of the second (predicted) measurement in repeated setups. As noted, one can prepare any given state, say, 
+ 
+13 The physical nature of this “amplification” is part of the problem, commonly, including by this author, seen as 
+unsolved (although there are claims to the contrary, for example, on lines of decoherence or consistent histories 
+approaches), of the transition from the quantum to the classical. The subject is beyond my scope here. Fortunately, 
+quantum phenomena and QM allow us to bypass this problem in quantum measurements or predictions, seen here as 
+the transition from the invisible to thought to the visible to thought (or vice versa in a preparation). As Bohr noted, 
+QM is “justified only by the possibility of disregarding in its domain of application the atomic structure of 
+measuring instruments themselves in the interpretation of the results of experiments” [Bohr 1937, p. 88]. This 
+disregard, as Bohr observed, may lead to new complexities in high-energy physics and QED. As he said, invoking, 
+again, a renunciation of visualization: “For a correlation of still deeper laws of nature involving not only the mutual 
+interaction of the so-called elementary constituents of nature but also the stability of their existence, this last 
+assumption can no longer be maintained, as we must be prepared for a more comprehensive generalization of the 
+complementary mode of description which will demand a still more radical renunciation of the so-called 
+visualizations” [Bohr 1937, p. 88]. As it happens, QFT (in high-energy regimes) still disregards “the atomic 
+structure of measuring instruments,” which may be responsible for the appearance of infinities and the necessity of 
+renormalization and other, still unresolved, complexities there. 
+
+ 
+19 
+that of a “spin-up,” as manifested in the corresponding measurement, even though one cannot do so in a 
+single experimental preparation but only by post-selecting the required preparation. By contrast, the 
+outcome of the second (predicted) measurement cannot be controlled at all, only allowing one to predict 
+the probability or, if the experiment is repeated, statistics of the outcome. 
+The statistics of the outcomes of multiply repeated experiments performed in both such experimental 
+settings will be the same. On the other hand, an individual quantum experiment cannot be reproduced, as. 
+is always possible to do so in classical physics, because the interference of measurement can be neglected 
+or controlled, at least in principle. All data observed in quantum experiments remains classical, by the 
+Bohr postulate, and hence visible to thought (or even to the immediate phenomenal perception) and can 
+be communicated unambiguously. Unlike in classical physics, however, this data cannot be recreated by a 
+different system, which combines a quantum object (in the present view, again, a concept only applicable 
+at the time of observation) and an apparatus, the observable part of which is described classically. This 
+situation embodies the no cloning theorem [Park 1970, Dieks 1982, Wootters and Zurek 1982]. 
+As noted, the probabilistic or statistical character of quantum predictions must, on experimental 
+grounds, hold in interpretations of QM or alternative theories of quantum phenomena (such as Bohmian 
+mechanics) that are classically causal. QM or QFT, in RWR interpretations, are not classically causal 
+because the ultimate nature of reality responsible for quantum phenomena is assumed to be beyond a 
+representation or conception. Classical causality would imply at least a partial conception and even 
+representation of this reality. These circumstances imply a different reason for the recourse to probability 
+in quantum theory in RWR interpretations. According to Bohr: 
+ 
+[I]t is most important to realize that the recourse to probability laws under such circumstances is essentially 
+different in aim from the familiar application of statistical considerations as practical means of accounting for 
+the properties of mechanical systems of great structural complexity. In fact, in quantum physics we are 
+presented not with intricacies of this kind, but with the inability of the classical frame of concepts to comprise 
+the peculiar feature of indivisibility, or “individuality,” characterizing the elementary processes. [Bohr 1987, v. 
+2, p. 34] 
+ 
+The “indivisibility” refers to the indivisibility of phenomena in Bohr’s sense, defined by the impossibility 
+of considering quantum objects independently from their interactions with these instruments. 
+“Individuality” refers to the assumption that each phenomenon is individual and unrepeatable, as well as 
+discrete relative to any other phenomenon, and correlatively, to the essential randomness of individual 
+quantum phenomena. Collectively they may not be strictly random by virtue of one or another form of 
+quantum correlations (such as EPR-type correlations, at stake in Bell’s or Kochen-Specker theorem), 
+which are, however, strictly quantum as well and not found in classical phenomena. This randomness is 
+not found in classical physics, because even when one must use probability there, at bottom one deals 
+with individual process that are classically causal and in fact deterministic. Hence, in classical physics, 
+randomness does not ultimately exist or is assumed ultimately not to exist; only probability does. In 
+principle, one can isolate an individual constituent of the structurally complex mechanical system, say, a 
+molecule of a gas, something that, as classical, is in its behavior, visible to thought, and predict its 
+behavior ideally exactly. It is, however, the “in principle” that is crucial, because this is never possible in 
+considering individual quantum systems, no matter how elementary. By the same token, such systems or 
+(since the term “system” is not ultimately applicable either, except at the time of measurement) the 
+ultimate nature of the reality considered can never be made visible to thought, which is, again, the reason 
+why they cannot be assumed to be classically causal or predicted deterministically. In fact, as explained, 
+the possibility of never observing quantum objects as isolated defined Bohr’s concept of a phenomenon, 
+and in the present (more radical) view, quantum objects are only defined, still as invisible to thought, at 
+the time of observation. Quantum physics, then, contains an essential randomness not found in classical 
+physics, which is at bottom classically causal and, when it comes to the behavior of its elemental 
+individual constituents, deterministic, thus making the recourse to probability a practical, epistemological 
+matter, as Bohr says. A coin toss is an example of a classical probabilistic system. The outcome can, in 
+
+ 
+20 
+practice, only be predicted probabilistically due to the mechanical complexity of the process, beginning 
+with the motion of the hand tossing it. However, this is still a classically causal process, the outcome of 
+which is determined and could, in principle, be predicted ideally exactly with sufficient technical and 
+computational capacities, which is, as explained, the meaning of “classically causal.” The recourse to 
+probability is practical, epistemological, due to our lack of knowledge concerning the underlying behavior 
+of the systems considered. In the case of any quantum system, no matter how simple, this idealization is 
+not possible: its behavior is not assumed to be classically causal in RWR interpretations, and as invisible 
+to thought, it cannot be so assumed.14 Quantum physics, however, only contains this randomness, rather 
+than is entirely random, because it allows for probabilistic or statistical predictions (purely random events 
+do not, which makes it impossible to handle them scientifically) and, more crucially, correlations. One of 
+the greatest mysteries of quantum phenomena is how random individual events can, under certain 
+circumstances, give rise to an order, even if only a (statistical) correlational order [Plotnitsky 2021a, pp. 
+253-256]. QM predicts these correlations, but at least in RWR interpretations, it does not explain them, 
+any more than it explains how any single outcome of an observation or measurement, comes about. The 
+emergence of either is invisible to thought. 
+I shall now explain Bohr’s concept of complementarity, especially, as it appears in his ultimate, 
+strong RWR interpretation, where it applies to phenomena in Bohr’s sense as outlined above. As defined 
+generally complementarity is characterized by: 
+ 
+(A) a mutual exclusivity of certain phenomena, entities, or conceptions; and yet  
+(B) the possibility of considering each one of them separately at any given point; and  
+ 
+14 One might further distinguish between indeterminacy, as a more general category, and randomness, as a most 
+radical form of indeterminacy, when a probability cannot be assigned to a possible event, which may also occur 
+unexpectedly. Both indeterminacy and randomness only refer to possible future events and define our expectations 
+concerning them. Once an event has occurred, it is determined. An indeterminate nature of events may either allow 
+for assuming an underlying classically causal architecture (which may be temporal) of the physical reality 
+responsible for this nature, whether this process is accessible to us or not, or disallow for making such an 
+assumption. The first case, as just explained, defines indeterminacy in classical physics, such as classical statistical 
+physics or chaos theory, or more radically in considering the so-called algorithmic complexity, such as Kolmogorov 
+complexity (also known as Solomonoff-Kolmogorov-Chaitin complexity), which may not be computable, but still 
+for practical, epistemological reasons. The second is found in QM or QFT in RWR interpretations. According to 
+Bohr, the idea of indeterminacy (or, again, randomness) apart from a classically causal order has “hardly been 
+seriously questioned until Planck’s discovery of the quantum of action” (Bohr 1938, p. 94). As he said on a later 
+occasion (in 1949): “[E]ven in the great epoch of critical [i.e., post-Kantian] philosophy in the former century, there 
+was only a question to what extent a priori arguments could be given for the adequacy of space-time coordination 
+and causal connection of experience, but never a question of rational generalizations or inherent limitations of such 
+categories of human thinking” (Bohr 1987, v. 2, p. 65). Even more radical philosophical questionings of the classical 
+idea or ideal of causality, such as those by David Hume, are those of our epistemological capacity to perceive the 
+underlying classically causal world, which would be presupposed at the ultimate level as inaccessible to us. It is 
+impossible to ascertain that an apparently random sequence of events, events that occurred apparently randomly, 
+was in fact random, rather than connected by some rule, such as that defined by classical causality, and there is no 
+mathematical proof that any “random” sequence is actually random (e.g., Aaronson 2013, pp. 71-92). The sequences 
+of indeterminate events that allow for probabilistic predictions concerning them is a different matter, although there 
+is still no guarantee that such sequences are not ultimately underlain by classically causal connections in the case of 
+quantum phenomena. Experimentally, again, quantum phenomena only preclude determinism, because identically 
+prepared quantum experiments in general lead to different outcomes. It follows that the claim of quantum 
+randomness can, in principle, be falsified, but establishing a classically causal theory or algorithm that reproduces 
+the indeterminate or random data in question, which becomes no longer indeterminate random. This would imply 
+that RWR interpretations, which precludes such connections, does not correspond to the ultimate nature of reality 
+responsible for quantum phenomena. See [D’Ariano 1922], which establishes the existence of a falsifiable quantum 
+random generator. In the present view, such a generator cannot be classical, because all classical (or relativistic) 
+theories of individual systems are deterministic, that is, can be so idealized as such. 
+
+ 
+21 
+(C) the necessity of considering all of them at different moments of time for a comprehensive account 
+of the totality of phenomena that one must consider in quantum physics. 
+ 
+The concept was never given by Bohr a single definition of this type. However, this definition may be 
+surmised from several of Bohr’s statements, such as: “Evidence obtained under different experimental 
+conditions cannot be comprehended within a single picture, but must be regarded as complementary in the 
+sense that only the totality of the phenomena [some of which are mutually exclusive] exhaust the possible 
+information about the objects” (e.g., [Bohr 1987, v. 2, p. 40]). In classical mechanics, we can comprehend 
+all the information about each object within a single picture because the interference of measurement can 
+be neglected. This allows us to identify the phenomenon with the object under investigation and establish 
+the quantities defining this information, such as its position and momentum, in the same experiment. In 
+quantum physics, this interference cannot be neglected and leads to different, in fact mutually exclusive, 
+experimental conditions for each measurement and their complementarity, in correspondence with the 
+uncertainty relations. The situation implies two incompatible pictures of what is observed, as phenomena, 
+in measuring instruments. Hence, the possible information about a quantum object, the information to be 
+found in measuring instruments, could only be exhausted by the mutually incompatible evidence obtained 
+under different experimental conditions. On the other hand, once made, either measurement, say, that of 
+the position, will provide the complete actual information (manifested in measuring instruments) about 
+the object, as complete as possible, at this moment in time. One could never obtain the complementary 
+information, provided by the momentum measurement, at this moment in time, because to do so one 
+would need simultaneously to perform a complementary experiment on it, which is impossible. 
+Thus, parts (B) and (C) of the above definition of complementarity are as important as part (A) and 
+disregarding them can lead to a misunderstanding of Bohr's concept, often misleadingly identified with 
+just (A). Bohr’s complementarity is not only about a mutual exclusivity of things, but also about 
+performing quantum experiments by human agents, in which a mutual exclusivity becomes necessary. 
+That we have a free (or at least sufficiently free) choice as concerns what kind of experiment we want to 
+perform is in accordance with the very idea of experimentation in science, including in classical physics 
+[Bohr 1935, p. 699]. However, contrary to the case of classical physics or relativity, implementing our 
+decision concerning what we want to do will allow us to make only certain types of predictions and will 
+irrevocably exclude certain other, complementary, types of possible predictions. In other words, we have 
+a freedom, at least a sufficient degree of freedom, of choice which experiment to perform in classical and 
+quantum physics alike. In classical physics (or relativity), however, it does not matter in fundamental 
+terms because all variables necessary for defining the future course of reality, in accord with classical 
+causality, can always be determined at any moment in time, as there is no complementarity or the 
+uncertainty relations. By contrast, by virtue of complementarity, it does matter in quantum physics: By 
+staging, by decision, our experiments in one complementarity way or the other, we define the course of 
+reality, even if only probabilistically, because, while we can control the set-up of the experiment, we 
+cannot control the outcome. Such uncontrollable outcomes are no longer a matter of surprise that nature 
+confronts us with but is instead what we expect from nature, or our interaction with nature, in quantum 
+experiments. It also follows that we always, at any point, have a freedom, in any event, a sufficient degree 
+of freedom to make this choice or to change our choice and thus a future course of reality. Beyond, as 
+discussed below, the Bohr-EPR debate concerning the EPR experiment, this aspect of complementarity is 
+related in Bell’s and the Kochen-Specker theorem, or the Conway-Kochen free will theorem. The latter 
+connections are, however, a separate subject beyond my scope here. 
+For the moment, more immediately complementarity is a reflection of the fact that, in a radical 
+departure from classical physics or relativity, the behavior of quantum objects of the same type, say, 
+electrons, or, again, the ultimate nature of reality responsible for quantum phenomena defined by such 
+objects, is not governed by the same physical law, especially a representational physical law, in all 
+possible contexts, specifically in complementary contexts. This leads to incompatible observable physical 
+effects in complementary contexts. On the other hand, the mathematical formalism of QM offers correct 
+probabilistic or statistical predictions of quantum phenomena in all contexts, in RWR interpretations 
+
+ 
+22 
+under the assumption that the ultimate nature of reality responsible for quantum phenomena is invisible to 
+thought.15 However, as Bohr observed, reiterating his argument concerning the nature of quantum 
+probability considered above:  
+ 
+Just in this last respect [of the renunciation in each experimental arrangement of the one or the other of two 
+aspects of the description of the physical phenomena] any comparison between quantum mechanics and 
+ordinary statistical mechanics,—however useful it may be for the formal presentation of the theory,—is 
+essentially irrelevant. Indeed we have in each experimental arrangement suited for the study of proper quantum 
+phenomena not merely to do with an ignorance of the value of certain physical quantities, but with the 
+impossibility of defining these quantities in an unambiguous way. [Bohr 1935, p. 699] 
+ 
+It might be noted that wave-particle complementarity, with which the concept of complementarity is often 
+associated, had not played a significant, if any, role in Bohr’s thinking, especially after the Como lecture. 
+Bohr was always aware of the difficulties of applying the concept of physical waves to quantum objects 
+or assuming both types of behavior, particle-like and wave-like, pertain to the same individual entities, 
+such as each photon or electron itself, considered independently. Bohr’s ultimate solution to the dilemma 
+of whether quantum objects are particles or waves was that they were neither, any more than anything 
+else, by the Heisenberg postulate. Instead, either “picture” refers to one of the two mutually exclusive sets 
+of discrete individual effects, described classically by the Bohr postulate, of the interactions between 
+quantum objects and measuring instruments, particle-like, which may be individual or collective, or 
+wave-like, which are always collective, composed of discrete individual effects. An example of the latter 
+are interference effects, composed of a large number of discrete traces of the collisions between the 
+quantum objects and the screen in the double-slit experiment in the corresponding setup (when both slits 
+are open and there are no means to know through which slit each object has passed). These two sets of 
+effects may be seen as complementary, also when it comes to calculating the probabilities or statistics for 
+each set of events, or, if one takes a Bayesian view, for each event of each set. The two types of effects 
+involved are mutually exclusive and require mutually exclusive experimental setups to be observed. In 
+classical physics, wave-like (radiation) and particle-like objects or (as they can be identified) phenomena 
+were treated by two mutually exclusive theories, which is not the same as being complementary in Bohr’s 
+sense. The latter must include (B) and (C) part of the concept, applicable to the same (quantum) objects or 
+the ultimate reality responsible for quantum phenomena, but leading two different phenomena by (A), 
+depending on which setup one decided to use, predicted, differently, by the same theory, QM or QFT.  
+I would like, in closing my discussion of the (strong) RWR view, as defined by the role of the 
+invisible to thought in quantum physics, briefly to reflect, from this perspective, on the EPR experiment 
+and the Bohr-EPR exchange concerning it. My reflections follow [Plotnitsky 2021a, pp. 227-272], which 
+offers a detailed discussion, although the angle of visible and invisible to thought is new. The case is, 
+however, both exemplary and highly significant in this context, as Bohr, as noted above, realized in 
+stressing the significance of the EPR experiment as “suited to emphasize how far, in quantum theory, we 
+are beyond the reach of pictorial visualization” (Bohr 1987, v. 2, p. 59). One might give a new angle on 
+and amplify Bohr’s point, by arguing that that in their argument, EPR in effect assume that the 
+independent reality of quantum objects is visible to thought. EPR’s argument is, however, based on 
+disregarding or at least not adequately considered the constitutive role of observational instruments in 
+defining quantum phenomena in the way Bohr argued to be necessary, based on an analysis of this role in 
+his reply [Bohr 1935]. In fact, while EPR do, unavoidably, refer to “measurement,” EPR do not 
+considered or even mention measuring instruments, the constitutive role of which in defining all physical 
+variable concerned would make it difficult or even impossible to assume that the ultimate nature of reality 
+responsible for quantum phenomena can be visible to thought. 
+ 
+15 This situation is also responsible for what is known as “contextuality,” which was considered from the RWR 
+perspective in [Plotnitsky 2019, Plotnitsky 2021a], and, along different lines, in [Jaeger 2019, Howard 2021]. See 
+also Khrennikov’s extended survey [Khrennikov 2022].  
+
+ 
+23 
+EPR advanced the following argument based on the criterion of reality they which, they thought, 
+equally applicable in classical and quantum physics: “If, without in any way disturbing a system, we can 
+predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there 
+exists an element of physical reality corresponding to this physical quantity” [Einstein et al, p. 138]. 
+While, however, this criterion is unproblematically applicable in classical physics, it is, Bohr contended, 
+“ambiguous” in the case of quantum phenomena, because of the role of measuring instruments in defining 
+all such quantities. EPR’s argued that it is possible to ascertain “an element of reality” pertaining to a 
+quantum object, the second, S2, object of the EPR pair (S1, S2), independently of any interaction between 
+S2 and a measuring instrument (thus “without in any way disturbing the system”). This association is 
+made possible by a prediction by means of QM, say, of variable q (like that associated with the position 
+operator Q) with “probability equal to unity,” a prediction based on the measurement performed on S1, as 
+is indeed possible, at least ideally or in principle. As earlier, I use capital letters, Q or P, to refer to the 
+operators in the Hilbert space considered, and small letters, q or p, to physical variables probabilistically 
+predicted by the formalism by using Q or P, which have no physical connections to q or p apart from 
+these predictions in RWR interpretations. EPR argue that because this prediction, “with probability equal 
+to unity,” is possible “without in any way disturbing” S2, this property could be ascertained as an element 
+of physical reality pertaining to S2, in accordance with their criterion. As such, it is in effect assumed to be 
+visible to thought and unambiguously communicable, even though it is not actually observed or 
+measured, and as such is not available to our immediate phenomenal perception at the time of prediction, 
+or at any time, unless a measurement is performed, or even then because we can only perceive what is 
+observed in measuring. While in the latter case it is in principle possible to associate such a measured 
+quantity with an element of reality pertaining to the object itself, one is, obviously, outside the situation 
+covered by EPR’s criterion, because we no longer deal with a prediction and of course disturb the object 
+by an observation. Hence, it is the concept of visible to thought that is crucial here.  
+Bohr counterargued that, while EPR’s claim would work in classical physics, the situation was 
+different in considering quantum phenomena, including those of the EPR type, because of the essential 
+role of observational instruments in the constitution of all quantum phenomena and, thus, in any 
+unambiguous application of the concept of reality or of an element of reality in quantum physics. This 
+role, he argued, must be taken into consideration even in the case of predictions “with probability equal to 
+unity” without a measurement previously performed on the system considered, S2, and instead by using a 
+measurement performed on S1, as is ideally possible in the EPR case. As, however, I noted earlier and as 
+Bohr argued in his reply, this prediction is not sufficient for assigning an element of physical reality to S2, 
+contrary to EPR’s claim based on their criterion of reality, assumed by then to equally apply in both 
+classical and quantum theory. This is, however, not the case. In classical physics, where one can, in 
+principle, always measure and define both variables simultaneously, by neglecting the interference of 
+observational instruments, it is possible to speak, at any moment in time, unambiguously of the reality in 
+terms of its physical elements, thus visible to thought, associated with both conjugate classical variables, 
+Q and P (as functions of real variables) and define them as pertaining to the object itself considered. 
+Everything, at any point, is always visible to thought. 
+Not so, in quantum physics. Let us assume that by measuring qS1 on S1 and using the formalism, 
+applied to Q (an operator in a complex Hilbert space), one makes a prediction, “with probability equal to 
+unity,” concerning qS2 associated, via a measuring instrument, with S2 at some future time, t. If one 
+measures q at time t, one then will indeed obtain the value qS2. However, one can, instead of q, always 
+measure at time t the complementary variable, p (which would relate to the momentum operator P in the 
+formalism, although one does not use in a measurement). If one does so, the value of q becomes 
+completely undetermined, ambiguous, by the uncertainty relations. Hence, this measurement of p would 
+preclude associating any physical reality with the predicted value qS2. Thus, qS2, as defined by this 
+prediction, may be visible to thought, but it can no longer correspond to any element of physical reality 
+that can be associated with S2, or at least there is no way to experimentally ascertain such a 
+correspondence. S2 is assumed to exist and hence be real, but to assume so is not the same as associating 
+
+ 
+24 
+an element of reality with it and thus making it visible to thought.16 This association is only possible if the 
+measurement, confirming the predicted value, qS2, is performed. Doing so, however, can be in principle 
+precluded by making a complementary measurement and, thus, in contrast to classical physics (where 
+both conjugate variables can always be assigned, corresponding to elements of reality, simultaneously), 
+disabling the association of the predicted value qS2 with S2. This is so even if one assumes that one can 
+associate an element of reality with S2 as such, rather than only with a classical observed part of a 
+measuring instrument, at the time of measurement. In other words, unless the corresponding measurement 
+is performed, qS2 can correspond to no elements of physical reality, and the possibility of establishing 
+such a correspondence can be denied if one measures p instead. This situation is captured by A. Peres’s 
+statement that “unperformed experiments have no result” [Peres 1978]. Bohr’s claim concerning “an 
+essential ambiguity” of EPR’s criterion is defined by this situation, not considered by EPR in advancing 
+this criterion, or by Einstein in his related arguments, based in the same criterion. As Bohr stated in the 
+passage of his reply cited above, in view of complementarity, “we have in each experimental arrangement 
+suited for the study of proper quantum phenomena not merely to do with an ignorance of the value of 
+certain physical quantities, but with the impossibility of defining these quantities in an unambiguous way” 
+(Bohr 1935, p. 699; also Bohr 1987, v. 2, p. 62). There is, he argues, absolutely no possibility to 
+unambiguously define both “elements of reality” in question for S2, “without in any way disturbing” it. 
+One can only do so for one or other of complementarity quantities, say, q, by making the corresponding 
+measurement on S1, qS1, and predicting the reality of the same type of element, qS2, for S2, still under the 
+assumption that one could in principle perform the corresponding measurement. That, however, 
+irrevocably precludes one from predicting the complementary element of reality, pS2, for S2, because any 
+measurement of p on S1 was precluded by measuring qS1. On the other hand, if one instead measures p on 
+S2, which of course would require disturbing S2, one in turn irrevocably precludes ascertaining qS2 as an 
+element of reality pertaining to S2. Locating this ambiguity enables Bohr to argue that QM can be seen as 
+both complete within its proper scope (as complete as nature allows our theory of low-energy quantum 
+phenomena to be) and local, insofar it does not entail any physical action at a distance, or at least that 
+EPR, who argued that QM is either incomplete or nonlocal in this sense, did not demonstrate otherwise, 
+as explained in detail in [Plotnitsky 2021a, pp. 227-272].17 
+The EPR-Bohr exchange was crucial for the development of Bohr’s thinking, leading him to his 
+ultimate, strong RWR interpretation and, correlatively, a deeper understanding of the nature of 
+complementarity as a physical concept. It compelled Bohr eventually to adopt the view that no 
+measurable quantity, even a single such quantity (rather than only both complementary quantities, as 
+precluded by the uncertainty relations) and hence no element of reality can be attributed to a quantum 
+object even at the time of measurement. While a quantum object was assumed by Bohr to exist and hence 
+be real independently of observation, any reference to the nature of its reality becomes ambiguous, 
+making Bohr speak of “the essential ambiguity involved in a reference to physical attributes of objects 
+when dealing with phenomena where no sharp distinction can be made between the behavior of the 
+objects themselves and their interaction with the measuring instruments” [Bohr 1987, v. 2, p. 61]. Such 
+ 
+16 It should be kept in mind that, as Schrödinger was the first to note in defining entanglement, in dealing with 
+entangled systems it is not possible to speak of the properties on each system separately or even (Schrödinger does 
+not appear to go that far, at least not expressly) even of two separate systems [Schrödinger 1935, pp. 160-161]. 
+However, once a measurement on S1 is performed (thus also establishing it as quantum object], S1 and S2 are no 
+longer entangled. This situation, it might be added, gives another justification to using the Dirac postulate, which 
+only defines either system as such at the time of measurement. 
+17 EPR were aware and assumed in their argument that both quantities cannot be measurement or predicted 
+simultaneously, and their criterion of reality allows for assigning both “elements of reality” to S2 without 
+simultaneously predicting both. They argued, however, that the only alternative to their argument is the assumption 
+of the nonlocality (an action at a distance) of QM or quantum phenomena [Einstein et al, p. 141]. Bohr 
+counterargued that this nonlocality, as well as the incompleteness of QM, can be avoided by virtue of the ambiguity, 
+and hence inapplicability, of EPR’s criterion of reality to quantum phenomena, as here discusses. A detailed 
+argument is offered in [Plotnitsky 2021a, pp. 227-272]. 
+
+ 
+25 
+attributes, as elements of reality, can only be unambiguously ascribed (under the constraint of the 
+uncertainty relations) to certain parts, elements, of quantum phenomena, defined by the observable parts 
+of measuring instruments. This fact makes these elements open to being described by classical physics. 
+While, however, Bohr associated the ultimate, invisible-to-thought, reality responsible for quantum 
+phenomena with quantum objects, in the present interpretation, by the Dirac postulate, the concept of a 
+quantum object is only applicable at the time of observation, still as an RWR concept, which precludes 
+associating any attributes, elements, with it. The character of the ultimate reality considered as invisible to 
+thought equally defines both interpretations arising from, and arguably reaching the most radical 
+manifestation of, “the spirit of Copenhagen,” in Heisenberg’s memorable phrase “der Kopenhagener 
+Geist der Quantenheorie,” honoring Bohr’s contribution to our understanding of quantum theory 
+[Heisenberg 1930, p. iv)].  
+The existence, at least a possible existence, of a reality invisible to thought (the Heisenberg 
+postulate), which is, at the same time, ultimately responsible for what is visible to thought in quantum 
+phenomena (the Bohr postulate), is what Bohr saw as “an epistemological lesson of quantum mechanics” 
+[Bohr 1987, v. 3, p. 12]. At least, this is an epistemological lesson of his interpretation of quantum 
+mechanics, to which the present interpretation adds the Dirac postulate. Perhaps, however, quantum 
+mechanics or physics in general cannot teach us lessons otherwise. It is just that there appears now (this 
+has not always been the case) to be more consensus, albeit not an entirely unanimous one either, as 
+concerns our interpretation of classical physics and relativity as realist theories. When it comes to QM or 
+QFT, the proliferation of diverse (and sometimes incompatible) interpretations and the debate concerning 
+them, still overshadowed by the Bohr-Einstein confrontation, continue with an undiminished intensity and 
+no end in sight. But then, the stakes are high: the future of our understanding of nature and thought alike. 
+ 
+3. The Bohr-Schrödinger exchange on classical concepts in quantum measurement  
+ 
+Bohr’s insistence, reflecting (in present terms) the Bohr postulate, on the indispensable role of classical 
+physical concepts in considering measuring instruments is often misunderstood, and the subject is 
+significant in the context of the cat experiment, which provides an additional reason for addressing this 
+insistence in detail in this article. It is instructive to consider in this connection Schrödinger’s comments 
+on this aspect of Bohr’s thinking in Schrödinger’s letter after reading Bohr’s reply to EPR (in a 
+prepublication version), while working on his cat-paradox paper. The exchange, relevant to Schrödinger’s 
+overall argument in his paper, might have also affected his comments on the cat experiment, although the 
+origin of the experiment appears to be a suggestion by Einstein [Fine and Ryckman 2020]. Schrödinger’s 
+(long) letter and Bohr’s (brief) reply in part resume an earlier exchange, in 1928-1929, on the subject 
+among Einstein, Schrödinger, and Bohr [Plotnitsky 2021a, pp. 32-34]. Schrödinger writes: 
+ 
+You [Bohr] have repeatedly expressed your definite conviction that measurements must be described in terms 
+of classical concepts. For example, on p. 61 of the volume published by Springer in 1931 [the original German 
+edition of [Bohr 1987, v. 1]]: “It lies in the nature of physical observation, that all experience must ultimately be 
+expressed in terms of classical concepts, neglecting the quantum of action” [Bohr 1987, v. 1, pp. 94-95]. And 
+ibid. p. 74 “the invocation of classical ideas, necessitated by the very nature of measurement” [Bohr 1987, v. 1, 
+p. 114]. And once again [in the reply to EPR] you talk about “the indispensable use of classical concepts in the 
+interpretation of all [proper] measurement” [Bohr 1935, p. 701, where the printed version adds “proper”]. True 
+enough, shortly thereafter you say: “The removal of any incompleteness in the present methods of atomic 
+physics … might indeed only be affected by a still more radical departure from the methods of description of 
+classical physics, involving the considerations of the atomic constitution of all measuring instruments, which it 
+has hitherto been possible to disregard in quantum mechanics.” 
+ 
+This might sound as if what was earlier characterized as inherent in the very nature of any physical 
+observation as an “indispensable necessity”, would on the other hand after all just be a, fortunately still 
+permissible, convenient way of conveying information, a way we presumably sometime will be forced to give 
+up. If this were your opinion, then I would gladly agree. However, the subsequent stringent and clear 
+comparison with the theory of relativity make me doubt whether, in what I just said, I have understood your 
+
+ 
+26 
+view correctly. Because, if we considered the theory of relativity as a conceptual edifice in itself, without any 
+relationships to quantum mechanics, we would presumably never be able to renounce the sharp separation 
+between space and time in any measurement. Still, it seems possible that in connection with the unavoidable 
+mutual modification of these two theories, both would be forced to shake off their classical eggshell—and that 
+this is what you mean. (Letter to Bohr, October 13, 1935 [Bohr 1972-1996. v. 7, p. 505]) 
+ 
+As Schrödinger admits (“it seems possible”), this may not be and, I would argue, is not what Bohr means. 
+First, especially given that Bohr’s argued in his reply to EPR that QM is a complete theory within its 
+scope (as complete as nature allows our theory of nonrelativistic quantum phenomena to be), it is clear 
+that “incompleteness” in Bohr’s passage cited by Schrödinger does not refer to QM. It refers to the fact 
+that at the time QFT was hardly adequately developed even in the case of QED. (H. Yukawa’s meson 
+theory of nuclear forces just introduced.) QED, too, only worked then to the first order of approximation, 
+beyond which QED led to the appearance of infinities, which were only handled by renormalization 
+fifteen years later. The passage in question was removed from Bohr in the published version of his 
+response to the EPR paper [Bohr 1935], as Bohr explained in his reply to Schrödinger’s letter. He said: “I 
+have left out the reference to the possible significance of the atomic constitution of all measuring 
+instruments for the solution of the still unexplained difficulties of electron theory [QED]. The reason is 
+that together with Rosenfeld I am just about to finish a paper about a measuring problem in electron 
+theory in which this question will be elucidated somewhat more fully” [Letter to Schrödinger, October 25, 
+1935, in Bohr 1972-1996, v. 7, p. 511].18 Schrödinger was aware that Bohr referred to the incompleteness 
+of QED and possibly QFT. It is clear, however, from this comment and related elaborations by Bohr 
+including in [Bohr 1987, v. 1, pp. 89-91, 115], to which Schrödinger refers in his letter, that the point is 
+not that the observable parts of measuring instruments should no longer be described by classical physics 
+in QFT. Speaking of “a still more radical departure [than in QM] from the method of description of 
+classical physics” only refers to a more radical situation in QFT as concerns a possible necessity, as 
+against QM, of considering the atomic structure of measuring instrument, along with its observable parts, 
+described classically. The latter aspect of quantum measurement would remain in place in QFT in Bohr’s 
+view, for the reasons discussed earlier in the present article and explained in Bohr’s reply to 
+Schrödinger’s letter, while the atomic constitution interaction may need to be considered in a relativistic 
+quantum theory. In fact, we still do not have a quantum theory that does so, and as currently constituted, 
+QFT still works in the absence of such an account, which may be responsible for its difficulties.19 It is not 
+clear either whether such a theory is possible or necessary. QFT does contain unresolved difficulties (even 
+apart from the absence of a quantum theory of gravity). It does work, however. In works well as a 
+predictive theory or, one might argue, a framework, something sometimes referred to in theoretical 
+physics as “phenomenology” (not to be confused with the used of the term in philosophy or when one 
+speaks of our phenomenal representations). QED is now the best confirmed physical theory ever as 
+concerns its predictions, probabilistic or statistical as they are, which predictions are, however, again what 
+quantum experiments allows us, as things stand now. 
+Schrödinger, by contrast, appeared to think that Bohr believes that such a theory, as well as relativity, 
+“would be forced to shake off their classical eggshells” of the description of measuring instruments, 
+possibly even in QM. But, as I argue, this not what Bohr thinks: classical “eggshells” are part of 
+phenomena, and unlike is classical physics, if one shakes them off or breaks them you will only create 
+new eggs with eggshells, without ever exposing, making visible, what is inside. One cannot make an 
+omelet out of the eggs of quantum phenomena, only new eggs. Any subdivision of a phenomenon can 
+only result in a new phenomenon or phenomena, still each with classically described “shells,” without 
+ever exposing quantum objects. As Bohr explained later: “The individuality of the typical quantum effects 
+finds its proper expression in the circumstance that any attempt of subdividing the phenomena will 
+ 
+18 This paper was not published and only became available in the same volume of Bohr’s collected works [Bohr 
+1972-1996, v. 7, pp.195-209]. 
+19 See, note 13 above. 
+
+ 
+27 
+demands a change in the experimental arrangement introducing new possibilities of interaction between 
+objects and measuring instruments which in principle cannot be controlled” [Bohr 1987, v. 2, p. 39]. 
+Hence, Bohr speaks of closed phenomena, or the wholeness or indivisibility of phenomena. Bohr says in 
+his letter to Schrödinger:  
+ 
+However, these considerations [of the atomic structure of measuring instruments] do not have any close 
+connection to the Einstein paradoxes and to the question of limitation of the [classically] causal description of 
+quantum phenomena. On this point I must confess that I cannot share your doubts. My emphasis of [sic: on] the 
+point that the classical description of experiments is unavoidable amounts merely to the seemingly obvious fact 
+that the description of any measuring arrangement must, in an essential manner, involve the arrangement of the 
+instruments in space and their functioning in time, if we shall be able to state anything at all about phenomena. 
+The argument here is of course first and foremost that in order to serve as measuring instruments, they cannot 
+be included in the realm of application proper to quantum mechanics. [Letter to Schrödinger, October 25, 1935, 
+Bohr 1972-1996, v. 7, p. 511] 
+ 
+In other words, measuring instruments in their observable parts are and must be visible to thought and 
+even to our immediate phenomenal perception, to “be able to state anything at all about phenomena” and 
+thus to unambiguously communicate our findings, along with the mathematics that predicts them, to meet 
+“basic requirements of science,” as Bohr said in his reply to EPR [Bohr 1935, p. 697]. On the other hand, 
+the ultimate nature of the reality responsible for observed phenomena may be and, in Bohr’s view, is 
+invisible to thought, and hence nothing about it can be communicated unambiguously or at all. The same 
+situation is found in high-energy (QFT) regimes, whether we will ever be able to include the atomic 
+constitution of measuring instruments in the theory or not. As Heisenberg says, following Bohr’s 
+argument, and aware of Bohr’s exchanges with both Einstein and Schrödinger on the subject: 
+ 
+Therefore, it has sometimes been suggested that one should depart from the classical concepts altogether and 
+that a radical change in the concepts used for describing the experiments might possibly lead back to a 
+nonstat[ist]ical [sic!], completely objective description of nature. . . . This suggestion, however, rests upon a 
+misunderstanding. The concepts of classical physics are just a refinement of the concepts of daily life and are an 
+essential part of the language which forms the basis of all natural science. Our actual situation in science is such 
+that we do use the classical concepts for the description of the experiments, and it was the problem of quantum 
+theory to find theoretical interpretations of the experiments on this basis. There is no use in discussing what 
+could be done if we were other beings than we are. At this point we have to realize, as von Weizsäcker has put 
+it, that “Nature is earlier than man, but man is earlier than natural science.” The first part of the sentence 
+justifies classical physics, with its ideal of complete objectivity. The second part tells us why we cannot escape 
+the paradox of quantum theory, namely, the necessity of using classical concepts. [Heisenberg 1962, p. 56] 
+ 
+There is indeed no paradox here. Classical concepts reflect the essential workings of our biological and 
+specifically neurological nature born with our evolutionary emergence as human animals. Our thinking, 
+as the product of this machinery, is classical in that it is consistent with and leads to the concepts of 
+classical physics. Any concept we form derive from and can only apply to observed phenomena, and 
+quantum phenomena are physically classical as observed phenomena. They are different from classical 
+phenomena because the data observed in them precludes us from describing how they come about by 
+classical physics (which incapacity led to quantum theory) or in RWR interpretations, any physical theory 
+or even making them available, visible, to thought. Such a conception, which would make the emergence 
+of these data visible to thought, may be precluded by the same evolutionary biological or neurological 
+nature of ours and, thus, by our classical thinking and language, developed in the interaction with 
+(classical) objects consisting of millions of atoms, rather than anything on the atomic scale (e.g., 
+[Heisenberg 1930, p. 11]). This is another manifestation (correlative to classical physics) of the fact that 
+human nature and thus our thought are “earlier than natural science” and limit the latter. QM or QFT, 
+however, allows one to probabilistically predict the data considered, without representing or even without 
+us conceiving of how these data come about, or at least it allows for (RWR) interpretations, according to 
+which QM or QFT does no more. 
+
+ 
+28 
+In classical physics we only need one theory for observing (or measuring), representing, and 
+predicting the phenomena considered, which can be identified with the object considered, the interference 
+of observation can be neglected. In both relativity and quantum theory (QM and QFT) we need classical 
+theory to observe and measure the phenomena considered and the measuring instruments, but by 
+relativistic and quantum theory, respectively, but with a crucial difference. In relativity we can, just in 
+classical physics, still neglect the interference of measurement and, as a result, represent the behavior of 
+the objects considered and predict this behavior, ideally deterministically. In quantum theory this 
+interference cannot be neglected, essentially defining quantum phenomena as different from the objects 
+considered, which makes our predictions, in general probabilistic, regardless of interpretation. In RWR 
+interpretations, quantum objects or in the present view (in which quantum objects are only defined at the 
+time of measurement by the Dirac postulate) the ultimate nature of reality responsible for quantum 
+phenomena is placed beyond representation or conception. It is true that some classical theories, such as 
+classical statistical mechanics or chaos theory, are probabilistic, but these theories are not fundamental 
+because they do not deal with the ultimate constitution of nature. As explained, fundamental physics, as 
+things stand now, requires three types of theories—classical, which do not consider the roles of both c and 
+h, relativistic (which are epistemologically classical), which do not consider the role of h but do that of c, 
+and quantum which must take into account h, and in high-energy regimes c, with both relativistic and 
+quantum theories still using classical physics in representing the observable parts of measuring 
+instruments and the outcomes of observations or measurements.  
+These considerations do not imply that new concepts, physical or (which is, however, not the issue at 
+the moment) mathematical, are not possible in quantum theory. QM and QFT or their understanding and 
+interpretation would not have been possible without the invention of new concepts, with Bohr’s concepts 
+of complementarity and phenomenon, or Schrödinger’s concept of entanglement, among them. The 
+question is whether one can avoid classical physical concepts or classical physics or whether new realist 
+concepts, describing the ultimate nature of the reality responsible for quantum phenomena are possible or 
+even necessary, as both Einstein and Schrödinger thought. On the first question, Bohr’s or the present 
+view is that classical concepts and classical physics cannot be avoided. On the second question, Bohr 
+answers or at least that of the present author would be that such new realist concepts or theories may not 
+be possible, which is not the same as are not possible. They might be possible. Complementarity and 
+phenomenon are nonrealist concepts as concerns the ultimate constitution of the reality responsible for 
+quantum phenomenon, but they contain realist components by involving the description of observed 
+phenomena by (“old”) classical concepts. Entanglement, defined as a concept mathematically, could, as 
+concerns the physical reality considered, be understood along RWR lines as well, in accord with Bohr’s 
+view of the EPR experiment, manifesting entanglement and “suited to emphasize how far, in quantum 
+theory, we are beyond the reach of pictorial visualization” [Bohr 1987, v. 2, p. 59]. Schrödinger himself 
+spoke of the “entanglement of predictions,” defined by the corresponding aspects of formalism, rather 
+than quantum objects [Schrödinger 1935, p. 161; emphasis added]. 
+Schrödinger was, however, not yet finished in his letter, and asked another question, which surprised 
+Bohr as revealing something in Bohr’s thinking, of which Bohr was not entirely aware himself at the 
+time, and which is perhaps the most intriguing part of Schrödinger’s letter: 
+ 
+However that may be [as concerns a possible removal of the classical description of observation in relativistic 
+quantum regimes], there must be clear and definite reasons which cause you repeatedly to declare that we must 
+interpret observations in classical terms, according to their very nature. Whenever you say that, you state it so 
+definitely and clearly, in the indicative, without any reservation like “probably”, or “it might be”, or “we must 
+be prepared”, as if this were the uttermost certainty in the world. It must be among your firmest convictions—
+and I cannot understand what it is based upon. 
+ 
+It could not be just the point (about which you talked so insistently to me already in 1926): that our 
+traditional language and inherited concepts were completely unsuited to describe the phenomena with which we 
+are concerned now. Because, in the course of the development of our science (and mathematics), from its 
+earliest beginning to the situation at the end of the nineteenth century this was certainly the case over and over 
+again. If the break with the old tradition seems greater now than ever before, then we should take into account 
+
+ 
+29 
+that a particular time perspective is responsible for forming the impression that developments in which we 
+ourselves take part, stands out as being more important and more essential that earlier ones, which we cite only 
+from history, and whose stages we get to know mostly in reverse order. In fact, it is often difficult for us to 
+imagine earlier ways of thinking. And although the difficulty of such a historical step back actually speaks most 
+eloquently of how significant [the step] must have seemed to the pioneers of their earlier advances, still now and 
+then we cannot avert feeling: “Incredible that, up to then, people were so narrow-minded!” Here, the 
+underestimation of the time perspective shows itself most clearly. 
+ 
+Thus I think that the fact that we have not adapted our thinking and our means of expression to the new 
+theory cannot possibly be the reason for the conviction that experiments must always be described in the 
+classical manner, thus neglecting the essential characteristics of the new theory. (Letter to Bohr, October 13, 
+1935 [Bohr 1972-1996. v. 7, pp. 508-509]) 
+ 
+Indeed, as Bohr’s reply to Schrödinger, cited above, suggests, this is not the reason. It is not a matter of a 
+going beyond a tradition, say as that of classical physics or even earlier quantum theory. It is difficult to 
+object, and Bohr would not, to what Schrödinger says on this point. Hence, it would also be difficult to 
+agree that Bohr was ever neglecting the essential characteristics of QM; quite the contrary, he affirmed 
+them, not the least, as essentially different from classical physical theories, both deterministic or 
+probabilistic. Bohr’s emphasis of the classical description of measuring instruments is itself one of the 
+essential characteristics of quantum theory, given that what is so classically observed can only be 
+predicted by quantum theory. It is not that “we have not adapted our thinking and our means of 
+expression to the new theory,” because in fact physicists had so adapted their thinking (in terms of 
+physical, mathematical, and even daily concepts), and Bohr was one of the first to do so. The reason for 
+the conviction that “the experiments must also be described in a classical manner” are, as stated by Bohr 
+in his reply: “the seemingly obvious fact that the description of any measuring arrangement must, in an 
+essential manner, involve the arrangement of the instruments in space and their functioning in time, if we 
+shall be able to state anything at all about phenomena.” Or, more accurately, what is observed in 
+experiments must be so described, because, as Bohr, added: “the argument here is of course first and 
+foremost that in order to serve as measuring instruments, they cannot be included in the realm of 
+application proper to quantum mechanics.” This, too, then, is one the most essential features of quantum 
+theory, which brings into our thought a relation to what is invisible to thought. But this relation would not 
+be possible without describing in a classical manner what is visible to thought in quantum experiments. 
+Bohr’s “must” in “we must interpret observations in classical terms” is stated “so definitively” and 
+“without any reservation” because, while QM could become obsolete one day (although, remaining in 
+place for a century now, not anytime soon in the author’s view) or the RWR view, possibly in favor of 
+realism, this necessity of interpreting observation in classical terms will remain. Schrödinger was astute to 
+notice Bohr’s “must,” as Bohr didn’t fail to acknowledge this point in his reply: “I found it most amusing 
+that you noticed—which I myself had not at all been aware of—that just on this point, and only on this 
+one, I do not say, ‘it might be’” (Letter to Bohr, October 13, 1935 [Bohr 1972-1996. v. 7, p. 512]). Bohr 
+will be more aware of this fact from this point on, with its significance even more pronounced in his 
+subsequent writings. Bohr’s “must be” reflects his assumption of the necessity of unambiguous, and in 
+this sense objective, communication of, along with the logical and mathematical structure of quantum 
+theory, the outcomes of experiments, insured by the classical description of the observable part of 
+measuring instruments, in accordance with the Bohr postulate. As he said later (in 1949): 
+ 
+It is decisive to recognize that, however far the phenomena transcend the scope of classical physical 
+explanations, the account of all evidence must be expressed in classical terms. The argument is simply that by 
+the word “experiment” we refer to a situation where we can tell others what we have done and we have learned 
+and that, therefore, the account of the experimental arrangement and the results of the observation must be 
+
+ 
+30 
+expressed in unambiguous language with suitable application of the terminology of classical physics. [Bohr 
+1987, v. 2, p. 39]20 
+ 
+Our expectations or probability assignments concerning such outcomes may be different, depending on 
+different information we have pertaining to a given experiment, and in this sense, they are subjective or 
+personal. The latter might be a better concept insofar as these assignments are shaped by things in the 
+world, such as measuring instruments or the world itself which is assumed in this article or by Bohr to 
+exist independently and thus to be external to an agent. Things are rarely, if ever, completely subjective, 
+permitting that such exterior factors are interiorized at the time of an assignment of one or another 
+probability to a future event. There is nothing paradoxical or inconsistent with Bohr’s claims in this 
+understanding of probability. In life, too, we can have different expectations concerning future events 
+given the information we possess (which may be different), although QM, as a mathematical-
+experimental science, gives us a precise probability calculus to predict quantum events, which is, again, 
+all it does in the present view. Life rarely gives us such means. 
+At the same time (hence the consistency with Bohr’s claims), any measurement, in any quantum 
+experiment that would be performed would give a definitive, visible and informationally communicable 
+outcome, and as such is classical. An agent cannot control it but can only predict it probabilistically by 
+means of QM (cum Born’s rule). One might be able to decide (although it may not be simply a matter of a 
+free choice by our consciousness or even unconscious) which observation or measurement to perform, for 
+example, one or the other complementary observation, but one cannot control the specific outcome of it as 
+concerns which value one obtains, even if one controls the preparation of the instrument that will register 
+that outcome. In addition, as noted, one can always perform an alternative, complementary, measurement 
+at the end point of the experiment, which will irrevocably disable the original estimate. (In classical 
+mechanics, one can, again, always measure and predict, deterministically, all variables necessary for 
+accounting, representationally, for the system considered.) This makes measurement objective in this 
+double sense—the lack of control of an outcome and the possibility of an unambiguous communication of 
+an outcome—but in the present view, only in this double sense, rather than objectively attributing 
+anything to nature itself, apart from its existence and, as part of it, human existence. Making an 
+observation or measurement is, as stated, a unique act or event of creation with a unique outcome that can 
+be performed by a particular agent or several agents and as such has subjective or, again, personal aspects, 
+including those shaping our decision concerning this action, a decision inherent in the very idea of 
+experiment [Bohr 1935, p. 699]. Once the measurement is performed, however, the outcome becomes 
+fixed as a permanent record, part of the archive of physical data, always classical and visible to thought or 
+even our immediate phenomenal perception. It may be unknown to others, but that it is not the same as 
+being or bound to remain subjective. It is true, too, that, as any record, it must still be experienced as such 
+by us or others to be meaningful. Thus, performing an act of observation or measurement is personal (if 
+sometimes determined collectively), but its outcome need not be. It can also be experienced differently by 
+different agents, and in this sense, it is always personal and, in the first place, human. 
+Science is a human enterprise. But sharing and communicating our estimates of possible events and 
+experiences is also human and doing so is helpful and even unavoidable in human life. Science capitalizes 
+ 
+20 As indicated above (note 8), it is possible to argue that, while necessary for the description of the observable 
+quantum phenomena and measurements associated with them, classical physics is not a separate theory but rather a 
+limit case of QM, thus eliminating it from the class of fundamental theory, which is, however, not the present view 
+or, I would argue, that of Bohr. Thus, in the present view, if considered by itself, the cat in the cat experiment, is 
+always a classical object that cannot be handled, either representationally or (which is only possibility in RWR 
+interpretations) predictively, by QM. QM is only applicable in the cat experiment because there is a properly 
+quantum aspect to it: the emission of a particle by the radioactive atomic substance used. As will be seen, the same 
+considerations apply in the Wigner’s friend experiment, sometimes used to argue that everything can be considered 
+as quantum, without any use of classical physics. Most of these arguments, moreover, contend or imply that the 
+observable parts of measuring instruments can also be handled by QM, without, in contrast to Schrödinger’s 
+(subtler) argument that the situation requires new concept beyond both classical and quantum physics.  
+
+ 
+31 
+on this fact and on the possibility that the communication involved may be made unambiguous, helped by 
+the use of mathematical symbols, central to modern physics, from Galileo on. These symbols, too, or their 
+organization are visible to thought and hence unambiguously communicable, including those of the 
+mathematical formalism of QM or QFT. Mathematics itself, as a discipline, depends on this fact. In 
+classical physics and relativity, however, how the outcomes of experiments come about is visible to 
+thought as well, and may be assumed to be independent of observation, for all practical purposes, but, in 
+the present view, still only for all practical purposes, defined by human agents and agencies, such as 
+science. Not so in quantum physics, essentially dealing with and fundamentally shaped by what is 
+invisible to thought. In quantum physics, the role of human agents and experimental technology cannot in 
+principle be neglected, as reflected in the nature of quantum probability, which, as discussed above, is no 
+longer due, as in classical physics when it uses probability, to our insufficient knowledge of how the 
+phenomena considered come about. At stake in RWR interpretations is the impossibility in principle of 
+any knowledge or even conception concerning how this happens, which makes probability fundamentally 
+irreducible. The mathematics of quantum mechanics is visible to thought, and as such is unambiguously 
+communicable. But how what this mathematics predicts (in general probabilistically) comes about, as 
+outcomes of quantum experiments, is not.  
+We do not know what Schrödinger thought upon receiving Bohr’s reply, although it appears that he 
+had never have accepted Bohr’s view concerning the irreducible role of classical concepts in quantum 
+theory. It is not clear, for example, to what degree, if any, the cat paradox or even his paper overall were 
+an attempt to show that new physical concepts may after all be necessary in quantum theory. As explained 
+below, it appears that, by saying that “the 𝜓-function of the entire system would express this by having in 
+it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts,” he assumes the 
+cat to be a quantum object [Schrödinger 1938, p. 157]. This view has a difficulty in the fact that, if we 
+open the box or use a box with glass walls, we will see the cat, the same cat, at any stage of the 
+experiment, while one can never so see a properly quantum object, such as an electron, for detecting 
+which one always needs an instrument. In the present view, moreover, a quantum object is a concept that 
+only applies at the time of the experiment by the Dirac postulate, and hence implies that each observation 
+concerns a different quantum object, although identifying these objects is permissible in low-energy (QM) 
+regimes, but not high-energy (QFT) regimens. In the cat experiment, we always see the same cat, which 
+can, as noted, change its state, but not its sameness, always visible to thought. Be it as it may on that 
+score, Schrödinger thought, as did Einstein, that new concepts associated with quantum objects and their 
+behavior might be necessary received a new support from the EPR experiment. Such concepts, they 
+thought, would ground a realist alternative to QM, viewed by Schrödinger as “perhaps after all a 
+convenient calculational trick” [Schrödinger 1935, p. 167]. It is difficult to assume that Einstein saw it as 
+anything more than that. Neither thought that QM was likely to be interpreted on realist lines, although 
+such interpretations have been advanced. For Bohr, as explained, the EPR experiment confirmed, in 
+accordance with his (strong RWR) interpretation, “how far, in quantum theory, we are beyond the reach 
+of pictorial visualization,” to the point of reaching what is invisible to thought [Bohr 1987, v. 2, p. 59]. 
+ 
+4. Schrödinger’s cat experiment through the optics of visible and invisible to thought 
+ 
+Schrödinger’s paper containing the cat experiment was a response to EPR’s paper, which it discusses at 
+some length, and was, arguably, most important for the concept of entanglement, introduced by 
+Schrödinger, and its overall discussion of QM, its main concern, as reflected in its title “The present 
+situation in quantum mechanics” [Schrödinger 1935]. His analysis is thoroughgoing and penetrating, even 
+though (or perhaps because( Schrödinger assessed QM, especially as interpreted along the (Copenhagen) 
+RWR lines, as “a doctrine born of distress” [Schrödinger 1935, p. 154]. He saw QM, if not necessarily as 
+incomplete insofar as concerns its capacity to predict all that was possible to predict (or else nonlocal), as 
+EPR argued, but then as “perhaps after all only a convenient calculational trick” [Schrödinger 1935, p. 
+167]. EPR’s experiment gave Schrödinger, as it did to Einstein, new hopes that an alternative realist 
+theory of quantum phenomena might be possible. The cat experiment was part of Schrödinger’s overall 
+
+ 
+32 
+analysis of QM, a relatively marginal part, which did not appear to have initially received much attention. 
+Neither did initially the paper itself, even the concept of entanglement, introduced there, a major 
+contribution to QM. During the last half a century or so, however, the cat experiment has been 
+interminably discussed in technical, philosophical, and popular literature, and has even acquired a semi-
+mythical status. There are many reasons for this upsurge of attention to it, such as its role in helping 
+realist or classical causal views of QM or fundamental physics, or countering the Bohr postulate, often 
+resisted as much as the lack of realism or classical causality (e.g., [Plotnitsky 2022b]). There is of course 
+also a narrative appeal to the experiment, especially in popular accounts, but not only there. 
+From the present perspective, there is nothing especially remarkable or revealing in the cat 
+experiment, or anything that would challenge Bohr’s or the present interpretation. There does not appear 
+to be any evidence that Bohr ever commented on the experiment or on Schrödinger’s paper. The letter 
+exchange, discussed above, between Schrödinger and Bohr concerning Bohr’s emphasis on the classical 
+description of measuring instruments is relevant to the cat experiment. But, as preceding discussion 
+makes clear, this exchange took place before Schrödinger’s paper was published and was about Bohr’s 
+views, rather than any aspect of Schrödinger’s paper. I’d surmise that Bohr would not find anything in the 
+paradox either of much interest or as challenging his views. I’d also surmise that he was likely to have 
+seen, as I do here, the cat as a classical and not a quantum object. As indicated above, while not assumed 
+by Bohr, the Dirac postulate, which only applies to quantum and not to classical objects, lends further 
+support to this view. This is because the postulate states that each quantum observation concerns a 
+different quantum object, while only allowing one to assume that successive observations deal with the 
+same quantum objects as a statistically permissible idealization of low energy (QM) regimes but not in 
+high-energy (QFT) regimes. By contrast, the cat is aways the same object (if in a different classical state) 
+at any stage of the experiment. At least, as I shall argue, it is difficult to assume otherwise. According to 
+Schrödinger, then: 
+ 
+One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following 
+diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a 
+tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays, but 
+also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay 
+releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for 
+an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would 
+have poisoned it. The 𝜓-function of the entire system would express this by having in it the living and the dead 
+cat (pardon the expression) mixed or smeared out in equal parts. [Schrödinger 1935, p. 157] 
+ 
+In the present interpretation the last sentence would not apply, at least as stated, and the preceding two 
+sentences, which are in accord with the present view, appear to contradict the last sentence. I shall explain 
+why this is so presently. First, however, Schrödinger adds an elaboration that is rarely discussed or given 
+proper attention, which provided a further context for his thought experiment. He says: “It is typical of 
+these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into 
+macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so 
+naively accepting as valid a ‘blurred model’ for representing reality” [Schrödinger 1935, p. 157]. A 
+blurred model is defined by a view of the 𝜓-function as “an imagined entity that images the blurring of all 
+variables at every moment [unless a measurement intervenes] just as clearly and faithfully as the classical 
+model [images] its sharp numerical values” [Schrödinger 1935, p. 156]. In other words, the problem 
+arises if one sees the 𝜓 -function as representing the independent behavior of quantum systems, in this 
+case as blurred. In the present view, the 𝜓-function does not “faithfully” represent the behavior of the 
+quantum object considered or the ultimate reality responsible for quantum phenomena, because it does 
+not represent this reality at all. It only provides an (discrete) “expectation-catalog” for possible future 
+experiments, as Schrödinger himself called it [Schrödinger 1935, p. 154]. In developing his wave 
+mechanics, Schrödinger initially aimed for a (wave-like) representation of the ultimate reality responsible 
+for quantum phenomena in his project for his wave mechanics that led him to his famous equation. He 
+
+ 
+33 
+had, however, long given up on the idea by this point in view of the difficulties of reconciling his wave 
+mechanics had with observable features of quantum phenomena, in particular their discreteness and the 
+probabilistic nature of predictions concerning them. His equation itself can of course be and has been 
+(immediately in the Göttingen-Copenhagen circles) interpreted so as to accommodate these features, 
+especially given M. Born’s probabilistic interpretation of the 𝜓-function, eventually part of RWR 
+interpretations of QM.  
+These interpretations, including the present one, would, however, contrary to Schrödinger’s 
+statement, preclude the claim that “the 𝜓-function of the entire system would express [the situation] by 
+having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts,” 
+unless Schrödinger meant that his claim only applies to blurred variables. His claim, however, appears to 
+be more general and applicable when one does not view the variable considered as blurred, as his 
+subsequent reference to the cat experiment indicates [Schrödinger 1935, p. 161]. In any event, in the 
+present view, the cat, as a classical object, would always be either dead or alive at any stage of the 
+experiment, as Schrödinger’s preceding sentences imply: “If one has left this entire system to itself for an 
+hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay 
+would have poisoned it.” Why then claim: “the 𝜓-function of the entire system would express this by 
+having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts”? It is 
+also, in principle, possible that by his phrasing “the 𝜓-function of the entire system would express this by 
+having in it” (emphasis added) Schrödinger only meant the mixing of amplitudes for these two outcomes 
+so that “the 𝜓-function of the entire system” contains both possible future outcomes, as would be the case 
+in the present view. Schrödinger, however, does not qualify his statement in this way. In the present view, 
+without any conflict with the first two sentences, which only refer to a classical object, what will be 
+“mixed” or superposed are mathematical state vectors in the formalism. This mixture enables the 
+probabilities of predicting the atomic decay involved, to which, as a quantum process, such terms as 
+“dead” or “alive,” or any other terms, do not apply. It is only because of this purely mathematical mixture 
+that one is able to estimate the probability of finding the cat dead or alive. The 𝜓-function has no 
+association with the cat apart from these predictions (via Born’s rule), and it never represents the state of 
+the cat, as a classical object. The 𝜓-function never represents the physical state of a quantum object 
+either, as would be implied, by suggesting that the cat is seen as a quantum object, by Schrödinger’s 
+formulation.  
+In the present view, the cat is never mixed or smeared in equal parts between the living and the dead 
+cat. It is either the alive or dead cat at any stage of the experiment. One merely does not know (after a 
+certain moment in time, while cat is inside the box) whether it is alive or dead until one opens the box. 
+There is nothing that can be said or thought of concerning the ultimate reality responsible for quantum 
+phenomena (including, quantum objects, in the present view defined only at the time of observation by 
+the Dirac postulate), including that responsible for the atomic decay in the cat experiment. By contrast, 
+there are always things we can say about any classical reality, involved in quantum experiment, as part of 
+what is, in principle observable, in them, as is the cat in the cat experiment.21 The former reality is 
+invisible to thought, the latter is visible to thought.  
+ 
+21 The emphasized phrase deliberately echoes Heisenberg’s famous and much misunderstood, especially along 
+empiricist (Machian-like) lines, opening claim in his first paper of QM to the effect that he aims to ground his new 
+mechanics in “the relationships between quantities which in principle are observable” [Heisenberg 1925, p. 263]. 
+The quantities in question are empirically observable in measuring instruments, but the relationships in question (the 
+word usually disregarded in empiricist readings of this statement) are the probabilistic relationships established by 
+his new mechanics. As Heisenberg said, shortly before completing his paper: “What I really like in this scheme is 
+that one can really reduce all interactions between atoms and the external world ... to transition probabilities” 
+[Heisenberg, Letter to Kronig, 5 June 1925; cited in Mehra and Rechenberg 2001, v. 2, p. 242]. By speaking of the 
+“interactions between atoms and the external world,” this statement suggests that QM was only predicting the effects 
+of these interactions observed in measuring instruments, without representing how these effects come about. As 
+explained, this procedure replaced measurement in the classical sense (of measuring some preexisting properties of 
+
+ 
+34 
+One could, however, in the arrangement of the cat experiment, consider the cat as part of an object 
+under investigation, concerning which the prediction in question is made by means of QM, but as 
+explained earlier, only as part of this object because a proper quantum object must be involved in order to 
+have a quantum experiment. The cat experiment is a quantum experiment because of the radioactive 
+decay and not because of the cat. Therefore, considering the cat as part of an object under investigation 
+does not change the point that the cat is a classical object, always visible to thought or to our immediate 
+sense perception (before and after the experiment, or throughout if, if the box has glass walls), but not a 
+quantum object, which is never available to our phenomenal perception and, in the present view, is 
+invisible to thought. One can at any point see the cat as such, independently of a quantum observational 
+device, by opening the box or, again, using the box with glass walls, but one can never see a quantum 
+object as such or rather (since it cannot be seen) establish its presence without a suitable observational 
+device. A quantum object cannot be observed as separated from the phenomenon considered, which is a 
+result of the interaction between this object and the instrument. It would, accordingly, be more reasonable 
+to see the cat as a classical object, while, within in the overall arrangement of the experiment, being part 
+of the object of investigation by quantum means, enabling one to predict its possible classical state of 
+being dead or alive at the final stage of experiment. 
+As such the cat is also an object that can be described by ordinary language, as opposed to a quantum 
+object like an electron, which is, in RWR interpretations, merely a name, without a concept attached to it. 
+This fact makes misleading using, as is done sometimes, such notations as |𝜓⟩ = 𝛼|𝑑𝑒𝑎𝑑⟩ + 𝛽|𝑎𝑙𝑖𝑣𝑒⟩, as 
+opposed to the something like |𝜓⟩ = 𝛼|ℎ⟩ + 𝛽|𝑣⟩). The later refers to state vectors, in a superposition, 
+used to predict definitive classical events (which are never in a superposition), such as, within the chain of 
+events in the arrangement, that of the cat being dead or alive in the cat experiment, but has no other 
+connections to either (physical) state of the cat. Technically, QM predicts the effects that quantum objects 
+(or in the present view, the ultimate reality responsible for quantum phenomena) can have on the classical 
+world we experience. These effects define quantum phenomena or events. As any observation in quantum 
+physics, opening the box in the cat experiment is the phenomenon that reveals a classical state of the 
+reality, a state in this case already established in advance, which includes the cat, either dead or alive. One 
+or another properly quantum object, such as a radioactive atom and a particle it emits is always necessary 
+to have such an effect, even if the object under investigation, as different from a measuring instrument, 
+can contain a classical object, such as the cat. Calculating the probability of any such prediction will have 
+to involve h because of the radioactive decay as involving properly quantum objects, while the cat is of no 
+help in estimating such probabilities. 
+To support the case just outlined more rigorously, I turn to Bohr’s argument in his reply to EPR, 
+concerning “discriminating in each experimental arrangement between those parts of the physical system 
+considered which are to be treated as measuring instruments and those which constitute the objects under 
+investigation,” and the question of the cut thus arising. According to Bohr: 
+ 
+This necessity of discriminating in each experimental arrangement between those parts of the physical system 
+considered which are to be treated as measuring instruments and those which constitute the objects under 
+investigation may indeed be said to form a principal distinction between classical and quantum-mechanical 
+description of physical phenomena. It is true that the place within each measuring procedure where this 
+discrimination is made is in both cases largely a matter of convenience. While, however, in classical physics the 
+distinction between object and measuring agencies does not entail any difference in the character of the 
+description of the phenomena concerned, its fundamental importance in quantum theory … has its root in the 
+indispensable use of classical concepts in the interpretation of all proper measurements, even though the 
+classical theories do not suffice in accounting for the new types of regularities with which we are concerned in 
+atomic physics. In accordance with this situation there can be no question of any unambiguous interpretation of 
+the symbols of quantum mechanics other than that embodied in the well-known rules which allow us to predict 
+ 
+quantum objects) with establishing, by using measuring instruments, quantum phenomena, which can be treated 
+classically without measuring the properties of quantum objects, a view was adopted and developed by Bohr. 
+
+ 
+35 
+the results to be obtained by a given experimental arrangement described in a totally classical way. [Bohr 1935, 
+p. 701; second emphasis added] 
+ 
+It is important to avoid two common misunderstandings of this and related statements by Bohr. The first 
+concerns measuring instruments, in view of Bohr’s insistence on the classical description of the 
+observable part of measuring instruments, a subject discussed in detail in Sections 2 and 3, beginning 
+with the fact that instruments have quantum parts through which they interact with quantum objects. The 
+second concerns quantum objects. Bohr’s statement does not mean that, while observable parts of 
+measuring instruments are described by classical physics, the independent behavior of quantum objects is 
+described by means of the quantum-mechanical formalism, which assumption would be in conflict with 
+the RWR interpretation held by Bohr. This type of (realist) view has been adopted by some, sometimes 
+under the heading of “the Copenhagen interpretation,” beginning, influentially, with Dirac’s and von 
+Neumann’s classic studies, with Dirac’s book originally published in 1930 and von Neumann’s (in 
+German, in 1932 [Dirac 1958, von Neumann 1955]. Both books, moreover, assume a classically causal 
+independent behavior of quantum objects, with probability brought in only by measurement. This was, 
+however, not Bohr’s view, especially at this stage of his thinking in 1935, or even almost immediately 
+after the Como lecture of 1927, which may be seen as having adopted, still ambivalently, this type of 
+view and which arguably influenced both Dirac and von Neumann in this regard [Plotnitsky 2016, pp. 
+198-211]. In the passage in question, Bohr only says that classical theories cannot account for how 
+quantum phenomena (physically described classically) come about or predict what is observed. He does 
+not say that the independent behavior of quantum objects or objects under investigation (which may not 
+be quantum but must contain quantum objects) is represented by the formalism of QM. In Bohr’s view, 
+the “symbols” of QM only have a probabilistically predictive role, without, by the Heisenberg postulate, 
+offering a representation of how quantum phenomena come about, while quantum phenomena themselves 
+are represented by classical physics. So, QM does not represent them either, by the Bohr postulate. Thus, 
+in Bohr’s interpretation, while predicting, in general probabilistically, the data observed as part of 
+phenomena, the formalism of QM is otherwise dissociated in physical terms from both the ultimate nature 
+of reality responsible for quantum phenomena and these phenomena themselves, which phenomena are 
+described by classical physics.  
+The circumstance that “the place within each measuring procedure where this discrimination is made 
+is … largely a matter of convenience” is related to, although is not quite the same as, the arbitrariness of 
+the cut or the Heisenberg cut, or sometimes the Heisenberg-von-Neumann cut, because Heisenberg and 
+von Neumann favored the term (each giving it a somewhat different meaning), not used as such by Bohr. 
+Bohr qualifies this claim, and this qualification is important, including in the context of the cat 
+experiment. While “it is true that the place within each measuring procedure where this discrimination is 
+made is … largely a matter of convenience,” it is true only largely but not completely, because “in each 
+experimental arrangement and measuring procedure we have only a free choice of this place within a 
+region where the quantum-mechanical description of the process concerned is effectively equivalent with 
+the classical description” [Bohr 1935, p. 701].22 Thus, the ultimate constitution of the physical reality or 
+quantum objects and in quantum part of the instruments interacting with quantum objects is never on the 
+measurement side of the event, and by the same token they can never serve as measuring instruments 
+either. As beyond representation or even conception, as invisible to thought, quantum objects cannot be 
+assigned any properties, even at the time of measurement. This impossibility is correlative to their 
+position of always being on the other (than measurement) side of the event. All observable properties, are, 
+by the Bohr postulate, only those of the observable parts of measuring instruments, described classically, 
+but appearing under the impact of quantum objects. QM only predicts these visible properties. In Bohr’s 
+or the present view, in part by virtue of associating the cut with the discrimination between what is 
+considered the object under investigation and what is considered as the measuring instruments, the cut 
+ 
+22 This situation may be seen as a manifestation of Bohr’s correspondence principle, according to which the 
+quantum-mechanical and the classical descriptions give the same predictions in the classical limit.  
+
+ 
+36 
+and its (within certain limits) shifting nature need not imply that the classical theory, say, classical 
+mechanics or classical electromagnetic theory is a special form of QM or (in the case of 
+electromagnetism) QFT. As emphasized throughout this article, in Bohr or the present view, or that of 
+Heisenberg, these are two different theories that deal with two different types of objects, even though 
+classical objects are still composed of quantum objects or, in the present view (because the concept of a 
+quantum object only applies at the time of observation) of the same ultimate reality. We do not know how 
+classical objects (which could be considered, at least in principle, independently of observational 
+technology) emerge from this this ultimate reality. It is true that a quantum system with a large number of 
+coherent (quantum) states behaves close to a classical system, close but a) not quite; and b) still it is only 
+a special class of quantum systems, which are still not the same as classical systems, which one uses to 
+describe classical objects, including measuring instruments.  
+As indicated earlier, the cut, as here understood, reflects the possibility of placing some classical parts 
+of the overall arrangement in a quantum experiment (including the cat experiment) on either side of the 
+cut, as “an object under investigation,” concerning which predictions can be made by QM. As I argue, 
+however, the arrangement must include a quantum object (like a particle in radioactive decay in the cat 
+experiment) for the experiment to be quantum, to have quantum effects. Thus, the cat is always visible at 
+least to our mind’s eye, even when, while inside the box, not actually available to our sense perception. A 
+quantum object is never available to such a perception, and one always needs an experimental device 
+(which could be multi-stage, as in the cat experiment) capable of interacting with this object to have a 
+quantum effect. This effect is manifested in classically observed phenomenon or event, such as the cat 
+being alive or dead, in this case in fact, rather than being a properly quantum effects, preceded and made 
+possible by another classically observed event, the breaking of the flask, which is more properly quantum 
+effect, due to the interaction between it and the emitted particle. This interaction is quantum, while the 
+rest is the chain of classical events triggered by it. Bohr, accordingly, does not call a composite object, 
+containing both classical and quantum objects, on the object side of the cut a “quantum object,” but the 
+“object under investigation.” Quantum objects, while they can also be objects under investigation, are 
+only those objects that strictly belongs to the ultimate reality responsible for quantum phenomena, and 
+they are always on the object, never measurement, side of the event, and hence a quantum object can 
+never be a measuring instrument either. Other parts of objects under investigation in quantum 
+experiments are physically classical objects, such as the cat or everything inside the box, except that part 
+of the flask that can interact with the emitted particle. While it can be part of an object of investigation in 
+a quantum experiment and treated by quantum means, if considered by itself, a classical object cannot be 
+treated as a quantum object. Any prediction or measurement associated with a quantum object, elemental 
+(such as an electron) or composite (such as a Josephson device), will always involve h, thus correlative to 
+what is invisible to thought in quantum physics, even though h itself is observed classically. Observing 
+the cat in the cat experiment does not require h because the cat is a classical object, which, again, cannot 
+be treated as a quantum object, only a quantum part of it can, like protons in its body in the MRI test. 
+On the other hand, in certain circumstances, a quantum object could be treated for all practical as a 
+classical object, but without ever being a classical object. Thus, as when it is far enough from the nucleus 
+(for large quantum numbers), an electron can be treated as behaving classically. This is, however, an 
+approximation or idealization which disregards possible quantum effects of this behavior. As Bohr noted, 
+also connecting this situation to “mechanical pictures” and “classical pictures,” as visible to thought: 
+ 
+[I]n the limit of larger quantum numbers where the relative difference between adjacent stational states vanishes 
+asymptotically, mechanical pictures of electronic motion [as orbits] may be rationally utilized [by the 
+correspondence principle]. It must be emphasized, however, that this connection cannot be regarded as a 
+gradual transition toward classical theory in the sense that the quantum postulate [as an essential discontinuity 
+and individuality of quantum phenomena] would lose its significance for high quantum numbers. On the 
+contrary, the conclusions obtained from the correspondence principle with the aid of classical pictures depends 
+just upon the assumptions of the conception of stationary and of [discrete] individual transition processes are 
+maintained even in this limit. [Bohr 1987, v. 1, p. 85]  
+
+ 
+37 
+ 
+By this point (in 1927), Bohr adopts the view, which, following Heisenberg, equally renounced both the 
+classical, orbital “picture” of stationary states, still assumed in Bohr’s 1913 theory, and any classical view 
+of the transitions, “quantum jumps,” between states, already abandoned by Bohr’s 1913 theory, the first 
+instance of the RWR view (even if only partially applied). The concept, while it enabled Bohr to account 
+for the stability of atoms (vs. E. Rutherford’s preceding view), was nevertheless incompatible with 
+classical mechanics and classical electrodynamics alike. Neither the time nor direction of each jump could 
+be explained, although it could be predicted probabilistically or statistically, which, thus, from a classical 
+perspective paradoxically, brought the atomic stability and quantum randomness together. This stability is 
+of course that of a dynamic system, which can change its states, although these changes could only be 
+registered in measuring instruments. Bohr’s conceptual framework makes the term “jump” misleading in 
+suggesting some representation of what happens. Electrons do not jump; quantum states (as physical 
+states) discontinuously change, and no representation of how they do this is available. What was 
+responsible for these changes was assumed to be real, but this reality was assumed to be at least beyond 
+representation, in accord with the weak RWR view, although intimating the strong RWR view, insofar as 
+no concept of how these transitions appeared to be possible to form either. In Heisenberg’s approach 
+leading him to his invention of QM, the same situation defined the case of electrons in stationary states. 
+Electrons were not moving in orbits around nuclei: their quantum states (associated with variables other 
+than energy, as the energy remained the same in a stationary state) were changing, with these changes 
+observable as discrete phenomena. In this view, there were only the states of quantum objects, manifested 
+in measuring instruments, and transitions between these states. This was a decisive shift in our 
+understanding of the nature of physical reality.23 One might say that, rather than making any transition to 
+a new energy, an electron in a given stationary state disappears and a new electron is born in this new 
+stationary state. Each corresponding measurement will detect a different electron, in accord with the Dirac 
+postulate. The wave function for an electron in an atom can be recast in terms of annihilation and creation 
+operators, used in QFT.  
+Bohr’s statement cited above concerning the behavior of electrons in the case of large quantum 
+numbers confirms his view of classical objects and processes as visible to thought or even our immediate 
+phenomenal perception, and the behavior of quantum objects as, at this point (in 1927), no longer at least 
+to our general phenomenal intuition, and if not yet invisible to thought, as Bohr came to understand the 
+situation by the late 1930s. The behavior in the limit of large quantum numbers can be treated for all 
+practical purposes as that of classical objects. This treatment is, however, merely a workable 
+approximation of what is the ultimate nature of the reality responsible for what is thus observed, a reality 
+invisible to thought, and one still require a measuring instrument for this observation. At bottom one still 
+deals with the combination of stationary states and discontinuous quantum jumps. These states are too 
+close to each other for this combination to be detected, but one would, as it were, register these states (as 
+invisible to thought electrons or photons they emit still cannot be “seen”) by “zooming” on them, if one 
+had an instrument to do so. Any such instrument would, however, need to be able, by interacting with 
+electrons or “emitted” photons, to register properly quantum effects. Technically, an “emission,” too, is a 
+classical concept, which cannot represent how photons are “emitted,” which is invisible to thought. All 
+we can see are traces of photons, or what we assume to be photons, traces manifested, literally visible, in 
+spectra. Similarly, a macro quantum object (still defined as such by its microscopic quantum 
+constitution), such as a Josephson device, can only be detected as quantum by means of a suitable 
+instrument. Otherwise, it will be observed as a classical object and as such as something (two 
+superconductors standing in a lab) available our immediate phenomenal perception. 
+Thus, in quantum physics, on the one hand, there is always a discrimination between an object and an 
+instrument, and, on the other, their indivisibility in quantum phenomena, or what Bohr calls the 
+wholeness of phenomena or its closed nature, from which one can never extract the object itself at the 
+time of measurement. Any investigation in quantum theory must involve this combination, which is also 
+ 
+23 I am indebted to Laurent Friedel on this point. 
+
+ 
+38 
+that of what is invisible to thought and as such cannot be communicated unambiguously, and what is 
+visible to thought, via observational instruments, and can be communicated unambiguously. This 
+situation thus sharply contrasts with that of classical physics or relativity, where the role of measuring 
+instruments can be neglected or controlled and where, as a result, which always deal with what is visible 
+to thought and, as such unambiguously communicable or sharable as information. If the object under 
+investigation is classical (visible, representable, with its character unambiguously communicable, and so 
+forth), like the cat in the cat experiment, it can always be considered independently apart from quantum 
+experiments and discussed unambiguously. There is, as noted, never any ambiguity in assessing the cat as 
+an independent object in the cat experiment but only two unambiguously defined possibilities, each 
+visible to our mind’s eye, of the cat being either dead or alive, with the probability defined by the 𝜓-
+function associated with the atomic decay and only secondarily to the state, always classical, of the cat. A 
+cat, inside or outside the box, is always a cat, dead or alive. As such it can only be seen as a physically 
+classical part of the arrangement, inside the box, before the interaction with the particle emitted by an 
+atom, which can never be observed (and terms like particle or emission cannot apply in any sense we can 
+attribute to these terms). The cat can be on both sides of the event (or the cut), but the radioactive decay 
+or the particle emitted by it can only be on one side, the side of the object, and never the measurement 
+side. This emission occurs or not regardless of the cat in the box, or the box, or the flask, all of which are 
+classical and are parts of the arrangement made by us, while the radioactive atom is prepared by nature. 
+The cat could be removed from the box in advance without affecting this possible quantum event. The 
+flask is the only classical object that interacts with the emitted particle, which enable one to register with 
+the presence of the emission, if it occurs.  
+Technically, one need not see the opening the box as a quantum experiment, as the properly quantum 
+experiment in the arrangement, which is the shattering of the flask, occurs (if it does) before the box is 
+opened, and then the outcome of this experiment leads to the event that classically affects the cat. The cat 
+is more like an “agent,” akin to (although of course not the same as) “Wigner’s friend” in a related 
+famous experiment, than an instrument, and is, again, never a properly quantum object.24 Neither, again, 
+is anything else in the arrangement, except the flask, inside the box. But, as explained, even if one does 
+see the whole arrangement and an instrument (an arrangement and an instrument made by us as humans), 
+the cat, just as the box, it is still only a classical part of this arrangement, always visible to our mind’s eye 
+ 
+24 Although it has additional complexities, the Wigner’s friend experiment [Wigner 1961] can be considered along 
+the lines of the argument advanced here. In Wigner’s scenario, “the friend” hidden from “Wigner” inside some lab 
+(just as the cat is hidden from the observer in the cat experiment), performs an experiment on a previously prepared 
+quantum system, S, with the outcome, which, unlike the initial preparation, is hidden from “Wigner” as well. 
+“Wigner” leaves the lab after the initial preparation, which enables one to associate with S (which is, in the present 
+view, not the same as assign to S) the wave function |𝜓⟩, known to both. QM can, then, be used by “Wigner” in 
+estimating this hidden outcome. This is, I would argue, possible while, just as in the present view of the cat 
+experiment, considering “the friend” as a classical object within the overall arrangement, which, as that of the cat 
+experiment, contains a proper quantum object, S. The case would require a separate analysis. I might note, however, 
+that most discussions of the Wigner’s friend experiment and the problems and paradoxes found in many of them, 
+beginning with Wigner’s own encounter assume that “the friend” (or sometimes “Wigner”) can be considered as a 
+quantum object. For more recent treatments, see [Pusey 2018, Bauman and Brukner 2020, DeBrota et al 2020], and 
+further references in these articles. It is not my aim to assess these arguments (sometimes questioning each other) as 
+concerns their effectiveness in resolving the “paradoxes” of Wigner’s experiment and Wigner’s own initial 
+argument, which, I would argue, in effect suggests, even if against Wigner’s own grain, the difficulty of assuming 
+that the friend is a quantum object. Another, related, feature of some of recent arguments, most especially [DeBrota 
+et al 2020], which is based in quantum Bayesianism (QBism), is their claim of the subjective nature not only of our 
+predictions, a view assumed here as well, but also of the outcomes of quantum measurements, a view not assumed 
+here. In the present view, following Bohr, these outcomes are objective in the sense of being unambiguously 
+communicable (with further qualifications given above). This assumption is correlative to that of the physically 
+classical description of the observable part of measuring instruments and quantum phenomena. In the present view, 
+“the friend” is a classical object, just as is the cat in the cat experiment, even though the arrangement considered can 
+be treated by QM as concerns “Wigner’s” estimates of the outcome of the friend’s measurement.  
+
+ 
+39 
+or directly if the box has glass walls, even if the arrangement requires us to use QM to predict what 
+happens once one opens the box. The cat or the friend in the Wigner’s friend experiments is not an 
+instrument either, and either arrangement implies the presence of an observational instrument capable of 
+interacting with quantum objects and registering (classically) the outcome of such interactions. The cat 
+does not, of course, consciously observe such an outcome in the way a human agent would. The cat can 
+only manifest this outcome by being dead or alive. One the other hand, the friend in Wigner’s friend 
+experiment does consciously observes it, which fact was central to Wigner’s original argument. This 
+difference, however, does not change the fact that the cat or the friend is a classical object or that both 
+experiments essentially depend on the role of properly quantum objects, which are never of the 
+measurement side of the event. Nor could they serve as measuring instruments, which requires to have a 
+classical part in order to be observed by an agent. Nor of course could quantum objects serve as agents. 
+ 
+5. Conclusion 
+ 
+I return, in closing, to Denmark, first, not to that of Bohr but that of Shakespeare and Hamlet three 
+centuries earlier, and the lines of the play, used as my first epigraph: 
+ 
+Hamlet [commenting on his dead father]: 
+My father—methinks I see my father. 
+Horatio: 
+Where, my lord? 
+Hamlet: 
+In my mind’s eye, Horatio. 
+[The Tragedy of Hamlet, Prince of Denmark, Act 1, Scene 2, ll. 183-185] 
+ 
+The reason for Horatio’s puzzlement is that he saw the ghost of Hamlet’s father and wondered if perhaps 
+Hamlet has already seen the ghost as well, which Hamlet’s response proves not to be the case. Hamlet’s 
+encounter with the ghost of his father is yet to come. At stake in this scene is Hamlet’s image of his father 
+in his mind’s eye, and thus as something visible to thought. This image shadows Hamlet and the play 
+from beginning to end, and Hamlet’s encounter with the ghost, dramatic and consequential for the play, 
+adds to the power as this image in and over Hamlet’s mind, but this power was already there all along. I 
+am, however, not concerned here with much discussed psychological, such as psychoanalytic, 
+implications of this power, but instead with the capacity of our thought, conscious and unconscious, to 
+create an image of the world and of objects in the world, on which Shakespeare capitalizes in Hamlet and 
+his other works. No less remarkable, however, is our thought’s capacity to think that which is beyond 
+thought, is invisible to thought, and hence has no image in our mind’s eye, such as the ultimate nature of 
+the reality responsible for quantum phenomena. Shakespeare might have realized this capacity of thought 
+at least to some degree, as suggested by Hamlet’s comment to Horatio after his encounter with the ghost:  
+ 
+Horatio: O day and night, but this is wondrous strange! 
+Hamlet: And therefore as a stranger give it welcome. 
+There are more things in heaven and earth, Horatio,  
+Than are dreamt of in your philosophy. 
+[The Tragedy of Hamlet, Prince of Denmark, Act I, Scene 4, ll. 165-166] 
+ 
+Some editions have “our philosophy.” “Your philosophy” makes Hamlet more suspicious of philosophy. 
+It makes him more akin to a quantum physicist, who can only estimate the probabilities of future events 
+defined by experiments Hamlet stages at the castle of Elsinore, a prominent aspect of the play. There are, 
+quantum physics may indeed be telling us, things in, or beyond, heaven and earth that we cannot dream of 
+or otherwise see in our mind’s eye, consciously or unconsciously. 
+Bohr is reported to have replied, after the rise of quantum physics but before QM was discovered, to 
+H. Høffding’s question “Where can the photon be said to be?” with “To be, to be, what does it mean to 
+
+ 
+40 
+be?” (cited in [Wheeler and Ford 1998, p. 131]). Bohr might have been echoing the most famous sentence 
+of Shakespeare’s Hamlet, “To be, or not to be, that is the question” (Act 3, ll. 1749), realizing that in 
+quantum physics one might want to or even must ask first “What does it mean to be?” (Hamlet’s famous 
+monologue is, too, about much more than merely deciding to live or die.) Høffding’s and Bohr’s 
+questions are still unanswered and, in Bohr’s ultimate, RWR-type, view, are unanswerable, when it comes 
+to quantum objects, such as photons. Even as invisible to thought, quantum objects are idealizations (and 
+hence still products of thought), in the present interpretation, ultimately only applicable at the time of 
+observation, even if Bohr himself did not go that far. Either way, such questions as “Where can something 
+be said to be?” or “When had something happened?” can only be asked about quantum phenomena, 
+observed in measuring instruments, and as such visible to thought, to our mind’s eye, or even to our 
+immediate perceptuon. Nature has no photons or electrons, any more than being or reality, including that 
+of the RWR-type. Admittedly, as are our thought and hence these concepts are created (we don’t know 
+how either) by our brains, which are part of by biological and neurological constitution, and in this sense 
+still by nature. This is, however, not the same as saying (as is done sometimes) that nature uses these 
+concepts through us. Rather, nature allows us to create concepts—daily, physical, philosophical, or 
+mathematical—and use them in considering our interactions with nature by means of technology, 
+beginning with that of our bodies, and again, our thought. It is this interaction and only this interaction 
+that enables us to idealize some part of the constitution of nature, even its ultimate constitution, as 
+something that is invisible to thought and hence that cannot appear in our mind’s eye. 
+This brings me to my second epigraph “To die for the invisible—this is metaphysics,” courtesy of E. 
+Levinas’s book, Totality and infinity: An essay on exteriority [Levinas 2012, p. 39] (originally published 
+in French in 1961). The book is about ethics (as is, along with its many other themes, Hamlet as well), 
+and as such it might appear distant from quantum physics. Levinas’s epistemology, however, advanced in 
+this book not only shares some of the philosophical genealogy, for example, in Kant’s philosophy, with 
+quantum theory, but might have more direct connections with it. Quantum theory and its epistemological 
+problems, and possibly Bohr’s ideas, were known to Levinas, as they were widely discussed on the 
+French intellectual scene to which Levinas’s work belongs. Levinas’s concept of exteriority, expressly 
+associated by him with “the invisible” [l’invisible], has manifested affinities with the idea of invisible to 
+thought and even strictly means, in the ethical domain, that which is invisible to thought. My main 
+interest here is the association, quite dramatic—“To die for the invisible!” No less!—of the invisible with 
+metaphysics, which I would like to connects to quantum physics and indeed to all modern physics, from 
+Galileo on, as a mathematical-experimental science. Insofar as one means by metaphysics something 
+exterior to nature or, to use the ancient Greek word, physis, especially referring by metaphysics to 
+something theological, modern physics excludes metaphysics. There is no metaphysics. On the other 
+hand, insofar, because the idea of physics as a mathematical-experimental science or the idea of nature in 
+the first place still belongs to thought, even when nature is invisible to thought, there is only metaphysics. 
+Modern physics navigates and negotiates between both, “no metaphysics” and “only metaphysics.” How 
+close we come, in modern physics, to understanding nature, including in its ultimate constitution, even if 
+the latter is ultimately invisible to thought, depends on our interactions with nature by means of 
+experimental technologies and mathematics. These interactions are part of nature, too, but a particular 
+part of it, specific to us, to our thinking and technologies, beginning with that of our bodies and brains, 
+which are responsible for our thought. Our thought, however, also has a capacity to reach what is beyond 
+it, is invisible to it, and to affirm it: to die for the invisible—this is metaphysics.  
+ 
+Acknowledgments. This paper was in part prompted by the question by Lorenzo Maccone 
+concerning the relationships between the cat paradox and Bohr’s complementarity, and the subsequent 
+exchanges, for which I thank him, even though our views concerning quantum theory are very different. I 
+am grateful to G. Mauro D’Ariano for exceptionally illuminating conversations concerning the subjects 
+considered the article and fundamental physics in general. I am also happy to thank Gregg Jaeger and 
+Andrei Khrennikov for many discussions on quantum foundations.  
+ 
+
+ 
+41 
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+ 
+
diff --git a/wtFPT4oBgHgl3EQf_TXz/content/tmp_files/2301.13219v1.pdf.txt b/wtFPT4oBgHgl3EQf_TXz/content/tmp_files/2301.13219v1.pdf.txt
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+DESY-23-015
+Exploring Inelasticity in the S-Matrix Bootstrap
+Ant´onio Antunesa,b, Miguel Costaa, Jos´e Pereiraa
+a Centro de F´ısica do Porto, Departamento de F´ısica e Astronomia,
+Faculdade de Ciˆencias da Universidade do Porto,
+Rua do Campo Alegre 687, 4169-007 Porto, Portugal
+b Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
+The modern S-Matrix Bootstrap provides non-perturbative bounds on low-energy aspects of scat-
+tering amplitudes, leveraging the constraints of unitarity, analyticity and crossing. Typically, the
+solutions saturating such bounds also saturate the unitarity constraint as much as possible, meaning
+that they are almost exclusively elastic. This is expected to be unphysical in d > 2 because of Aks’
+theorem. We explore this issue by adding inelasticity as an additional input, both using a primal
+approach in general dimensions which extends the usual ansatz, and establishing a dual formulation
+in the 2d case. We then measure the effects on the low-energy observables where we observe stronger
+bounds than in the standard setup.
+Introduction. Scattering amplitudes are some of the
+most studied observables in quantum field theory. They
+encode the probability amplitudes of transitions between
+asymptotic states with a definite number of particles.
+The simplest non-trivial amplitude
+in⟨p1, p2|p3, p4⟩out,
+corresponding to 2-2 scattering has been extensively
+studied for decades, notably through Feynman pertur-
+bation theory, which extracts the connected amplitude
+through the LSZ procedure which takes as input a four-
+point correlation function. Additionally, modern on-shell
+perturbative techniques formulate these amplitudes in
+terms of simpler objects and give illuminating insights on
+the structure of the answers [1]. However, these methods
+are of limited applicability and general methods to tackle
+the non-perturbative case are lacking.
+A glimmer of hope is provided by the S-matrix Boot-
+strap, which takes the 2-2 amplitude and imposes severe
+constraints on its form by imposing elementary proper-
+ties that should hold non-perturbatively: unitarity, an-
+alyticity and crossing symmetry [2–33]. To understand
+the space of possible scattering amplitudes, one shoots
+towards its boundaries, by maximizing along some finite-
+dimensional observable space. Natural candidates are the
+residues or cubic couplings g associated to a bound-state
+of mass mb, such that T(s, t) ∼ g2/(s − m2
+b), or the low-
+energy expansion around the crossing symmetric point
+s = t = u = 4m2/3, where one defines the coefficients
+Λab = ∂a
+s ∂b
+t T(4/3, 4/3, 4/3) ,
+(1)
+where we set m = 1. Maximizing or minimizing such ob-
+servables gives a finite-dimensional slice of the infinite-
+dimensional space of scattering amplitudes, but also
+yields explicit amplitudes saturating the bounds. Typ-
+ically, such solutions saturate the unitarity constraints
+[2].
+For example, in 2d, where we can write the full
+S-matrix, including the disconnected contribution as
+S(s) = 1 + iT(s)/(
+�
+s(s − 4)), one finds solutions that
+satisfy
+|S(s)| ≈ 1,
+s ⩾ 4 ,
+(2)
+saturating the unitarity condition |S(s)| ⩽ 1 up to very
+high energies.
+This saturation is expected in the so-
+called elastic unitarity region, where the only possible
+end-states are precisely the two-particle states one starts
+with. On the other hand, at large enough energies, new
+normal thresholds will emerge, for example at s = (nm)2,
+corresponding to the n−particle threshold or at 4m2
+b cor-
+responding to the continuum of two-bound state pairs.
+The fact that |S(s)| = 1 implies that the amplitude is
+purely elastic, meaning that there is no particle produc-
+tion S2→X̸=2 = 0. This is, of course, the key property of
+integrable field theories in 2 dimensions, which evidently
+saturate the corresponding bounds.
+On the other hand, in higher dimensions, Aks’ theorem
+ensures that non-trivial scattering implies non-vanishing
+particle production, albeit only asymptotically at large
+energies [34]. A quantitative lower bound on this pro-
+duction was derived by Roy and Martin [35], stating that
+σinel(s) ≳ const
+s5/2 Exp
+�
+−9√s
+8
+ln s
+�
+.
+(3)
+On the other hand, solutions to the higher dimensional
+S-Matrix bounds in practice also remain elastic up to very
+high energies [3]. Decomposing onto partial amplitudes
+of well-defined angular momentum ℓ, one observes
+|Sℓ(s)| ≈ 1,
+s ⩾ 4,
+(4)
+as can be seen for example in figure 1, for the case of
+maximizing the ”quartic coupling” Λ00. Presumably, for
+the physics of inelasticity and particle production to be
+more easily captured by the bootstrap, the constraints as-
+sociated to higher-point amplitudes should be included.
+For example, by including 3-particle states, one can re-
+quire that the matrix of inner products between all 2 and
+3 particle states is positive semi-definite.
+This would
+then impose non-trivial constraints between 4-, 5- and
+6-point amplitudes. In practice however, technical con-
+trol over higher-point amplitudes is very limited non-
+perturbatively and their analytic and crossing properties
+are an important open problem.
+arXiv:2301.13219v1  [hep-th]  30 Jan 2023
+
+2
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+ϕ
+0.990
+0.992
+0.994
+0.996
+0.998
+1.000
+Nmax = 4
+Nmax = 6
+Nmax = 8
+Nmax = 10
+Nmax = 12
+FIG. 1: The absolute value of S0 for several Nmax. φ is
+the argument of the complex ρ variable defined below,
+such that s correspondingly increases from 4 to infinity.
+In this paper, we take a more pragmatic and ex-
+ploratory approach. We consider the following question:
+Given some postulated inelasticity profile, how much do
+the S-matrix bounds actually change? We will see that
+the answer is certainly non-zero and we will quantify it
+in several examples. Along the way, we will develop an
+ansatz that correctly captures the additional thresholds
+and speculate on less standard maximization problems
+which might inherently capture particle production [36].
+There are a few motivations to do this. First, in grav-
+itational scattering there is a natural behaviour for the
+inelastic part of the amplitude, stemming from black-
+hole production and subsequent Hawking radiation. In-
+deed, one expects the elastic part of the amplitude to be
+exponentially small at large s, as a consequence of the
+Bekenstein-Hawking formula [33]. Secondly, in realistic
+particle-physics setups, as in pion scattering in QCD, one
+finds that the physical theories are often not so close to
+the bounds derived by the S-matrix Bootstrap [7, 15].
+Information regarding the total cross-section, which of
+course captures the inelastic physics as well, could help
+bring QCD closer to the bound.
+Finally, in the sim-
+plest non-integrable QFT, 2d Ising field theory, recent
+advances in quantum spin chain simulations are on the
+brink of allowing us to study 2 → 3 scattering [37]. How
+such knowledge could improve the Bootstrap bounds and
+zoom in on a solution of the theory is an interesting ques-
+tion as well.
+Primal
+in
+2d.
+We begin from the standard
+Mandelstam-like ρ−ansatz for a 2-dimensional S-matrix
+of 2−2 scattering of mass m particles with a bound state
+of mass mb [3]
+S(s) =
+�
+−Jρ g2
+ρ
+ρ(s) − ρ(m2
+b) + s ↔ t
+�
++
+Nmax
+�
+a,b=0
+c(ab) ρa
+sρb
+t
+(5)
+where
+ρs =
+√4 − s0 − √4 − s
+√4 − s0 + √4 − s,
+s = s0(1 − ρs)2 + 16ρs
+(1 + ρs)2
+, (6)
+This ansatz uses an analytic extension which assumes
+only the normal thresholds in s > 4m2 and t > 4m2 si-
+multaneously. Each ρ variable then maps the cut plane
+to a disk with the center at s0 which is henceforth set to
+2. Imposing the relation s + t = 4m2 yields a crossing-
+symmetric S-matrix with the expected analytic proper-
+ties.
+To make it a full-fledged S-matrix, one imposes
+unitarity for physical scattering energies
+|S(s)| ⩽ 1 ,
+s ⩾ 4m2 .
+(7)
+This statement follows from the elementary quantum me-
+chanical principle of probability conservation
+�
+X
+|S2→X|2 = 1 ,
+(8)
+which of course bounds the absolute value of the 2 − 2
+component S(s) by 1. As mentioned above, integrable
+theories satisfy the above principle with a single term,
+the elastic component. Generic theories, e.g. φ4 theory,
+will admit for example a S2→4 component which will then
+ensure that the absolute value of S(s) is strictly less than
+1. We parametrize this by an inelasticity profile β(s),
+thereby refining the bootstrap problem to
+|S(s)|2 ⩽ β(s),
+s ⩾ 4m2;
+β(s) ⩾ 0;
+(9)
+and we require the profile to have the following structure:
+β(s) =
+�
+1,
+4 ⩽ s ⩽ 16
+profile(s)
+s > 16
+(10)
+We are hence trying to impose elastic unitarity, mean-
+ing we only allow particle production after the 4-particle
+threshold energy has been achieved.
+Of course, this
+choice is arbitrary and we can replace 16 by s∗, a tunable
+threshold value, which could be a 3-particle or 2-bound
+state threshold. We will study the effect of this param-
+eter later on. We will see that complying with elastic
+unitarity |S(s)| = 1 for s < s∗ is challenging for the
+usual ansatz (5), and this will be one of the motivations
+to construct a more general parametrization.
+We can now try to study standard maximization prob-
+lems, for example, maximizing the cubic coupling to the
+bound state. The solution to this problem is given by
+the sine-Gordon breather S-matrix, but by imposing a
+non-trivial profile, we will explore the interior of this
+bound. For an alternative way to explore this interior
+see Appendix A. In 2-dimensions, we are blessed with an
+explicit solution to the S-matrix which reconstructs the
+phase given its absolute value [2, 20]. It is given by
+S
+Sel
+= exp
+�� ∞
+4
+ds′
+2πi
+�
+st
+s′t′
+�log(β(s′))
+s − s′
++ s ↔ t
+��
+(11)
+
+3
+0.2
+0.4
+0.6
+0.8
+1.0 α
+18.0
+18.5
+19.0
+19.5
+20.0
+20.5
+g1
+2
+Analytical
+Numerical
+FIG. 2: Direct comparison between profile β(s) and the
+analytical results
+10
+20
+30
+40
+50
+60s
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Absolute
+Real
+Imaginary
+FIG. 3: S-matrix components for the inelastic profile
+β(s) with α = 0.5 and the analytical results in red
+dashes lines.
+and will allow us to check our results. We will of course
+not have such a luxury in four dimensions. Let us con-
+sider then two simple profiles:
+β(s) =
+�
+1, 4 ⩽ s ⩽ s∗
+1 − α, s > s∗
+β(e) =
+�
+1, 4 ⩽ s ⩽ s∗
+e−α√s−s∗ s > s∗
+(12)
+Using the standard maximization algorithms, we find the
+results of figure 2, which used Nmax = 5 and a grid of 100
+points for unitarity. Very clearly, the standard algorithm
+is under-performing. Looking with more detail into the
+numerical solution, we find the S-matrix components of
+figure 3.
+It is easy to see that not only the threshold
+discontinuity is not correctly captured, but that it is also
+quite difficult to satisfy elastic unitarity and to saturate
+the inelasticity profile we imposed. Similar results are
+found for the other profile, and for other problems, for
+example extremization of S(s = 2) subject to the β(s)
+constraints.
+To capture the effects of the additional threshold, we
+first introduce a simple generalization of the ρ variables,
+which map the cut plane starting at the new threshold
+s∗ to the disk. They manifestly satisfy the original ana-
+0.2
+0.4
+0.6
+0.8
+1.0 α
+18.0
+18.5
+19.0
+19.5
+20.0
+20.5
+g1
+2
+Analytical
+Numerical
+FIG. 4: Direct comparison between the profile β(s) and
+the analytical results. The bound-state mass is fixed,
+m2
+1 = 3, Nmax = 5 and ¯Nmax = 2.
+lyticity assumptions and are given by
+¯ρs ≡
+√s∗ − s0 − √s∗ − s
+√s∗ − s0 + √s∗ − s,
+(13)
+It is then natural to extend the ansatz for the S-matrix
+to have the form
+Simproved(s) = S(s) +
+¯
+Nmax
+�
+a,b=0
+d(ab) ¯ρa
+s ¯ρb
+t ,
+(14)
+where we introduced an independent cutoff ¯Nmax. As for
+the standard ρ variable, the ¯ρ satisfy the simple algebraic
+identity
+¯ρs + ¯ρt + 4¯ρs¯ρt + ¯ρ2
+s ¯ρt + ¯ρs¯ρ2
+t = 0 ,
+(15)
+which allows us to eliminate several of the coefficients in
+(14). In fact we will keep exactly the same a, b as we
+do for the coefficients in (5).
+It is easy to check that
+additional terms involving products of the different ρ’s
+and similar variables parametrizing a discontinuity be-
+tween 4 and s∗ are redundant, and don’t further improve
+the bounds [38]. We can now repeat the cubic coupling
+maximization, yielding figure 4. Not only do the milder
+discontinuities get a significant enhancement, but also
+the sharper ones as α → 1 appear to be getting quali-
+tatively reproduced. A look at the S-matrix extremizing
+the bounds, figure 5, also reveals that the additional dis-
+continuity is correctly captured, and elastic unitarity is
+satisfied until much closer to the threshold. In fact, we
+improve the convergence by at least an order of magni-
+tude. It turns out that this ansatz will work remarkably
+well in four dimensions, but we first establish even more
+accurate results in two dimensions by resorting to the
+dual formalism [18, 21, 24].
+Dual in 2d. In terms of optimization language, the
+primal problem is
+maximize
+{T (s), g2
+1}
+g2
+1
+(16)
+
+4
+20
+40
+60
+80
+100
+-1.0
+-0.5
+0.5
+1.0
+Abs
+Re
+Im
+FIG. 5: S-matrix components for the inelastic profile
+β(s) using the new Ansatz of (14). In red are the
+components of the analytical solution. Here, the
+parameters used were m2
+1 = 3, Nmax = 10 and
+¯Nmax = 10 and α = 0.6.
+subject to the constraints
+A(s) ≡ T(s) +
+g2
+1
+s − m2
+1
+−
+� ∞
+4m2
+dz
+π
+Im T(z)
+s − z
+− (s ↔ t) = 0
+(17)
+and the new unitarity condition, which is given by
+U(s) ≡ 2
+�
+s(s − 4) (1 − β(s)) + 2 Im T(s)
+−
+1
+2
+�
+s(s − 4)
+|T(s)|2 ⩾ 0 .
+(18)
+To achieve a dual formulation we build the Lagrangian
+density L(T, ω, λ),
+L(T, ω, λ) = g2
+1 +
+� ∞
+4m2 ds ω(s)A(s) + λ(s)U(s) ,
+(19)
+where λ ⩾ 0. Using the weak duality principle and in-
+troducing the function W(s) satisfying ImW(s) = ω(s),
+which does not depend on the form of the unitarity con-
+dition, we arrive at the expression
+L(T, ω, λ) =
+� ∞
+4m2 ds Im [W(s)T(s)] + λ(s)U(s)
+(20)
+provided that Re W(m2
+1) = −1/π. We conclude that the
+correct dual optimization problem is
+min
+{W } D(W) =
+� ∞
+4m2
+ds
+ρ2
+11(s) [|W(s)|β(s) + Re W(s)] ,
+(21)
+where ρ−2
+11 = 2
+�
+s(s − 4) subject to the condition that
+W(m2
+1) = − 1
+π ,
+(22)
+where β(s) are the profiles used above. Lastly, it is nec-
+essary to propose an ansatz to carry out the numerical
+extremization. The naive ρ(s) expansion can still be used
+W(s) =
+1
+s(4m2 − s)
+Nmax
+�
+n=1
+an(ρ(s)n − ρ(t)n) ,
+(23)
+0.2
+0.4
+0.6
+0.8
+1.0 α
+18.0
+18.5
+19.0
+19.5
+20.0
+20.5
+g1
+2
+Analytical
+Numerical
+FIG. 6: Comparison between the analytical expression
+(11) and the data for β(s) with some tiny error bars.
+5
+10
+15
+20
+25
+30 s
+-1.0
+-0.5
+0.5
+1.0
+Real
+Imaginary
+Absolute
+FIG. 7: S-matrix components for the inelastic profile
+β(s) for the dual approach vs. the analytical
+components, red dashed curves. m2
+1 = 3, Nmax = 40,
+α = 0.5 and s∗ = 16 were used.
+where Nmax is again the numerical cutoff. The fact that
+the same Ansatz was used even though the S-matrix en-
+codes additional thresholds is remarkable when compared
+to the primal formalism. This follows from the fact that
+the absolute value of the S-matrix is fixed in the dual pro-
+cedure because of the strong duality principle. Namely
+T(s) =
+i
+ρ2
+11(s)
+�
+1 + W ∗(s)
+|W(s)|
+�
+β(s)
+�
+,
+(24)
+which has the correct absolute value regardless of W.
+Revisiting the cubic coupling maximization yields figure
+6. Here we performed a simple extrapolation in Nmax
+which agrees with the analytic result to one part in 10−4
+and estimated some (very small) error bars. A look at S-
+Matrix components, figure 7, confirms the expectations.
+The absolute value exactly matches the profile β(s), and
+the real and imaginary parts nicely follow the expected
+singular behavior.
+Primal in 4d.
+The natural way to generalize the
+inelastic constraints discussed in 2 dimensions above, is
+to restrict the absolute values of the partial amplitudes
+Sℓ = 1 + i
+�
+(s − 4)/s fℓ, where fℓ are the partial waves
+
+5
+●
+●
+●
+●
+●
+●
+●
+●
+■
+■
+■
+■
+■
+■
+■
+■
+★
+★
+★
+★
+★
+★
+★
+6
+8
+10
+12
+14
+2.55
+2.60
+2.65
+2.70
+Nmax
+λmax
+●
+In., ℓmax = 10
+■
+In., ℓmax = 12
+★
+In., ℓmax = 14
+Elastic, ℓmax = 10
+Elastic, ℓmax = 12
+Elastic, ℓmax = 14
+FIG. 8: Result of maximizing the quartic coupling
+subject to (26), for profile β(s)
+0 (s) with α = 0.5 and
+s∗ = 16. The dashed lines correspond to the solution
+with unitarity only.
+normalized as in [16]. We then impose
+|Sℓ(s)|2 ⩽ βℓ(s),
+s ⩾ 4.
+(25)
+In this section we will study step profiles, identical to
+the ones used in the previous sections. In practice, we
+will use non-trivial inelasticity profiles only for the low
+spin partial-waves and require standard unitarity for the
+remaining ones.
+To use (25) in terms of semi-definite
+programming we rewrite it in terms of a matrix inequality
+�
+�
+1 −
+Im aℓ
+1+√
+βℓ(s)
+Re aℓ
+Re aℓ
+βℓ(s) − 1 +
+�
+1 +
+�
+βℓ(s)
+�
+Im aℓ
+�
+� ⪰ 0
+(26)
+where aℓ =
+�
+(s − 4)/s fℓ. In 4d, we will stick to the sim-
+plest maximization problem, which can be studied even
+for a model without bound states: Maximization of the
+quartic coupling λ ≡ Λ00/32π. Motivated by the 2d ex-
+plorations, we start from an ansatz
+T(s, t, u) = κ
+�
+1
+ρs − 1 + cross.
+�
++
+Nmax
+�
+a,b,c=0
+αabc ρa
+sρb
+tρc
+u
++
+¯
+Nmax
+�
+a,b,c=0
+βabc ¯ρa
+s ¯ρb
+t ¯ρc
+u
+(27)
+where we use the ¯ρ variables discussed above, and add
+a pole at threshold, parametrized by κ, which is known
+to capture the physics of the extremal S-Matrix, in the
+standard case [3]. First, we consider a step profile for the
+spin zero partial wave β(s)
+0 (s), with α = 0.5 and s∗ = 16.
+Even when we don’t use the new terms in the ansatz,
+i.e.
+when ¯Nmax = 0, we find a substantial difference
+compared to the ”elastic” case, as can be seen in fig-
+ure 8: It seems that λ roughly achieves a plateau, which
+FIG. 9: Result of maximizing the quartic coupling
+subject to (26), for profile β(s)
+0 (s) with α = 0.8 and
+s∗ = 16. We find a plateau λ ≈ 2.61 around ¯Nmax ≈ 5.
+is clearly below the original bounds, where no inelastic-
+ity was imposed. However, extending the ansatz shows
+that this difference is somewhat overestimated.
+Vary-
+ing the parameter ¯Nmax, with Nmax = 14, one produces
+figure 9, where we considered a somewhat sharper discon-
+tinuity with α = 0.8. Clearly, the quartic coupling now
+plateaus at a slightly higher value, λmax ≈ 2.61 , which is
+quickly achieved by adding merely a few ¯ρ terms. We can
+now also look at the partial amplitudes, and observe how
+the particle production is realized, as shown in figure 10.
+Plotting the spin zero amplitude S0, we can observe the
+important effect of the more general ansatz. Not only
+is the transition smoother and less oscillatory but the
+constraints are more sharply saturated, and in particular
+elastic unitarity is satisfied remarkably well.
+Having established the reliability of the new ansatz
+(27), we now give a more interesting application. For the
+profile β(s)
+0 , we study how the maximum quartic coupling
+varies as we tune both the strength of the discontinuity
+α and the start of the inelastic threshold s∗. The results
+are presented in figure 11 [39]. As expected, both making
+the discontinuity softer (decreasing α), and pushing the
+inelastic threshold further way (increasing s∗) make the
+bounds approach from below the value obtained imposing
+simply unitarity, λmax ≈ 2.66. We also note that for this
+profile it is possible to accurately replicate very sharp
+discontinuities, as we find stable results, even for α as
+small as 0.01.
+As a final application of the methods we developed,
+we also consider the effect of inelasticity on the spin 2
+partial wave f2. Maximizing the quartic coupling λ with
+profile β(s)
+2
+with parameters α = 0.8 and s∗ = 16 leads
+to effects that are more severe than the corresponding
+spin 0 problem.
+A plateau is only achieved for larger
+values of ¯Nmax, and the maximum value of the coupling
+drops considerably to about λmax ≈ 2.5. This is reason-
+able, since we are pushing against the Froissart-Gribov
+tendency to lower ImfJ as J increases, reducing the solu-
+tion space and hence, λmax. An interesting question for
+
+12.8253.0
+max
+13.5
+14.0
+2.64
+2.62
+Amax
+2.60
+10
+5
+N
+max6
+20
+40
+60
+80
+100s
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Re
+Im
+Abs
+α0 = 0.8
+20
+40
+60
+80
+100s
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Re
+Im
+Abs
+α0 = 0.8
+FIG. 10: Spin zero partial amplitude maximizing the
+quartic coupling subject to 26, for profile β(s)
+0 (s),
+Nmax = 14, ℓmax = 14 and α = 0.8 with ¯Nmax = 0 (top)
+and ¯Nmax = 9 (bottom).
+FIG. 11: Maximum quartic coupling λmax as a function
+of α and s∗, for a profile β(s)
+0 . The results have
+converged since changing ℓmax from 10 to 14 leaves the
+bounds unchanged. Here, Nmax = 10 and ¯Nmax = 6.
+the future is to understand what happens in the physical
+cases where the inelasticity decays asymptotically with
+spin, as in the black-hole production case, or with Dragt
+behavior [40].
+Such an understanding would require a
+reliable way to estimate the finite spin effects.
+Total Cross-section.
+Having obtained improved
+bounds on the couplings and captured the extra thresh-
+old, we procceed to analyze the total cross-section σt,
+or equivalently the forward limit T(s, 0) of the resulting
+amplitude. The relation between the two is simply
+Im T(s, 0) =
+�
+s(s − 4) σt =
+�
+ℓ
+16π(2ℓ + 1)Imfℓ , (28)
+●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
+●
+●
+●
+●
+●
+●
+●
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★
+★★★★★★★★★★★★★★★★★★★★★★★★★ ★ ★ ★ ★ ★
+★
+★
+★
+★
+★
+★
+20
+40
+60
+80
+100s
+20
+40
+60
+80
+100
+Im T(s,0)
+Unitarity
+●
+Solution
+▲
+Elastic
+Inelastic
+★
+Sum
+FIG. 12: Comparison between the solution where only
+unitarity is imposed, Unitarity; the numerical solution
+with inelasticity imposed in S0, Solution; the elastic
+part of the amplitude obtained from the optical
+theorem, Elastic; the imposed inelastic profile, Inelastic;
+and the sum of elastic and inelastic components Sum.
+and of course, we can decompose the cross-section into
+its elastic and inelastic parts. Notably, we can use the
+optical theorem to obtain the elastic part of the ampli-
+tude in the forward limit in terms of an integral of its
+square
+Im T(s, 0) ⩾
+1
+64π
+�
+s − 4
+s
+� 1
+−1
+dz |T(s, t(z))|2 ,
+(29)
+with z the cosine of the scattering angle. The inequality
+is saturated in the elastic unitarity region, and after that
+there are inelastic terms further contributing. This can
+also be used as a check to the validity of the bounds ob-
+tained when imposing inelasticity. We plot the forward
+amplitude corresponding to maximizing the quartic cou-
+pling with α = 0.2 and s∗ = 16 in figure 12. We also
+computed the integral in the optical theorem numeri-
+cally, and determined the inelastic contribution to the
+total cross section through the sum over partial waves in
+(28). We find good agreement between the sum of the
+elastic and inelastic pieces and the resulting cross-section
+obtained directly through the maximization process, as
+shown in figure 12.
+We find that the forward amplitude always exceeds its
+elastic counterpart denoted by the solid blue line, which
+is obtained by maximizing the quartic coupling imposing
+only unitarity. Indeed, this also suggests a natural way
+to parametrize the inelasticity: inputing a lower bound
+on the total cross-section, which is a simple generaliza-
+tion of imposing positivity in the forward limit.
+This
+constraint mixes information about all the partial waves,
+so it should lead to weaker bounds than imposing in-
+elasticity for each partial wave as in (26). In practice,
+
+0.2
+α 0.4
+0.6
+0.8
+(max = 10
+2.65
+(max = 12
+max2.60
+(max = 14
+16
+15
+14
+s*
+13
+127
+●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+50
+100
+150
+200
+s
+20
+40
+60
+80
+100
+Im T(s,0)
+●
+●
+165
+170
+175
+180
+185
+190
+195
+200
+0
+1
+2
+3
+4
+5
+6
+7
+s
+Unitarity
+●
+Solution
+▲
+Elastic
+Total cross-sec.
+FIG. 13: Comparison between the solution with only
+unitarity imposed, Unitarity; the solution with σt that
+follows from β(s)
+0 (s) with α = 0.2, Solution; the elastic
+part of the amplitude from the optical theorem, Elastic;
+and the imposed cross-section total cross-sec.
+to fully control the inelasticity, one needs to include the
+quadratic elastic piece which leads to a constraint that is
+hard to use. On the other hand, one can impose
+�
+Im T(s, 0) −
+�
+s(s − 4) σt 0
+0
+1
+�
+⪰ 0
+(30)
+where σ is just treated as a fixed function of s. In prac-
+tice, this just leads to saturation of this profile once the
+elastic cross-section exceeds this value, as shown in fig-
+ure 13. We also point out that due to the nature of the
+ρ expansion, the forward limit of the amplitude always
+decays at large s. It would be nice to study extensions
+which allow for Froissart-like behavior.
+Discussion. In this paper, we explored how inequal-
+ities stronger than unitarity, i.e. imposing an inelastic-
+ity profile, yields stronger bounds for the low-energy ob-
+servables one studies with the S-matrix bootstrap. To
+correctly capture the new thresholds, we introduced the
+¯ρ variables, which drastically improve convergence and
+allowed for the satisfaction of elastic unitarity. As ex-
+pected, we found that the maximum coupling decreases
+as we increase the amount of inelasticity, which can be
+achieved by making the discontinuity sharper or by low-
+ering the energy of the first inelastic threshold, as quanti-
+fied in figure 11. We also found that the same amount of
+inelasticity imposed on a higher spin partial wave further
+decreases the maximum coupling.
+The previous analysis of the cross-section suggests
+that it could be possible to obtain inelastic S-matrices by
+maximizing the total cross-section. Of course, to maxi-
+mize the total cross-section at a fixed value of the energy
+is not expected to be a physically well posed problem.
+The existence of resonances allows for sharp peaks in the
+cross-section, which are obviously physical and should
+not be excluded.
+A naive use of the maximization
+methods, leads to results that do not seem to converge
+as one increases Nmax, confirming the above expectation.
+Curiously, for fixed Nmax, and varying s, the values
+seem to roughly follow a Froissart-like trend. It would
+be interesting to understand how this would change for
+ans¨atze that allow for amplitudes that grow at infinity.
+On the other hand, suitable integrals of the cross section
+over energy should eliminate the problem mentioned
+above, and might prove to be an interesting observable.
+We thank C. Bercini, M. Correia, V. Goncalves, A.
+Guerrieri, A. Hebbar, A. Homrich, J. Penedones and P.
+Vieira for illuminating discussions and useful comments
+and M. Correia and A. Hebbar for useful comments on
+the draft. This research received funding from the Si-
+mons Foundation grant 488637 (Simons collaboration on
+the non-perturbative bootstrap).
+AA received funding
+from the German Research Foundation DFG under Ger-
+many’s Excellence Strategy – EXC 2121 Quantum Uni-
+verse – 390833306. Centro de F´ısica do Porto is partially
+funded by Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT)
+under the grant UID04650-FCUP.
+Appendix A. In the main text, we used inelasticity
+as a way to probe the interior of the bootstrap bounds.
+Here, we study a simpler problem which also allows us
+to visit non-extremal theories with respect to maximiz-
+ing the cubic coupling. We fix the cubic coupling to be
+0 < g1 < gmax, and then maximize and minimize the
+quartic coupling S(2). The result is figure 14. At the tip
+of maximum coupling, the solution is unique and is sat-
+urated by sine-Gordon. In the other extreme g1 = 0, the
+bound state decouples, and the maximum and minimum
+theories are just a free boson and fermion respectively.
+In between, there are non-trivial solutions, generated by
+products of CDD factors. One of the solutions can be
+seen as adding the T ¯T deformation in the vicinity of the
+sine-Gordon point.
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+S(2)
+FIG. 14: Plot of g1 vs. S(2) for m1 =
+√
+3. The blue
+curve represents the maximum value of S(2) while the
+orange represents the minimum.
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+
diff --git a/wtFPT4oBgHgl3EQf_TXz/content/tmp_files/load_file.txt b/wtFPT4oBgHgl3EQf_TXz/content/tmp_files/load_file.txt
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@@ -0,0 +1,633 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf,len=632
+page_content='DESY-23-015  Exploring Inelasticity in the S-Matrix Bootstrap  Ant´onio Antunesa,b, Miguel Costaa, Jos´e Pereiraa  a Centro de F´ısica do Porto, Departamento de F´ısica e Astronomia,  Faculdade de Ciˆencias da Universidade do Porto,  Rua do Campo Alegre 687, 4169-007 Porto, Portugal  b Deutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 85, 22607 Hamburg, Germany  The modern S-Matrix Bootstrap provides non-perturbative bounds on low-energy aspects of scat-  tering amplitudes, leveraging the constraints of unitarity, analyticity and crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Typically, the  solutions saturating such bounds also saturate the unitarity constraint as much as possible, meaning  that they are almost exclusively elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' This is expected to be unphysical in d > 2 because of Aks’  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We explore this issue by adding inelasticity as an additional input, both using a primal  approach in general dimensions which extends the usual ansatz, and establishing a dual formulation  in the 2d case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We then measure the effects on the low-energy observables where we observe stronger  bounds than in the standard setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Scattering amplitudes are some of the  most studied observables in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' They  encode the probability amplitudes of transitions between  asymptotic states with a definite number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  The simplest non-trivial amplitude  in⟨p1, p2|p3, p4⟩out,  corresponding to 2-2 scattering has been extensively  studied for decades, notably through Feynman pertur-  bation theory, which extracts the connected amplitude  through the LSZ procedure which takes as input a four-  point correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Additionally, modern on-shell  perturbative techniques formulate these amplitudes in  terms of simpler objects and give illuminating insights on  the structure of the answers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' However, these methods  are of limited applicability and general methods to tackle  the non-perturbative case are lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  A glimmer of hope is provided by the S-matrix Boot-  strap, which takes the 2-2 amplitude and imposes severe  constraints on its form by imposing elementary proper-  ties that should hold non-perturbatively: unitarity, an-  alyticity and crossing symmetry [2–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' To understand  the space of possible scattering amplitudes, one shoots  towards its boundaries, by maximizing along some finite-  dimensional observable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Natural candidates are the  residues or cubic couplings g associated to a bound-state  of mass mb, such that T(s, t) ∼ g2/(s − m2  b), or the low-  energy expansion around the crossing symmetric point  s = t = u = 4m2/3, where one defines the coefficients  Λab = ∂a  s ∂b  t T(4/3, 4/3, 4/3) ,  (1)  where we set m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Maximizing or minimizing such ob-  servables gives a finite-dimensional slice of the infinite-  dimensional space of scattering amplitudes, but also  yields explicit amplitudes saturating the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Typ-  ically, such solutions saturate the unitarity constraints  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  For example, in 2d, where we can write the full  S-matrix, including the disconnected contribution as  S(s) = 1 + iT(s)/(  �  s(s − 4)), one finds solutions that  satisfy  |S(s)| ≈ 1,  s ⩾ 4 ,  (2)  saturating the unitarity condition |S(s)| ⩽ 1 up to very  high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  This saturation is expected in the so-  called elastic unitarity region, where the only possible  end-states are precisely the two-particle states one starts  with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' On the other hand, at large enough energies, new  normal thresholds will emerge, for example at s = (nm)2,  corresponding to the n−particle threshold or at 4m2  b cor-  responding to the continuum of two-bound state pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  The fact that |S(s)| = 1 implies that the amplitude is  purely elastic, meaning that there is no particle produc-  tion S2→X̸=2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' This is, of course, the key property of  integrable field theories in 2 dimensions, which evidently  saturate the corresponding bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  On the other hand, in higher dimensions, Aks’ theorem  ensures that non-trivial scattering implies non-vanishing  particle production, albeit only asymptotically at large  energies [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' A quantitative lower bound on this pro-  duction was derived by Roy and Martin [35], stating that  σinel(s) ≳ const  s5/2 Exp  �  −9√s  8  ln s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  (3)  On the other hand, solutions to the higher dimensional  S-Matrix bounds in practice also remain elastic up to very  high energies [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Decomposing onto partial amplitudes  of well-defined angular momentum ℓ, one observes  |Sℓ(s)| ≈ 1,  s ⩾ 4,  (4)  as can be seen for example in figure 1, for the case of  maximizing the ”quartic coupling” Λ00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Presumably, for  the physics of inelasticity and particle production to be  more easily captured by the bootstrap, the constraints as-  sociated to higher-point amplitudes should be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  For example, by including 3-particle states, one can re-  quire that the matrix of inner products between all 2 and  3 particle states is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  This would  then impose non-trivial constraints between 4-, 5- and  6-point amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In practice however, technical con-  trol over higher-point amplitudes is very limited non-  perturbatively and their analytic and crossing properties  are an important open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='13219v1  [hep-th]  30 Jan 2023  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content='994  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='998  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='000  Nmax = 4  Nmax = 6  Nmax = 8  Nmax = 10  Nmax = 12  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 1: The absolute value of S0 for several Nmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' φ is  the argument of the complex ρ variable defined below,  such that s correspondingly increases from 4 to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  In this paper, we take a more pragmatic and ex-  ploratory approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We consider the following question:  Given some postulated inelasticity profile, how much do  the S-matrix bounds actually change?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We will see that  the answer is certainly non-zero and we will quantify it  in several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Along the way, we will develop an  ansatz that correctly captures the additional thresholds  and speculate on less standard maximization problems  which might inherently capture particle production [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  There are a few motivations to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' First, in grav-  itational scattering there is a natural behaviour for the  inelastic part of the amplitude, stemming from black-  hole production and subsequent Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In-  deed, one expects the elastic part of the amplitude to be  exponentially small at large s, as a consequence of the  Bekenstein-Hawking formula [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Secondly, in realistic  particle-physics setups, as in pion scattering in QCD, one  finds that the physical theories are often not so close to  the bounds derived by the S-matrix Bootstrap [7, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Information regarding the total cross-section, which of  course captures the inelastic physics as well, could help  bring QCD closer to the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Finally, in the sim-  plest non-integrable QFT, 2d Ising field theory, recent  advances in quantum spin chain simulations are on the  brink of allowing us to study 2 → 3 scattering [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' How  such knowledge could improve the Bootstrap bounds and  zoom in on a solution of the theory is an interesting ques-  tion as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Primal  in  2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  We begin from the standard  Mandelstam-like ρ−ansatz for a 2-dimensional S-matrix  of 2−2 scattering of mass m particles with a bound state  of mass mb [3]  S(s) =  �  −Jρ g2  ρ  ρ(s) − ρ(m2  b) + s ↔ t  �  +  Nmax  �  a,b=0  c(ab) ρa  sρb  t  (5)  where  ρs =  √4 − s0 − √4 − s  √4 − s0 + √4 − s,  s = s0(1 − ρs)2 + 16ρs  (1 + ρs)2  , (6)  This ansatz uses an analytic extension which assumes  only the normal thresholds in s > 4m2 and t > 4m2 si-  multaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Each ρ variable then maps the cut plane  to a disk with the center at s0 which is henceforth set to  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Imposing the relation s + t = 4m2 yields a crossing-  symmetric S-matrix with the expected analytic proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  To make it a full-fledged S-matrix, one imposes  unitarity for physical scattering energies  |S(s)| ⩽ 1 ,  s ⩾ 4m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  (7)  This statement follows from the elementary quantum me-  chanical principle of probability conservation  �  X  |S2→X|2 = 1 ,  (8)  which of course bounds the absolute value of the 2 − 2  component S(s) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' As mentioned above, integrable  theories satisfy the above principle with a single term,  the elastic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Generic theories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' φ4 theory,  will admit for example a S2→4 component which will then  ensure that the absolute value of S(s) is strictly less than  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We parametrize this by an inelasticity profile β(s),  thereby refining the bootstrap problem to  |S(s)|2 ⩽ β(s),  s ⩾ 4m2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  β(s) ⩾ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  (9)  and we require the profile to have the following structure:  β(s) =  �  1,  4 ⩽ s ⩽ 16  profile(s)  s > 16  (10)  We are hence trying to impose elastic unitarity, mean-  ing we only allow particle production after the 4-particle  threshold energy has been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Of course, this  choice is arbitrary and we can replace 16 by s∗, a tunable  threshold value, which could be a 3-particle or 2-bound  state threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We will study the effect of this param-  eter later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We will see that complying with elastic  unitarity |S(s)| = 1 for s < s∗ is challenging for the  usual ansatz (5), and this will be one of the motivations  to construct a more general parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  We can now try to study standard maximization prob-  lems, for example, maximizing the cubic coupling to the  bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The solution to this problem is given by  the sine-Gordon breather S-matrix, but by imposing a  non-trivial profile, we will explore the interior of this  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' For an alternative way to explore this interior  see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In 2-dimensions, we are blessed with an  explicit solution to the S-matrix which reconstructs the  phase given its absolute value [2, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' It is given by  S  Sel  = exp  �� ∞  4  ds′  2πi  �  st  s′t′  �log(β(s′))  s − s′  + s ↔ t  ��  (11)  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0 α  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  g1  2  Analytical  Numerical  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 2: Direct comparison between profile β(s) and the  analytical results  10  20  30  40  50  60s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  Absolute  Real  Imaginary  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 3: S-matrix components for the inelastic profile  β(s) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5 and the analytical results in red  dashes lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  and will allow us to check our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We will of course  not have such a luxury in four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Let us con-  sider then two simple profiles:  β(s) =  �  1, 4 ⩽ s ⩽ s∗  1 − α, s > s∗  β(e) =  �  1, 4 ⩽ s ⩽ s∗  e−α√s−s∗ s > s∗  (12)  Using the standard maximization algorithms, we find the  results of figure 2, which used Nmax = 5 and a grid of 100  points for unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Very clearly, the standard algorithm  is under-performing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Looking with more detail into the  numerical solution, we find the S-matrix components of  figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  It is easy to see that not only the threshold  discontinuity is not correctly captured, but that it is also  quite difficult to satisfy elastic unitarity and to saturate  the inelasticity profile we imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Similar results are  found for the other profile, and for other problems, for  example extremization of S(s = 2) subject to the β(s)  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  To capture the effects of the additional threshold, we  first introduce a simple generalization of the ρ variables,  which map the cut plane starting at the new threshold  s∗ to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' They manifestly satisfy the original ana-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0 α  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  g1  2  Analytical  Numerical  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 4: Direct comparison between the profile β(s) and  the analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The bound-state mass is fixed,  m2  1 = 3, Nmax = 5 and ¯Nmax = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  lyticity assumptions and are given by  ¯ρs ≡  √s∗ − s0 − √s∗ − s  √s∗ − s0 + √s∗ − s,  (13)  It is then natural to extend the ansatz for the S-matrix  to have the form  Simproved(s) = S(s) +  ¯  Nmax  �  a,b=0  d(ab) ¯ρa  s ¯ρb  t ,  (14)  where we introduced an independent cutoff ¯Nmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' As for  the standard ρ variable, the ¯ρ satisfy the simple algebraic  identity  ¯ρs + ¯ρt + 4¯ρs¯ρt + ¯ρ2  s ¯ρt + ¯ρs¯ρ2  t = 0 ,  (15)  which allows us to eliminate several of the coefficients in  (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In fact we will keep exactly the same a, b as we  do for the coefficients in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  It is easy to check that  additional terms involving products of the different ρ’s  and similar variables parametrizing a discontinuity be-  tween 4 and s∗ are redundant, and don’t further improve  the bounds [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We can now repeat the cubic coupling  maximization, yielding figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Not only do the milder  discontinuities get a significant enhancement, but also  the sharper ones as α → 1 appear to be getting quali-  tatively reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' A look at the S-matrix extremizing  the bounds, figure 5, also reveals that the additional dis-  continuity is correctly captured, and elastic unitarity is  satisfied until much closer to the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In fact, we  improve the convergence by at least an order of magni-  tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' It turns out that this ansatz will work remarkably  well in four dimensions, but we first establish even more  accurate results in two dimensions by resorting to the  dual formalism [18, 21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Dual in 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In terms of optimization language, the  primal problem is  maximize  {T (s), g2  1}  g2  1  (16)  4  20  40  60  80  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  Abs  Re  Im  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 5: S-matrix components for the inelastic profile  β(s) using the new Ansatz of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In red are the  components of the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Here, the  parameters used were m2  1 = 3, Nmax = 10 and  ¯Nmax = 10 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  subject to the constraints  A(s) ≡ T(s) +  g2  1  s − m2  1  −  � ∞  4m2  dz  π  Im T(z)  s − z  − (s ↔ t) = 0  (17)  and the new unitarity condition, which is given by  U(s) ≡ 2  �  s(s − 4) (1 − β(s)) + 2 Im T(s)  −  1  2  �  s(s − 4)  |T(s)|2 ⩾ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  (18)  To achieve a dual formulation we build the Lagrangian  density L(T, ω, λ),  L(T, ω, λ) = g2  1 +  � ∞  4m2 ds ω(s)A(s) + λ(s)U(s) ,  (19)  where λ ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Using the weak duality principle and in-  troducing the function W(s) satisfying ImW(s) = ω(s),  which does not depend on the form of the unitarity con-  dition, we arrive at the expression  L(T, ω, λ) =  � ∞  4m2 ds Im [W(s)T(s)] + λ(s)U(s)  (20)  provided that Re W(m2  1) = −1/π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We conclude that the  correct dual optimization problem is  min  {W } D(W) =  � ∞  4m2  ds  ρ2  11(s) [|W(s)|β(s) + Re W(s)] ,  (21)  where ρ−2  11 = 2  �  s(s − 4) subject to the condition that  W(m2  1) = − 1  π ,  (22)  where β(s) are the profiles used above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Lastly, it is nec-  essary to propose an ansatz to carry out the numerical  extremization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The naive ρ(s) expansion can still be used  W(s) =  1  s(4m2 − s)  Nmax  �  n=1  an(ρ(s)n − ρ(t)n) ,  (23)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0 α  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  g1  2  Analytical  Numerical  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 6: Comparison between the analytical expression  (11) and the data for β(s) with some tiny error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  5  10  15  20  25  30 s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  Real  Imaginary  Absolute  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 7: S-matrix components for the inelastic profile  β(s) for the dual approach vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' the analytical  components, red dashed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' m2  1 = 3, Nmax = 40,  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5 and s∗ = 16 were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  where Nmax is again the numerical cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The fact that  the same Ansatz was used even though the S-matrix en-  codes additional thresholds is remarkable when compared  to the primal formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' This follows from the fact that  the absolute value of the S-matrix is fixed in the dual pro-  cedure because of the strong duality principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Namely  T(s) =  i  ρ2  11(s)  �  1 + W ∗(s)  |W(s)|  �  β(s)  �  ,  (24)  which has the correct absolute value regardless of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Revisiting the cubic coupling maximization yields figure  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Here we performed a simple extrapolation in Nmax  which agrees with the analytic result to one part in 10−4  and estimated some (very small) error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' A look at S-  Matrix components, figure 7, confirms the expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  The absolute value exactly matches the profile β(s), and  the real and imaginary parts nicely follow the expected  singular behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Primal in 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  The natural way to generalize the  inelastic constraints discussed in 2 dimensions above, is  to restrict the absolute values of the partial amplitudes  Sℓ = 1 + i  �  (s − 4)/s fℓ, where fℓ are the partial waves  5 ■  ■  ■  ■  ■  ■  ■  ■  ★  ★  ★  ★  ★  ★  ★  6  8  10  12  14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='60  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='70  Nmax  λmax In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=', ℓmax = 10  ■  In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=', ℓmax = 12  ★  In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=', ℓmax = 14  Elastic, ℓmax = 10  Elastic, ℓmax = 12  Elastic, ℓmax = 14  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 8: Result of maximizing the quartic coupling  subject to (26), for profile β(s)  0 (s) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5 and  s∗ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The dashed lines correspond to the solution  with unitarity only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  normalized as in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We then impose  |Sℓ(s)|2 ⩽ βℓ(s),  s ⩾ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  (25)  In this section we will study step profiles, identical to  the ones used in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In practice, we  will use non-trivial inelasticity profiles only for the low  spin partial-waves and require standard unitarity for the  remaining ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  To use (25) in terms of semi-definite  programming we rewrite it in terms of a matrix inequality  �  �  1 −  Im aℓ  1+√  βℓ(s)  Re aℓ  Re aℓ  βℓ(s) − 1 +  �  1 +  �  βℓ(s)  �  Im aℓ  �  � ⪰ 0  (26)  where aℓ =  �  (s − 4)/s fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In 4d, we will stick to the sim-  plest maximization problem, which can be studied even  for a model without bound states: Maximization of the  quartic coupling λ ≡ Λ00/32π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Motivated by the 2d ex-  plorations, we start from an ansatz  T(s, t, u) = κ  �  1  ρs − 1 + cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  �  +  Nmax  �  a,b,c=0  αabc ρa  sρb  tρc  u  +  ¯  Nmax  �  a,b,c=0  βabc ¯ρa  s ¯ρb  t ¯ρc  u  (27)  where we use the ¯ρ variables discussed above, and add  a pole at threshold, parametrized by κ, which is known  to capture the physics of the extremal S-Matrix, in the  standard case [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' First, we consider a step profile for the  spin zero partial wave β(s)  0 (s), with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5 and s∗ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Even when we don’t use the new terms in the ansatz,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  when ¯Nmax = 0, we find a substantial difference  compared to the ”elastic” case, as can be seen in fig-  ure 8: It seems that λ roughly achieves a plateau, which  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 9: Result of maximizing the quartic coupling  subject to (26), for profile β(s)  0 (s) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8 and  s∗ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We find a plateau λ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='61 around ¯Nmax ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  is clearly below the original bounds, where no inelastic-  ity was imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' However, extending the ansatz shows  that this difference is somewhat overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Vary-  ing the parameter ¯Nmax, with Nmax = 14, one produces  figure 9, where we considered a somewhat sharper discon-  tinuity with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Clearly, the quartic coupling now  plateaus at a slightly higher value, λmax ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='61 , which is  quickly achieved by adding merely a few ¯ρ terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We can  now also look at the partial amplitudes, and observe how  the particle production is realized, as shown in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Plotting the spin zero amplitude S0, we can observe the  important effect of the more general ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Not only  is the transition smoother and less oscillatory but the  constraints are more sharply saturated, and in particular  elastic unitarity is satisfied remarkably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Having established the reliability of the new ansatz  (27), we now give a more interesting application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' For the  profile β(s)  0 , we study how the maximum quartic coupling  varies as we tune both the strength of the discontinuity  α and the start of the inelastic threshold s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The results  are presented in figure 11 [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' As expected, both making  the discontinuity softer (decreasing α), and pushing the  inelastic threshold further way (increasing s∗) make the  bounds approach from below the value obtained imposing  simply unitarity, λmax ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We also note that for this  profile it is possible to accurately replicate very sharp  discontinuities, as we find stable results, even for α as  small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  As a final application of the methods we developed,  we also consider the effect of inelasticity on the spin 2  partial wave f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Maximizing the quartic coupling λ with  profile β(s)  2  with parameters α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8 and s∗ = 16 leads  to effects that are more severe than the corresponding  spin 0 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  A plateau is only achieved for larger  values of ¯Nmax, and the maximum value of the coupling  drops considerably to about λmax ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' This is reason-  able, since we are pushing against the Froissart-Gribov  tendency to lower ImfJ as J increases, reducing the solu-  tion space and hence, λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' An interesting question for  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  max  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='64  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='62  Amax  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='60  10  5  N  max6  20  40  60  80  100s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  Re  Im  Abs  α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8  20  40  60  80  100s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  Re  Im  Abs  α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 10: Spin zero partial amplitude maximizing the  quartic coupling subject to 26, for profile β(s)  0 (s),  Nmax = 14, ℓmax = 14 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8 with ¯Nmax = 0 (top)  and ¯Nmax = 9 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 11: Maximum quartic coupling λmax as a function  of α and s∗, for a profile β(s)  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The results have  converged since changing ℓmax from 10 to 14 leaves the  bounds unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Here, Nmax = 10 and ¯Nmax = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  the future is to understand what happens in the physical  cases where the inelasticity decays asymptotically with  spin, as in the black-hole production case, or with Dragt  behavior [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Such an understanding would require a  reliable way to estimate the finite spin effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Total Cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Having obtained improved  bounds on the couplings and captured the extra thresh-  old, we procceed to analyze the total cross-section σt,  or equivalently the forward limit T(s, 0) of the resulting  amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The relation between the two is simply  Im T(s, 0) =  �  s(s − 4) σt =  �  ℓ  16π(2ℓ + 1)Imfℓ , (28)  ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★  ★★★★★★★★★★★★★★★★★★★★★★★★★ ★ ★ ★ ★ ★  ★  ★  ★  ★  ★  ★  20  40  60  80  100s  20  40  60  80  100  Im T(s,0)  Unitarity Solution  ▲  Elastic  Inelastic  ★  Sum  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 12: Comparison between the solution where only  unitarity is imposed, Unitarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' the numerical solution  with inelasticity imposed in S0, Solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' the elastic  part of the amplitude obtained from the optical  theorem, Elastic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' the imposed inelastic profile, Inelastic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  and the sum of elastic and inelastic components Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  and of course, we can decompose the cross-section into  its elastic and inelastic parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Notably, we can use the  optical theorem to obtain the elastic part of the ampli-  tude in the forward limit in terms of an integral of its  square  Im T(s, 0) ⩾  1  64π  �  s − 4  s  � 1  −1  dz |T(s, t(z))|2 ,  (29)  with z the cosine of the scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The inequality  is saturated in the elastic unitarity region, and after that  there are inelastic terms further contributing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' This can  also be used as a check to the validity of the bounds ob-  tained when imposing inelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We plot the forward  amplitude corresponding to maximizing the quartic cou-  pling with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='2 and s∗ = 16 in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We also  computed the integral in the optical theorem numeri-  cally, and determined the inelastic contribution to the  total cross section through the sum over partial waves in  (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We find good agreement between the sum of the  elastic and inelastic pieces and the resulting cross-section  obtained directly through the maximization process, as  shown in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  We find that the forward amplitude always exceeds its  elastic counterpart denoted by the solid blue line, which  is obtained by maximizing the quartic coupling imposing  only unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Indeed, this also suggests a natural way  to parametrize the inelasticity: inputing a lower bound  on the total cross-section, which is a simple generaliza-  tion of imposing positivity in the forward limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  This  constraint mixes information about all the partial waves,  so it should lead to weaker bounds than imposing in-  elasticity for each partial wave as in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In practice,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='2  α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='8  (max = 10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='65  (max = 12  max2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='60  (max = 14  16  15  14  s*  13  127  ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲  50  100  150  200  s  20  40  60  80  100  Im T(s,0) 165  170  175  180  185  190  195  200  0  1  2  3  4  5  6  7  s  Unitarity Solution  ▲  Elastic  Total cross-sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 13: Comparison between the solution with only  unitarity imposed, Unitarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' the solution with σt that  follows from β(s)  0 (s) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='2, Solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' the elastic  part of the amplitude from the optical theorem, Elastic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  and the imposed cross-section total cross-sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  to fully control the inelasticity, one needs to include the  quadratic elastic piece which leads to a constraint that is  hard to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' On the other hand, one can impose  �  Im T(s, 0) −  �  s(s − 4) σt 0  0  1  �  ⪰ 0  (30)  where σ is just treated as a fixed function of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In prac-  tice, this just leads to saturation of this profile once the  elastic cross-section exceeds this value, as shown in fig-  ure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We also point out that due to the nature of the  ρ expansion, the forward limit of the amplitude always  decays at large s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' It would be nice to study extensions  which allow for Froissart-like behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In this paper, we explored how inequal-  ities stronger than unitarity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' imposing an inelastic-  ity profile, yields stronger bounds for the low-energy ob-  servables one studies with the S-matrix bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' To  correctly capture the new thresholds, we introduced the  ¯ρ variables, which drastically improve convergence and  allowed for the satisfaction of elastic unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' As ex-  pected, we found that the maximum coupling decreases  as we increase the amount of inelasticity, which can be  achieved by making the discontinuity sharper or by low-  ering the energy of the first inelastic threshold, as quanti-  fied in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We also found that the same amount of  inelasticity imposed on a higher spin partial wave further  decreases the maximum coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  The previous analysis of the cross-section suggests  that it could be possible to obtain inelastic S-matrices by  maximizing the total cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Of course, to maxi-  mize the total cross-section at a fixed value of the energy  is not expected to be a physically well posed problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  The existence of resonances allows for sharp peaks in the  cross-section, which are obviously physical and should  not be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  A naive use of the maximization  methods, leads to results that do not seem to converge  as one increases Nmax, confirming the above expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Curiously, for fixed Nmax, and varying s, the values  seem to roughly follow a Froissart-like trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' It would  be interesting to understand how this would change for  ans¨atze that allow for amplitudes that grow at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  On the other hand, suitable integrals of the cross section  over energy should eliminate the problem mentioned  above, and might prove to be an interesting observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  We thank C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Bercini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Correia, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Goncalves, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Guerrieri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Hebbar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Homrich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Vieira for illuminating discussions and useful comments  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Correia and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Hebbar for useful comments on  the draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' This research received funding from the Si-  mons Foundation grant 488637 (Simons collaboration on  the non-perturbative bootstrap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  AA received funding  from the German Research Foundation DFG under Ger-  many’s Excellence Strategy – EXC 2121 Quantum Uni-  verse – 390833306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Centro de F´ısica do Porto is partially  funded by Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT)  under the grant UID04650-FCUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In the main text, we used inelasticity  as a way to probe the interior of the bootstrap bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Here, we study a simpler problem which also allows us  to visit non-extremal theories with respect to maximiz-  ing the cubic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' We fix the cubic coupling to be  0 < g1 < gmax, and then maximize and minimize the  quartic coupling S(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The result is figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' At the tip  of maximum coupling, the solution is unique and is sat-  urated by sine-Gordon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' In the other extreme g1 = 0, the  bound state decouples, and the maximum and minimum  theories are just a free boson and fermion respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  In between, there are non-trivial solutions, generated by  products of CDD factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' One of the solutions can be  seen as adding the T ¯T deformation in the vicinity of the  sine-Gordon point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='2  0  5  10  g1  S(2)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 14: Plot of g1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' S(2) for m1 =  √  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' The blue  curve represents the maximum value of S(2) while the  orange represents the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  8  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Travaglini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Brandhuber, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Dorey, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' McLough-  lin,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Abreu,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Bern,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Bjerrum-Bo  hr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Bl¨umlein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Britto and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Carrasco, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' A 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='44, 443001 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1088/1751-  8121/ac8380 [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='13011 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Paulos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Toledo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' van  Rees  and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Vieira,  JHEP  11,  143  (2017)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP11(2017)143 [arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='06110 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Paulos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Toledo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' van  Rees  and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Vieira,  JHEP  12,  040  (2019)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP12(2019)040 [arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='06765 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Homrich,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Toledo,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' van  Rees  and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Vieira,  JHEP  11,  076  (2019)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP11(2019)076 [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='06905 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [5] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Doroud and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Elias Mir´o, JHEP 09, 052 (2018)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP09(2018)052 [arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='04376 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' He, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Irrgang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Kruczenski, JHEP 11, 093  (2018) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP11(2018)093 [arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='02812  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Guerrieri,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Penedones  and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Vieira,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  122,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='24,  241604  (2019)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='241604 [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='12849  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' C´ordova and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Vieira,  JHEP 12,  063 (2018)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP12(2018)063 [arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='11143 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Paulos and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Zheng, JHEP 05, 145 (2020)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP05(2020)145 [arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='11429 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Gabai  and  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Yin,  JHEP  10,  168  (2022)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP10(2022)168 [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='00710 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Elias Mir´o, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Guerrieri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Hebbar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Vieira, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 123, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='22, 221602 (2019)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='221602 [arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='08098  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' C´ordova,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' He,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Kruczenski and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Vieira,  JHEP 04,  142 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP04(2020)142  [arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='06495 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Bercini,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Fabri,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Homrich and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Vieira,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  D  101,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='4,  045022  (2020)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='045022  [arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='06453  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Karateev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Kuhn and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones, JHEP 07, 035  (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP07(2020)035 [arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='08940  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Guerrieri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Vieira, JHEP  06,  088  (2021)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP06(2021)088  [arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='02802 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Correia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Sever and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Zhiboedov, JHEP 03, 013  (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP03(2021)013 [arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='08221  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Bose,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Haldar,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Sinha,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Sinha  and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Tiwari,  SciPost  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  9,  081  (2020)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='21468/SciPostPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='081  [arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='12213  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Guerrieri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Homrich and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Vieira, JHEP 11, 084  (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP11(2020)084 [arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='02770  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Hebbar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Karateev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones, JHEP 01, 060  (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP01(2022)060 [arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='11708  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Tourkine and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Zhiboedov, JHEP 07, 228 (2021)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP07(2021)228 [arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='05211 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' He and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Kruczenski,  JHEP 08,  125 (2021)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP08(2021)125 [arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='11484 [hep-  th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Guerrieri,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Penedones  and  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Vieira,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content='8,  081601  (2021)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='081601 [arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='02847  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content=' Guerrieri, JHEP 10, 126 (2021)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP10(2021)126 [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content=' Sever, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content='25,  251601  (2021)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='251601  [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='10257 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Chen,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Fitzpatrick  and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  Karateev,  JHEP 02,  146 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1007/JHEP02(2022)146  [arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='10286 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content=' Kruczenski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Penedones and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' van Rees,  [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='02421 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Karateev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='114014  [arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='07140 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [36] See also [20] for an alternative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [37] In 2d, it is also instructive to think of the multi-particle  analysis of [4] from the point of view of a single ampli-  tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Focusing on the 11 → 11 amplitude, the 11 → 22  component plays the role of an inelastic profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' It is an in-  teresting question to understand how much stronger the  multi-correlator bounds are compared to the single cor-  relator bootstrap with the inelasticity imposed by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [38] Strictly speaking, the ¯ρ’s can also be expressed in terms of  the ordinary ρ’s but this requires infinitely many terms,  and therefore including them substantially improves the  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='  [39] For this plot we used only ¯ρ variables with threshold at  s∗ = 16, to avoid recomputing expensive partial wave  integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' Minor changes are to be expected if one adjusts  the ¯ρ variables for the respective value of s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content=' 156,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='5,  1588-1593 (1967)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
+page_content='1588' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQf_TXz/content/2301.13219v1.pdf'}
diff --git a/yNAzT4oBgHgl3EQftP0T/content/tmp_files/2301.01671v1.pdf.txt b/yNAzT4oBgHgl3EQftP0T/content/tmp_files/2301.01671v1.pdf.txt
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+arXiv:2301.01671v1  [math.LO]  4 Jan 2023
+SUMS OF TRIPLES IN ABELIAN GROUPS
+IDO FELDMAN AND ASSAF RINOT
+Abstract. Motivated by a problem in additive Ramsey theory, we extend
+Todorˇcevi´c’s partitions of three-dimensional combinatorial cubes to handle ad-
+ditional three-dimensional objects. As a corollary, we get that if the continuum
+hypothesis fails, then for every Abelian group G of size ℵ2, there exists a col-
+oring c : G → Z such that for every uncountable X ⊆ G and every integer k,
+there are three distinct elements x, y, z of X such that c(x + y + z) = k.
+1. Introduction
+By Hindman’s celebrated theorem (see [HS12, Corollary 5.9]), for every partition
+of an infinite commutative cancellative semigroup (G, +) into two cells A and B,
+there exists an infinite subset X ⊆ G such that the set of its finite sums
+FS(X) := {x1 + · · · + xn | x1, . . . , xn are distinct elements of X & n ∈ N \ 2}
+is completely contained in A or completely contained in B. Equivalently, for every
+coloring c : G → 2, there exists an infinite X ⊆ G such that c ↾ FS(X) is constant.
+Hindman’s theorem does not generalize to the uncountable, as it follows from a
+theorem of Milliken (see [Mil78, Theorem 9]) that the following assertion is consis-
+tent with the usual axioms of set theory: for every (not necessarily Abelian) group
+(G, ∗) whose size is a regular uncountable cardinal, there is a coloring c : G → G
+such that c ↾ FS2(X) is onto G for every X ⊆ G of size |G|, where this time
+FSn(X) := {x1 ∗ · · · ∗ xn | x1, . . . , xn are distinct elements of X}.
+A few years ago, starting with a paper by Hindman, Leader and Strauss [HLS17],
+the study of higher analogs of Hindman’s theorem regained interest. We mention
+only a few results that are relevant to this paper:
+(1) Improving upon a theorem from [HLS17], Komj´ath [Kom16], and inde-
+pendently Soukup and Weiss [SW16], proved that there exists a coloring
+c : R → 2 such that for every uncountable X ⊆ R and every i ∈ {0, 1},
+there are x ̸= y in X such that c(x + y) = i.
+(2) Solving a problem of Weiss, Komj´ath [Kom20] proved that there exists a
+coloring c : R → 2 such that for every uncountable X ⊆ R and every
+i ∈ {0, 1}, there are x ̸= y in X such that c(|x − y|) = i. As for dimension
+d > 1, assuming the continuum hypothesis, there exists a coloring c : R → 2
+such that for every uncountable X ⊆ Rd and every i ∈ {0, 1}, there are
+x ̸= y in X such that c(∥x − y∥) = i.
+(3) In [FBR17], Fern´andez-Bret´on and Rinot proved that there exists a coloring
+c : R → N such that for every X ⊆ R of size |R| and every i ∈ N, there are
+x ̸= y in X such that c(x + y) = i.
+Date: Preprint as of January 4, 2023. For the latest version, visit http://p.assafrinot.com/57.
+2010 Mathematics Subject Classification. Primary 03E02; Secondary 03E75, 03E35, 05A17.
+1
+
+2
+IDO FELDMAN AND ASSAF RINOT
+(4) By [FBR17], for class many cardinals κ (including κ = ℵn for every positive
+integer n), for every commutative cancellative semigroup G of size κ, there
+exists a coloring c : G → G such that for all X, Y ⊆ G of size κ and every
+g ∈ G, there are x ∈ X and y ∈ Y such that c(x + y) = g.1
+(5) By [FBR17], for every commutative cancellative semigroup G, there exists
+a coloring c : G → N such that c ↾ FS(X) is onto N for every uncountable
+X ⊆ G. It is also consistent that the same holds after replacing N by R.
+Note that in the results listed in (1), (2) and (5), the triggering set X may have
+cardinality smaller than that of G, whereas in (3) and (4), |X| coincides with |G|.
+Another important difference is that unlike the results of (1)–(4), in (5), no bound
+is asserted on the length of the sums needed to generate all the infinite colors. This
+raises a natural question whose simplest instance reads as follows.
+Question. Suppose that (G, +) is an Abelian group of size ℵ2.
+Must there exist a positive integer n and a coloring c : G → N such that c ↾
+FSn(X) is onto N for every uncountable X ⊆ G?
+A moment’s reflection makes it clear that an affirmative answer (even for one
+particular group G) immediately implies the relation ℵ2 ↛ [ℵ1]n
+ℵ0 from the classical
+study of partition relations for cardinal numbers [EHR65]. By a theorem of Erd˝os
+and Rado, the above relation may consistently fail for n = 2, and it is a remarkable
+theorem of Todorˇcevi´c [Tod94] that it does hold for n = 3. The first main result of
+this paper gives a consistent extension of Todorˇcevi´c’s theorem.
+Theorem A. If the continuum hypothesis fails, then for every Abelian group (G, +)
+of size ℵ2, there exists a coloring c : G → N such that for every uncountable
+X ⊆ G and every i ∈ N, there are three distinct elements x, y, z of X such that
+c(x + y + z) = i.
+Theorem A is not limited to Abelian groups. In fact, it works for all so-called
+well-behaved magmas, as follows.
+Definition. A magma is a structure (G, ∗), where ∗ is a binary operation. We say
+that it is well-behaved iff there exists a map ϕ : G → [G]<ω such that:2
+• G is countable-to-one;
+• for all x ̸= y in G, ϕ(x) △ ϕ(y) ⊆ ϕ(x ∗ y) ⊆ ϕ(x) ∪ ϕ(y).
+Every infinite commutative cancellative semigroup (G, +) is well-behaved (see,
+e.g., [FBR17, Lemma 2.2]). Also, every free group (G, ∗) is well-behaved, as wit-
+nessed by the map that sends a word to the set of its letters.
+As a third ex-
+ample, consider the magma appearing in result (2) above, namely, (R, d) where
+d(x, y) := |x − y|. Indeed, viewing R as a Q-vector space over some Hamel basis
+B, any x ∈ R \ {0} is the unique linear combination �
+i≤n qivi of nonzero rational
+numbers q0, . . . , qn, and an injective sequence ⟨vi | i ≤ n⟩ of elements of B. So
+ϕ : R → [R]<ω sending x to the unique {vi | i ≤ n} (and sending 0 to the emptyset)
+is countable-to-one, and for all x ̸= y, ϕ(x) △ ϕ(y) ⊆ ϕ(|x − y|) ⊆ ϕ(x) ∪ ϕ(y).
+The full statement of Theorem A reads as follows.
+Theorem A′. For every infinite cardinal µ such that µ<µ < µ+ < 2µ, for every
+well-behaved magma (G, ∗) of size µ++, there is a coloring c : G → N such that for
+1More is true, see [FBR17, Corollary 4.5].
+2Here, [G]<ω denotes the collection of all finite subsets of G.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+3
+every X ⊆ G of size µ+ and every i ∈ N, there are three distinct elements x, y, z of
+X such that c(x ∗ y ∗ z) = i.3
+While not so explicit, the approach of going through well-behaved magmas is
+already present in [FBR17].
+In particular, the coloring of result (4) attains all
+possible colors not only over evaluations of the form x + y, but also over any
+nontrivial Q-combination of x and y, such as |x − y|.
+This suggests that it is
+possible to obtain a coloring simultaneously witnessing result (1) together with
+the first half of (2). Indeed, Komj´ath’s theorems follow from the following finding
+(using θ := ℵ0):
+Theorem B. For every infinite cardinal θ such that 2<θ = θ, for every set G with
+θ < |G| ≤ 2θ, and every map ϕ : G → [G]<ω, there exists a corresponding coloring
+c : G → 2 satisfying the following.
+For every binary operation ∗ on G, if ϕ witnesses that (G, ∗) is well-behaved,
+then for every X ⊆ G of size θ+ and every i ∈ {0, 1}, there are x ̸= y in X such
+that c(x ∗ y) = i.
+The proofs of Theorems A′ and B are obtained in a few steps. As a first step,
+we consider a coloring principle Sn(κ, λ, θ) that is sufficient to imply that any well-
+behaved magma (G, ∗) of size κ admits a coloring c : G → θ that takes on every
+possible color on FSn(X) for every set X ⊆ G of size λ. The next step is the
+introduction of an extraction principle Extractn(κ, λ, . . .) that is sufficient for the
+reduction of Sn(κ, λ, θ) into a rectangular-type strengthening κ
+sup
+�−→ [λ, λ]n
+θ of the
+classical partition relation κ ↛ [λ]n
+θ . This leaves us with two independent tasks:
+proving instances of Extractn(κ, λ, . . .), and proving instances of κ
+sup
+�−→ [λ, λ]n
+θ . The
+harder task is the latter, and the second main result of this paper is an extension of
+Todorˇcevi´c’s theorem [Tod94] that Chang’s conjecture fails iff ω2 ↛ [ω1]3
+ω1 holds.
+Here ω2 ↛ [ω1]3
+ω1 is improved to ω2
+sup
+�−→ [ω1, ω1]3
+ω1. Specifically:
+Theorem C. The following are equivalent:
+(1) (ℵ2, ℵ1) ։ (ℵ1, ℵ0) fails;
+(2) There exists a coloring c : [ω2]3 → ω1 with the property that for all disjoint
+A, B ⊆ ω2 of order-type ω1 such that sup(A) = sup(B), for every color
+τ < ω1, there is (α, β, γ) ∈ [A ∪ B]3 \ ([A]3 ∪ [B]3) such that c(α, β, γ) = τ.
+1.1. Organization of this paper. In Section 2, we provide some necessary pre-
+liminaries.
+In Section 3, we recall the definition of a weak Kurepa tree and study related
+objects such as the branch spectrum T (µ, θ). This will play a role in both getting
+instances of Extractn(κ, λ, . . .) and of κ
+sup
+�−→ [λ, λ]n
+θ .
+In Section 4, we prove that Sn(κ, λ, θ) implies that any well-behaved magma
+(G, ∗) of size κ admits a coloring with the strong properties mentioned earlier.
+It is proved that in the special case of λ = κ, S2(κ, λ, θ) already follows from
+κ ↛ [λ; λ]2
+θ, and that, in general, Sn(κ, λ, θ) follows from κ
+sup
+�−→ [λ, λ]n
+θ together with
+Extractn(κ, λ, ω, ω). We then use tree combinatorics to obtain sufficient conditions
+for Extractn(κ, λ, . . .) to hold. The definitions of Extractn(κ, λ, θ, χ) and κ
+sup
+�−→
+[λ, λ]n
+θ will be found in this section as Definitions 4.17 and 4.20.
+3As ∗ is not assumed to be associative, the claim is that we get c(x ∗ y ∗ z) = i for both
+implementations of x ∗ y ∗ z.
+
+4
+IDO FELDMAN AND ASSAF RINOT
+In Section 5, we prove the general case of Theorem C in which ℵ2 is substituted
+by the double successor of a cardinal µ satisfying µ<µ = µ. The proof is a bit long,
+since the analysis goes through a division into a total of six cases and subcases.
+In Section 6, we verify that Todorˇcevi´c’s theorems on the correspondence between
+unstable sets and oscillation remains valid in the rectangular context. We then
+combine it with the results of Section 5 and get that λ+ sup
+�−→ [λ, λ]3
+ω holds for every
+successor λ = µ+ of an infinite cardinal µ = µ<µ.
+In Section 7, we obtain the intended applications in additive Ramsey theory.
+Theorem A′ is gotten as a corollary of the results of Sections 4 and 6, and Theorem B
+is gotten as a corollary of a theorem asserting that S2(κ, µ+, 2) holds whenever there
+exists a weak µ-Kurepa tree with κ-many branches.
+2. Preliminaries
+In this section, κ, λ, µ, θ, χ stand for nonzero cardinals, and n stand for a positive
+integer. We let Hκ denote the collection of all sets of hereditary cardinality less
+than κ. We write [κ]λ := {A ⊆ κ | |A| = λ} and [κ]<λ := {A ⊆ κ | |A| < λ}. Let
+Eκ
+χ := {α < κ | cf(α) = χ}, and define Eκ
+≤χ, Eκ
+<χ, Eκ
+≥χ, Eκ
+>χ, Eκ
+̸=χ analogously.
+For two distinct functions f, g ∈ θµ, write f <lex g to mean that f(δ) < g(δ) for
+the least δ < λ such that f(δ) ̸= g(δ). For functions f, g ∈ ≤θµ, we write f ⊑ g to
+mean that dom(f) ≤ dom(g) and g ↾ dom(f) = f.
+For sets of ordinals A1, . . . , An, we define
+A1 ⊛ · · · ⊛ An := {(α1, . . . , αn) ∈ A1 × · · · × An | α1 < · · · < αn}.
+By convention, whenever we write (α1, . . . , αn) ∈ [A]n (as opposed to {α1, . . . , αn} ∈
+[A]n), we mean that (α1, . . . , αn) ∈ A ⊛ · · · ⊛ A.
+For a set of ordinals A, we write ssup(A) := sup{α + 1 | α ∈ A}, acc+(A) :=
+{α < ssup(A) | sup(a ∩ α) = α > 0}, and acc(A) := A ∩ acc+(a). For two sets of
+ordinals A and B, we write A < B to mean that A × B coincides with A ⊛ B.
+Definition 2.1 (Positive round-bracket relations, [EHR65, §3]). κ → (λ)n
+θ asserts
+that for every coloring c : [κ]n → θ, there exists A ⊆ κ of order-type λ such that c
+is constant over [A]n.
+Definition 2.2 (Negative square-bracket relations, [EHR65, §18]). A coloring c :
+[κ]n → θ is said to witness:
+• κ ↛ [λ]n
+θ iff c[[A]n] = θ for every A ∈ [κ]λ;
+• κ ↛ [λ1, . . . , λn]n
+θ iff c[A1 × · · · × An] = θ for every ⟨Ai | 1 ≤ i ≤ n⟩ ∈
+�n
+i=1[κ]λi;
+• κ ↛ [λ1; . . . ; λn]n
+θ iff c[A1 ⊛ · · · ⊛ An] = θ for every ⟨Ai | 1 ≤ i ≤ n⟩ ∈
+�n
+i=1[κ]λi.
+Note that (κ ↛ [λ; . . . ; λ]n
+θ ) =⇒ (κ ↛ [λ, . . . , λ]n
+θ ) =⇒ (κ ↛ [λ]n
+θ ).
+Definition 2.3 (Fiber maps). Given a coloring of pairs c : [κ]2 → θ and some
+β < κ, we sometimes write cβ for the βth-fiber map of c, that is, for the unique
+map cβ : β → θ to satisfy cβ(α) = c(α, β) for every α < β.
+We say that c has injective fibers iff cβ is injective of every β < κ.
+Definition 2.4 ([LHR18]). U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 →
+θ such that for every σ < χ, every pairwise disjoint subfamily A ⊆ [κ]σ of size κ,
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+5
+for every τ < θ, there exists B ⊆ A of size µ such that min(c[a × b]) > τ for all
+a ̸= b from B.
+Remark 2.5. Of special interest are witnesses c : [κ]2 → θ to U(κ, µ, θ, χ) that are
+moreover subadditive, i.e., satisfying that for all α < β < γ < κ, the following hold:
+• c(α, γ) ≤ max{c(α, β), c(β, γ)};
+• c(α, β) ≤ max{c(α, γ), c(β, γ)}.
+These colorings are studied in [LHR23], and they will show up here in Section 5.
+Given a coloring c : [κ]2 → θ and a subset X ⊆ κ of order-type λ, we say that
+“c ↾ [X]2 witnesses U(λ, µ, θ, χ)” if for the order-preserving bijection π : λ ↔ X,
+the coloring d : [λ]2 → θ defined via d(α, β) := c(π(α), π(β)) is a witness for
+U(λ, µ, θ, χ). To be able to express that this happens globally, we introduce the
+following 5-cardinal extension of the principle of Definition 2.4.
+Definition 2.6. U(κ, λ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such
+that for every σ < χ, every pairwise disjoint subfamily A ⊆ [κ]σ of size λ, for every
+τ < θ, there exists B ⊆ A of size µ such that min(c[a × b]) > τ for all a ̸= b from B.
+Fact 2.7 ([Tod07, Lemma 9.2.3]). For every regular uncountable cardinal λ, if
+U(λ+, λ, 2, λ, 2) holds, then there exists a subadditive witness to U(λ+, λ, λ, λ, ω).
+Finally, we arrive at the notion motivating this paper.
+Definition 2.8. For a magma (G, ∗), we write G ↛ [λ]FSn
+θ
+to assert that there
+exists a coloring c : G → θ with the property that for every subset A ⊆ G of size λ
+and every prescribed color τ < θ, there is an injective sequence ⟨ai | 1 ≤ i ≤ n⟩ of
+elements of A such that c(a1 ∗ · · · ∗ an) = τ for all implementations of a1 ∗ · · ·∗ an.4
+In the special case of n = 2, [FBR17, Corollary 4.5] and [RZ21, Corollary 2.20]
+provide sufficient conditions for G ↛ [λ]FSn
+θ
+to follow from |G| ↛ [λ]n
+θ for all values
+of θ. Higher dimensional reductions are out of reach at present.
+2.1. Walks on ordinals. In this subsection, we provide a minimal background
+on walks on ordinals. This background is only necessary for Section 6, hence the
+exposition here is quite succinct. A thorough treatment may be found in [Tod07].
+For the rest of this subsection, κ denotes a regular uncountable cardinal, and we
+fix some C-sequence over κ, that is, a sequence ⃗C = ⟨Cβ | β < κ⟩ such that, for
+every β < κ, Cβ is closed subset of β with sup(Cβ) = sup(β).
+Definition 2.9 (Todorˇcevi´c). From ⃗C, derive maps Tr : [κ]2 → ωκ, ρ2 : [κ]2 → ω,
+and tr : [κ]2 → <ωκ, by letting for all α < β < κ:
+• Tr(α, β) : ω → κ is defined by recursion on n < ω:
+Tr(α, β)(n) :=
+
+
+
+
+
+β,
+n = 0
+min(CTr(α,β)(n−1) \ α),
+n > 0 & Tr(α, β)(n − 1) > α
+α,
+otherwise
+• ρ2(α, β) := min{l < ω | Tr(α, β)(l) = α};
+• tr(α, β) := Tr(α, β) ↾ ρ2(α, β).
+4The issue of implementation arises from the fact that we do not assume ∗ to be associative,
+e.g., it is possible that (a1 ∗ a2) ∗ a3 ̸= a1 ∗ (a2 ∗ a3).
+
+6
+IDO FELDMAN AND ASSAF RINOT
+To explain: Given a pair of ordinals α < β below κ, one would like to walk from
+β down to α. This is done by recursion, letting β0 := β, and βn+1 := min(Cβn \ α),
+thus, obtaining an ordinal βn+1 such that α ≤ βn+1 ≤ βn. Since the ordinals are
+well-founded, there must exist some integer k such that βk+1 = α, so that, the
+walk is β = β0 > β1 > · · · > βk+1 = α. This walk is recorded by Tr(α, β), since,
+for every n ≤ k, we have that Tr(α, β) = βn, and for every n > k, we have that
+Tr(α, β) = α. The length of the walk is recorded by the positive integer ρ2(α, β).
+Now, since Tr(α, β) is eventually constant with value α, its nontrivial part is those
+ordinals greater than α, i.e., β0 > β1 > · · · > βk; this is recorded by tr(α, β).
+Definition 2.10 ([Rin14, Definition 2.8]). Define a function λ2 : [κ]2 → κ via
+λ2(α, β) := sup(α ∩ {sup(Cη ∩ α) | η ∈ Im(tr(α, β))}).
+Note that λ2(α, β) < α whenever 0 < α < β < κ, since tr(α, β) is a finite
+sequence.
+Fact 2.11 ([LHR18, Lemma 4.7]). Suppose that λ2(α, β) < ǫ < α < β < κ.
+Then tr(ǫ, β) end-extends tr(α, β), and one of the following cases holds:
+(1) α ∈ Im(tr(ǫ, β)); or
+(2) α ∈ acc(Cð) for ð := min(Im(tr(α, β))).
+Definition 2.12 ([RZ21, Definition 2.10]). For every (α, β) ∈ [κ]2, we define an
+ordinal ðα,β ∈ [α, β] via:
+ðα,β :=
+�
+min(Im(tr(α, β))),
+α ∈ acc(Cmin(Im(tr(α,β))));
+α,
+otherwise;
+Remark 2.13. It is easy to see that sup(Cðα,β) = sup(α) for all α < β < κ, and it
+follows from Fact 2.11 that
+tr(ǫ, β) = tr(ðα,β, β)⌢ tr(ǫ, ðα,β),
+whenever λ2(α, β) < ǫ < α < β < κ.
+Fact 2.14 (Todorˇcevi´c, [Tod07, §9]). If κ = λ+ for a regular cardinal λ and
+otp(Cβ) ≤ λ for all β < κ, then there exists a subadditive coloring ρ : [κ]2 → λ with
+the property that ρ(α, β) ≥ otp(Cη ∩ α) for all α < β < κ and η ∈ Im(tr(α, β)).
+3. Weak Kurepa trees and the branch spectrum
+In this section, µ denotes a cardinal and θ denotes an ordinal.
+Definition 3.1. T (µ, θ) denotes the collection of all subsets T ⊆ <θµ such that
+the following two hold:
+(1) T is downward-closed, i.e, for every t ∈ T , {t ↾ α | α < θ} ⊆ T ;
+(2) for every α < θ, the set Tα := T ∩ αµ is nonempty and has size < µ.
+We say that T is a tree of height θ if there exists a cardinal µ such that T ∈
+T (µ, θ).5 Note that θ is uniquely determined. For such a tree T , we shall refer to
+Tα as the αth-level of T , and the set {b ∈ θµ | ∀α < θ (b ↾ α ∈ Tα)} of all branches
+through T is denoted by B(T ). Also, for all f, g ∈ ≤µθ, we let
+∆(f, g) :=
+�
+min{δ ∈ dom(f) ∩ dom(g) | f(δ) ̸= g(δ)},
+if f ⊈ g & g ⊈ f;
+min{dom(f), dom(g)},
+otherwise.
+5There is no loss of generality here, see [BR21, Lemma 2.5(2)].
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+7
+Definition 3.2. T ∈ T (µ, θ) is said to be normal iff for all α < β < θ and t ∈ Tα,
+there exists t′ ∈ Tβ with t ⊑ t′.
+Definition 3.3. Given a tree T and a subset B ⊆ B(T ), we consider the subtree:
+T ⇝B := {t ∈ T | |{b ∈ B | t ⊑ b}| = |B|}.
+Lemma 3.4. Suppose that T ∈ T (µ, θ), and λ is an infinite regular cardinal.
+(1) If λ ≥ µ, then for every B ∈ [B(T )]λ, T ⇝B is in T (µ, θ) and is normal;
+(2) If λ ≥ max{µ, |θ|+}, then for all A, B ∈ [B(T )]λ, there are s ∈ T and i ̸= i′
+such that s⌢⟨i⟩ ∈ T ⇝A and s⌢⟨i′⟩ ∈ T ⇝B.
+Proof. (1) Suppose that B ∈ [B(T )]λ and λ ≥ µ. It is clear that ∅ ∈ T ⇝B. Thus,
+to prove that T ⇝B has height θ and is normal, let α < β < θ and t ∈ (T ⇝B)α, and
+we shall show that there exists t′ ∈ (T ⇝B)β extending t.
+By the choice of t, B′ := {b ∈ B | t ⊑ b}| has size λ. Since T ∈ T (µ, θ), it is
+the case that 0 < |Tβ| < |B′| = cf(|B′|), and then the pigeonhole principle provides
+t′ ∈ Tβ such that {b ∈ B′ | t′ ⊑ b} has size λ. Evidently, t′ is as sought.
+(2) Suppose that A, B ∈ [B(T )]λ and λ ≥ max{µ, |θ|+}. By possibly passing to
+λ-sized subsets of A and B, we may assume that A ∩ B = ∅. Let ⟨aj | j < λ⟩ be
+some injective enumeration of A, and likewise let ⟨bj | j < λ⟩ be some injective
+enumeration of B. For each j < λ, as aj ̸= bj, we may let δj := ∆(aj, bj) + 1. As λ
+is a regular cardinal greater than |θ|, we may fix some J ∈ [λ]λ on which the map
+j �→ δj is constant with value, say, δ. As Tδ+1 has size < µ ≤ λ, we may moreover
+assume that the map j �→ ((aj ↾ δ + 1), (bj ↾ δ + 1)) is constant over J, with value,
+say, (s⌢⟨i⟩, s⌢⟨i′⟩). Then, we are done.
+□
+Corollary 3.5. Suppose that T ∈ T (λ, θ) ∩ P(<θ2), where λ = cf(λ) > cf(θ) ≥ ω.
+Suppose that we are given i < 2 and X ∈ [B(T )]λ. Then, for λ-many x ∈ X, there
+are cofinally many δ < θ such that the following two hold:
+(1) x(δ) = i;
+(2) {y ∈ X | ∆(x, y) = δ} has size λ.
+Proof. Suppose not.
+In particular, the set Y of all x ∈ X for which there are
+boundedly many δ < θ satisfying Clauses (1) and (2) has size λ.
+So, for each
+x ∈ Y , the following ordinal is smaller than θ:
+ǫx := sup{δ < θ | x(δ) = i & |{y ∈ X | ∆(x, y) = δ}| = λ}.
+As |Y | = cf(λ) > cf(θ), we may find some ǫ < θ such that Z := {x ∈ Y | ǫx = ǫ}
+has size λ. As |Tǫ+1| < λ, we may also find some t ∈ Tǫ+1 such that Zt := {x ∈ Z |
+t ⊑ x} has size λ. Now, by appealing to Lemma 3.4(2) with µ := λ, A := Zt and
+B := Zt, we may find s ∈ T and j < 2 such that ˆA := {a ∈ A | s⌢⟨j⟩ ⊑ a} and
+ˆB := {b ∈ B | s⌢⟨1 − j⟩ ⊑ b} are both of size λ. As A = B and by possibly
+switching the roles of ˆA and ˆB, we may assume that j = i. Denote δ := dom(s).
+For all a ∈ ˆA and b ∈ ˆB, since a, b ∈ Zt, both s⌢⟨i⟩ and s⌢⟨1 − i⟩ are compatible
+with t, so that δ = dom(s) ≥ dom(t) > ǫ. Now, for every x ∈ ˆA, it is the case that
+x(δ) = i and {y ∈ X | ∆(x, y) = δ} covers ˆB, but | ˆB| = λ, so we got a contradiction
+to the fact that δ > ǫ = ǫx.
+□
+Lemma 3.6. Suppose that T ∈ T (µ, µ), and µ is a regular uncountable cardinal.
+Suppose also that ⟨bξ | ξ < µ⟩ is an injective enumeration of some B ∈ [B(T )]µ.
+
+8
+IDO FELDMAN AND ASSAF RINOT
+For every ⟨tα | α < µ⟩ ∈ �
+α<µ(T ⇝B ∩ αµ), for club many α < µ,
+sup({∆(bβ, tα) | β < α} ∩ α) = α.
+In particular, for club many α < µ,
+sup{γ < µ | α ∈ acc+({∆(bβ, bγ) | β < α})} = µ.
+Proof. The ‘In particular’ part follows the main claim together with Lemma 3.4(1),
+using λ := µ. Next, to prove the main claim, let ⟨tα | α < µ⟩ ∈ �
+α<µ(T ⇝B ∩ αµ).
+Denote Γα := {γ < µ | tα ⊑ bγ}. Consider the club
+C := {α ∈ acc(µ) | ∀¯α < α [min(Γ¯α \ ¯α) < α]}.
+Note that for every α < µ, Dα := {∆(bβ, tα) | β ∈ α \ Γα} is a subset of α.
+Claim 3.6.1. The following set covers a club in µ:
+A := {α < µ | sup(Dα) = α}
+Proof. Suppose not. Fix an ordinal ǫ < µ for which the following set is stationary:
+S := {α ∈ C | sup(Dα) = ǫ}.
+There are two cases to consider:
+◮ Suppose that there exists a function b : µ → µ such that Sb := {α ∈ S |
+b ↾ α = tα} is cofinal in µ. Pick ¯α ∈ Sb \ (ǫ + 1). Since Γ¯α has more than one
+element, we may now find β ∈ Γ¯α such b ̸= bβ. Then ǫ < ¯α ≤ ∆(bβ, b) < µ. Pick
+α ∈ Sb above max{∆(bβ, b), β}. Then ǫ < ∆(bβ, tα) < α, contradicting the fact
+that sup(Dα) = ǫ.
+◮ Suppose the first case fails. First, since |Tǫ+1| < µ, pick a node t ∈ Tǫ+1 such
+that S′ := {α ∈ S | tα ↾ (ǫ+1) = t} is stationary. Since for every function b : µ → µ,
+the set Sb := {α ∈ S | b ↾ α = tα} is bounded in µ, we may now pick a pair
+(¯α, α) ∈ S′ such that t¯α ̸⊑ tα, so that ǫ < ∆(t¯α, tα) < ¯α. Let β := min(Γ¯α\¯α). Then
+bβ ↾ ¯α = t¯α and hence ∆(bβ, tα) = ∆(t¯α, tα). That is, ǫ < ∆(bβ, tα) < ¯α ≤ β < α,
+contradicting the fact that sup(Dα) = ǫ.
+□
+Let α ∈ A. Recall that Γα has size µ, and note that, for every γ ∈ Γα,
+{∆(bβ, bγ) | β < α} ∩ α = {∆(bβ, tα) | β < α} ∩ α = Dα,
+so we are done.
+□
+Lemma 3.7. Suppose that T ∈ T (2µ, µ) ∩ P(<µµ), where µ is an infinite regular
+cardinal. For every B ∈ [B(T )]µ, there exist B′ ∈ [B]µ and θ ≤ µ such that:
+(1) For every B′′ ∈ [B′]µ, T ⇝B′′ is in T (µ, θ) and is normal;
+(2) If θ < µ or if T contains no µ-Aronszajn subtrees, then |B(T ⇝B′′)| = µ for
+every B′′ ∈ [B′]µ;
+(3) If θ < µ, then |{b ∈ B′ | ssup{∆(t, b ↾ θ) | t ∈ T ⇝B′ & t ̸⊑ b} < θ}| < µ.
+Proof. Let B ∈ [B(T )]µ.
+Claim 3.7.1. If T ⇝B ∈ T (µ, µ), then the pair (B′, θ) := (B, µ) is as sought.
+Proof. Suppose that T ⇝B is in T (µ, µ). For every B′′ ∈ [B]µ, T ⇝B′′ = (T ⇝B)⇝B′′,
+and hence Lemma 3.4(1) implies that T ⇝B′′ ∈ T (µ, µ) and is normal. In addition,
+if there exists some B′′ ∈ [B]µ such that |B(T ⇝B′′)| < µ, then looking at B′′′ :=
+B′′ \ B(T ⇝B′′), we get that T ⇝B′′′ is a µ-Aronszajn subtree of T .
+□
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+9
+From now on, suppose that T ⇝B ∈ T (2µ, µ) \ T (µ, µ). Let θ < µ be the least
+such that (T ⇝B)θ has size ≥ µ. Let ⟨ti | i < µ⟩ be an injective sequence of elements
+of (T ⇝B)θ. For each i < µ, pick bi ∈ B such that ti ⊑ bi, and set B′ := {bi | i < µ}.
+To see that the pair (B′, θ) is as sought, let B′′ ∈ [B′]µ.
+Claim 3.7.2. T ⇝B′′ is in T (µ, θ) and is normal.
+Proof. For every α ∈ [θ, µ), it is the case that for every t ∈ (T ⇝B′′)α, there exists
+a unique i < µ such that ti ⊑ t, and hence {b ∈ B′′ | t ⊑ b} ⊆ {bi} is finite.
+Therefore, (T ⇝B′′)α is empty. In addition, as B′′ ⊆ B′ ⊆ B, it is the case that
+|(T ⇝B′′)α| ≤ |(T ⇝B)α| < µ for all α < θ.
+Clearly, ∅ ∈ T ⇝B′′. Finally, let α < β < µ with t ∈ (T ⇝B′′)α, and we shall find
+t′ ∈ (T ⇝B′′)β extending t. By the choice of t, B∗ := {b ∈ B′′ | t ⊑ b}| has size µ.
+By the minimality of θ, the map b �→ b ↾ β from B∗ to Tβ cannot have an image of
+size µ, and hence there exists B∗∗ ∈ [B∗]µ on which the said map is constant, with
+some value, say t′. Clearly, t′ is as sought.
+□
+Claim 3.7.3. |B(T ⇝B′′)| = µ.
+Proof. Suppose not. In particular, I := {i < µ | bi ∈ B′′ & ti /∈ B(T ⇝B′′)} has size
+µ. It follows that there exists an α < θ such that Iα := {i ∈ I | (ti ↾ α) /∈ T ⇝B′′}
+has size µ. However, µ is a regular cardinal greater than |Tα| ≥ |(T ⇝B′′)α|, and
+hence there must exist some s ∈ (T ⇝B′′)α such that {i ∈ Iα | (ti ↾ α) = s} has size
+µ. This is a contradiction.
+□
+Claim 3.7.4. |{i < µ | ssup{∆(t, ti) | t ∈ T ⇝B′ & t ̸⊑ ti} < θ}| < µ.
+Proof. Suppose not, and pick ǫ < θ such that the following set has size µ:
+I := {i < µ | ssup{∆(t, ti) | t ∈ T ⇝B′ & t ̸⊑ ti} = ǫ}.
+Then pick s ∈ Tǫ such that {i ∈ I | ti ↾ ǫ = s} has size µ. Finally, as in the proof
+of Lemma 3.4(2), we may find some s′ ∈ T ⇝B′ extending s and j ̸= j′ such that
+{i ∈ I | s′⌢⟨j⟩ ⊑ ti} and {i ∈ I | s′⌢⟨j′⟩ ⊑ ti} are both of size µ. In particular,
+there exist i ̸= i′ in I such that ti ∩ ti′ = s′. So ∆(s′⌢⟨i′⟩, ti) ≥ ǫ, contradicting the
+fact that i ∈ I.
+□
+This completes the proof.
+□
+Definition 3.8. Let µ denote an infinite cardinal.
+(1) A weak µ-Kurepa tree is a tree T of height µ, of size µ, satisfying |B(T )| > µ;
+(2) A µ-Kurepa tree is a tree T of height µ for which {α < µ | |Tα| > |α|} is
+nonstationary, and |B(T )| > µ.
+Remark 3.9. As in Exercise 34 of [Kun80, §II], if there exists a µ-Kurepa tree
+(resp. weak µ-Kurepa tree), then there exists one which is a subset of <µ2.
+Definition 3.10 (Branch spectrum). T (µ, θ) stands for the collection of all cardi-
+nals κ for which there exists T ∈ T (µ, θ) with κ ≤ |B(T )|.
+Proposition 3.11. Let µ and θ denote infinite cardinals. Then:
+(1) sup(T (µ, θ)) ≤ µθ;
+(2) If θ is the least cardinal to satisfy µθ > µ, then max(T (µ+, θ)) = (µ+)θ;
+(3) If there exists a weak µ-Kurepa tree, then µ+ ∈ T (µ+, µ);
+
+10
+IDO FELDMAN AND ASSAF RINOT
+(4) If there exists a µ-Kurepa tree, then µ+ ∈ T (µ, µ);
+(5) If µ is a strong limit, then 2µ ∈ T (µ, cf(µ)).
+Proof. Clear.
+□
+Remark 3.12. T (µ, θ) need not have a maximal element. By [Po´o21], it is consistent
+for T (ω1, ω1) to have ℵω2 has a supremum that is not attained. Note, however, that
+T (µ, θ) is closed under unions of length < cf(µ), and hence T (µ, θ) is < cf(µ)-closed.
+Proposition 3.13. For every κ ∈ T (µ, θ), there exists a coloring c : [κ]2 → θ
+witnessing U(κ, λ, λ, θ, 2) for every regular cardinal λ ∈ [µ, κ].
+Proof. Given κ ∈ T (µ, θ), let us fix T ∈ T (µ, θ) admitting an injective sequence ⟨bξ |
+ξ < κ⟩ consisting of elements of B(T ). Define c : [κ]2 → θ via c(α, β) := ∆(bα, bβ).
+Now, given τ < θ and A ∈ [κ]λ for a regular cardinal λ ∈ [µ, κ], since |Tτ+1| < µ, it
+is possible to find x ∈ Tτ+1 for which B := {α ∈ A | x ⊑ bα} has size λ. Evidently,
+c(α, β) > τ for all α ̸= β from B.
+□
+4. Coloring well-behaved magmas
+In this section, we obtain sufficient conditions for G ↛ [λ]FSn
+θ
+to hold. To ease
+on the reader, we start with the special case of n = 2. The upcoming Lemma 4.2
+reduces this case to the following simple combinatorial principle.
+Definition 4.1. S2(κ, λ, θ) asserts the existence of a coloring d : [κ]<ω → θ such
+that, for every X ⊆ [κ]<ω of size λ and every prescribed color τ < θ, there exist
+two distinct x, y ∈ X such that d(z) = τ whenever (x △ y) ⊆ z ⊆ (x ∪ y).
+Lemma 4.2 ([FBR17, Theorem 4.7]). Suppose that S2(κ, λ, θ) holds, for given
+cardinals θ ≤ λ ≤ κ with λ regular and uncountable.
+Then G ↛ [λ]FS2
+θ
+holds for every well-behaved magma (G, ∗) with |G| = κ.
+Proof. Let d be coloring witnessing S2(κ, λ, θ).
+Suppose that (G, ∗) is a well-
+behaved magma with |G| = κ.
+By identifying [G]<ω with [κ]<ω, we may thus
+fix a map ϕ : G → [G]<ω such that:
+• ϕ is <λ-to-one;
+• for all x ̸= y in G, ϕ(x) △ ϕ(y) ⊆ ϕ(x ∗ y) ⊆ ϕ(x) ∪ ϕ(y).
+Define a coloring c : G → θ by letting c := d ◦ ϕ. To see that c is as sought,
+let X ∈ [G]λ. As ϕ is <λ-to-one, X := {ϕ(x) | x ∈ X} has size λ. Thus, given
+a prescribed color τ < θ, we may find x, y ∈ X with ϕ(x) ̸= ϕ(y) such that
+d(z) = τ whenever (ϕ(x) △ ϕ(y)) ⊆ z ⊆ (ϕ(x) ∪ ϕ(y)). In particular, x ̸= y and
+c(x ∗ y) = d(ϕ(x ∗ y)) = τ.
+□
+The question arises: How do one obtain instances of S2(. . .)?
+The proof of
+[FBR17, Lemma 3.4] makes it clear that the following holds:
+Fact 4.3. Suppose that λ is a regular uncountable cardinal and that θ is an infinite
+cardinal. Then Pr1(κ, λ, θ, ω) implies S2(κ, λ, θ).
+Remark 4.4. The principle Pr1(κ, λ, θ, ω) is a particular strengthening of κ ↛
+[λ; λ]2
+θ.
+Since it will not play a role in this paper, we omit its definition, and
+settle for pointing out the following corollary. By a theorem of Fleissner [Fle78,
+§5], for every regular uncountable cardinal κ, in the forcing extension for adding
+κ-many Cohen reals, Pr1(κ, ω1, ω, ω) holds. It thus follows that if κ is a regular
+cardinal ≥ c, then after adding κ-many Cohen reals, S2(2ℵ0, ℵ1, ℵ0) holds.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+11
+In case that λ = κ, we can now improve Fact 4.3, as follows.
+Theorem 4.5. Suppose that κ is a regular uncountable cardinal and that θ is an
+infinite cardinal. Then κ ↛ [κ; κ]2
+θ implies S2(κ, κ, θ).
+Proof. Suppose that c : [κ]2 → θ is a coloring witnessing κ ↛ [κ; κ]2
+θ.
+Fix a
+bijection π : θ ↔ θ × ω, and then find c0 : [κ]2 → θ and c1 : [κ]2 → ω such that
+π(c(α, β)) = (c0(α, β), c1(α, β)) for every (α, β) ∈ [κ]2.
+Define a coloring d : [κ]<ω → θ, as follows. For z ∈ [κ]<2, just let d(z) := 0. Next,
+for z ∈ [κ]<ω of size ≥ 2, first let ⟨αi | i < |z|⟩ denote the increasing enumeration
+of z, and then let d(z) := c0(αjz, αjz+1), for
+jz := min{j < |z| − 1 | c1(αj, αj+1) = max{c1(αi, αi+1) | i < |z| − 1}}.
+To see this works, suppose that we are given a κ-sized family X ⊆ [κ]<ω, and a
+prescribed color τ < θ. By thinning out, we may assume that X forms an head-
+tail-tail ∆-system with some root r. By further thinning out, we may assume the
+existence of some n < ω such that c1“[x]2 ⊆ n for all x ∈ X. Split X into two κ-sized
+sets X = X0 ∪ X1. Set A := {min(x \ r) | x ∈ X0} and B := {max(x \ r) | x ∈ X1}.
+As c witnesses κ ↛ [κ; κ]2
+θ, fix (α, β) ∈ A ⊛ B such that c(α, β) = π−1(τ, n).
+Pick the unique x, y ∈ X such that α = min(x \ r) and β = max(x \ r).
+As
+X0 ∩ X1 = ∅, x ̸= y. Consequently, x \ r < y \ r. Now fix an arbitrary set z such
+that (x △ y) ⊆ z ⊆ (x ∪ y). Clearly |z| ≥ 2. Let ⟨αi | i < |z|⟩ denote the increasing
+enumeration of z. For every i < |z|, if {αi, αi+1} ⊆ x then c1(αi, αi+1) < n, and
+likewise, if {αi, αi+1} ⊆ y then c1(αi, αi+1) < n. As x \ r < y \ r and {α, β} ⊆ z,
+it follows that there exists j < |z| such that aj = α and αj+1 = β. For this j,
+we would have c1(αj, αj+1) = c1(α, β) = n. Altogether, d(z) = c0(α, β) = τ, as
+sought.
+□
+As for the general case (i.e., λ ≤ κ), we now present an extraction principle that
+is sufficient to derive S2(κ, λ, θ) from κ ↛ [λ; λ]2
+θ. Roughly speaking, the upcoming
+principle asserts the existence of a map e : [κ]<ω → [κ]2 that, in some scenarios,
+manages to extract two distinguished points e(z) from any given set z ∈ [κ]<ω.
+Definition 4.6. Extract2(κ, λ, θ, χ) asserts the existence of a map e : [κ]<ω → [κ]2
+satisfying that for every sequence ⟨xγ | γ < λ⟩ of subsets of κ, every r ∈ [κ]<θ, and
+every nonzero σ < χ such that:
+(1) for every (γ, γ′) ∈ [λ]2, xγ ∩ xγ′ ⊆ r;
+(2) for every γ < λ, yγ := xγ \ r has order-type σ,
+there exist j < σ and disjoint cofinal subsets Γ0, Γ1 of λ satisfying the following:
+(a) For every (γ, γ′) ∈ [Γ0 ∪ Γ1]2, yγ(j) < yγ′(j);
+(b) For every (γ, γ′) ∈ (Γ0 ⊛Γ1)∪(Γ1 ⊛Γ0), for every z ∈ [xγ ∪xγ′]<ω covering
+{yγ(j), yγ′(j)}, we have
+e(z) = (yγ(j), yγ′(j)).
+Remark 4.7. Without loss of generality, we may assume that e(z) ∈ [z]2 for every
+z ∈ [κ]<ω of size ≥ 2. Also note that Extract2(κ, κ, cf(κ), 2) is a theorem of ZFC.
+Lemma 4.8. Suppose that λ is a regular uncountable cardinal and that θ is an
+arbitrary cardinal. If κ ↛ [λ; λ]2
+θ and Extract2(κ, λ, ω, ω) both hold, then so does
+S2(κ, λ, θ).
+
+12
+IDO FELDMAN AND ASSAF RINOT
+Proof. Suppose that c : [κ]2 → θ is a witness for κ ↛ [λ; λ]2
+θ, and that e : [κ]<ω →
+[κ]2 is a witness for Extract2(κ, λ, ω, ω). We claim that d := c ◦ e is a witness for
+S2(κ, λ, θ). To this end, suppose that we are given a subfamily X ⊆ [κ]<ω of size
+λ, and a prescribed color τ < θ. As λ is regular and uncountable, by the ∆-system
+lemma, we may find a sequence ⟨xγ | γ < λ⟩ consisting of elements of X, some
+r ∈ [κ]<ω, and a nonzero σ < χ such that:
+(1) for every (γ, γ′) ∈ [λ]2, xγ ∩ xγ′ = r;
+(2) for every γ < λ, yγ := xγ \ r has order-type σ.
+It thus follows from the choice of e that we may pick some integer j < σ and cofinal
+subsets Γ0, Γ1 of λ satisfying the following:
+(a) For every (γ, γ′) ∈ [Γ0 ∪ Γ1]2, yγ(j) < yγ′(j);
+(b) For every (γ, γ′) ∈ (Γ0 ⊛Γ1)∪(Γ1 ⊛Γ0), for every z ∈ [xγ ∪xγ′]<ω covering
+{yγ(j), yγ′(j)}, we have
+e(z) = (yγ(j), yγ′(j)).
+Put A := {yγ(j) | γ ∈ Γ0} and B := {yγ(j) | γ ∈ Γ1}. By the choice of c, we may
+find (α, β) ∈ A⊛B such that c(α, β) = τ. Pick γ ∈ Γ0 such that yγ(j) = α, and pick
+γ′ ∈ Γ1 such that yγ′(j) = β. As α < β, Clause (a) implies that (γ, γ′) ∈ (Γ0 ⊛ Γ1).
+As xγ ∩ xγ′ = r, we infer that {α, β} ⊆ xγ △ xγ′. So, for every set z such that
+(xγ △ xγ′) ⊆ z ⊆ (xγ ∪ xγ′), we get that
+d(z) = c(e(z)) = c(α, β) = τ,
+as sought.
+□
+Motivated by the preceding reduction, one would like to see how to get Extract2(κ, λ,
+ω, ω). The next lemma provides a sufficient condition.
+Lemma 4.9. Suppose that κ ∈ T (µ, θ). Then there exists a map e : [κ]<ω → [κ]2
+witnessing Extract2(κ, λ, cf(θ), ω) for every regular cardinal λ with max{µ, θ+} ≤
+λ ≤ κ.
+Proof. As κ ∈ T (µ, θ), let us fix T ∈ T (µ, θ) admitting an injective sequence ⟨bξ |
+ξ < κ⟩ consisting of elements of B(T ). For notational simplicity, we shall write
+∆(α, β) for ∆(bα, bβ). First, for every z ∈ [κ]<ω, let:
+• Mz := {(α, β) ∈ [z]2 | ∆(α, β) = max(∆“[z]2)}, and
+• M ∗
+z := {(α, β) ∈ Mz | α = min{α′ | (α′, β′) ∈ Mz}}.
+Then, pick any function e : [κ]<ω → [κ]2 satisfying that for every z ∈ [κ]<ω:
+• for every z ∈ [κ]<ω of size ≥ 2, e(z) ∈ [z]2;
+• if Mz∗ is a singleton, then e(z) is its unique element.
+To see that e is a sought, suppose that λ is a regular cardinal satisfying max{µ, θ+} ≤
+λ ≤ κ, and that we are given ⟨xγ | γ < λ⟩, r ∈ [κ]<cf(θ) and σ < ω as in Defini-
+tion 4.6. By the pigeonhole principle and the Dushnik-Miller theorem, we may find
+a cofinal subset Γ ⊆ λ, an ordinal δ < θ, and a sequence ⟨tj | j < σ⟩ of nodes in
+Tδ+1 such that for every (γ, γ′) ∈ [Γ]2:
+(I) sup(∆“[r ∪ yγ]2) = δ;
+(II) for every j < σ, byγ(j) ↾ (δ + 1) = tj;
+(III) for every j < σ, yγ(j) < yγ′(j).
+Claim 4.9.1. There exist Γ0, Γ1 ∈ [Γ]λ and a sequence ⟨(sj, ij, i′
+j) | j < σ⟩ of
+triples in T × µ × µ such that, for every j < σ:
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+13
+• for every γ ∈ Γ0, sj⌢⟨ij⟩ ⊑ byγ(j),
+• for every γ ∈ Γ1, sj⌢⟨i′
+j⟩ ⊑ byγ(j), and
+• ij ̸= i′
+j.
+Proof. We shall define by recursion a sequence of pairs ⟨(Aj, Bj) | j ≤ σ⟩ such that,
+for all j < σ, Aj+1 ∈ [Aj ∩ Γ]λ and Bj+1 ∈ [Bj ∩ Γ]λ.
+We commence by letting both A0 and B0 be Γ. Now, for every j < σ for which
+the pair (Aj, Bj) has already been defined, we do the following. Set Aj := {yγ(j) |
+γ ∈ Aj} and Bj := {yγ(j) | γ ∈ Bj}. By Clause (III), Aj and Bj have size λ. So, by
+Lemma 3.4(2), we may find sj ∈ T and ij ̸= i′
+j such that {α ∈ Aj | sj⌢⟨ij⟩ ⊑ bα}
+and {β ∈ Bj | sj⌢⟨i′
+j⟩ ⊑ bβ} are both of size λ. Then, let Aj+1 := {γ ∈ Aj |
+sj⌢⟨ij⟩ ⊆ byγ(j)} and Bj+1 := {γ ∈ Bj | sj⌢⟨1 − ij⟩ ⊆ byγ(j)}.
+Clearly, Γ0 := Aσ and Γ1 := Bσ are as sought.
+□
+Let Γ0, Γ1 and ⟨(sj, ij, i′
+j) | j < σ⟩ be given by the preceding claim. Note that
+Γ0 is disjoint from Γ1. Set δ∗ := max{dom(sj) | j < σ} and j∗ := min{j < σ |
+dom(sj) = δ∗}.
+Claim 4.9.2. Let (γ, γ′) ∈ (Γ0 ⊛ Γ1) ∪ (Γ1 ⊛ Γ0). Let z ∈ [xγ ∪ xγ′]<ω be such that
+{yγ(j∗), yγ′(j∗)} ⊆ z. Then e(z) = (yγ(j∗), yγ′(j∗)).
+Proof. It is clear that 2 ≤ otp(z) < ω, so that Mz is nonempty. Let (α, β) ∈ [z]2.
+By the choice of z, we must analyze the following cases:
+(1) Suppose that α ∈ r.
+As β ∈ r ∪ yγ ∪ yγ′, it follows from Clause (I) that ∆(α, β) ≤ δ.
+(2) Suppose that α ∈ yγ.
+(a) If β ∈ r ∪ yγ, then it follows from Clause (I) that ∆(α, β) ≤ δ;
+(b) If β ∈ yγ′, then let jα, jβ < σ be such that, α = yγ(jα) and β = yγ′(jβ).
+There are two possible options:
+(i) If jα = jβ = j, then by Clause (II), fα ↾ (δ + 1) = tj = fβ ↾
+(δ + 1). So ∆(α, β) > δ.
+(ii) If jα ̸= jβ, then by Clauses (I) and (II),
+∆(yγ′(jα), β) ≤ δ < ∆(α, yγ′(jα)),
+and hence ∆(α, β) = ∆(yγ′(jα), β) ≤ δ.
+(3) If α ∈ yγ′, then the analysis is analogous to that of (2).
+Altogether, so far we have shown that
+∅ ⊊ Mz ⊆ {(yγ(j), yγ′(j)) | j < σ}.
+Recalling that (γ, γ′) ∈ (Γ0 ⊛ Γ1) ∪ (Γ1 ⊛ Γ0), we infer from the choice of δ∗ that
+∅ ⊊ Mz ⊆ {(yγ(j), yγ′(j)) | j < σ, dom(sj) = δ∗}.
+So, since {yγ(j∗), yγ′(j∗)} ⊆ z, it is the case that M ∗
+z = {(yγ(j∗), yγ′(j∗))}. In
+particular, e(z) = (yγ(j∗), yγ′(j∗)), as sought.
+□
+This completes the proof.
+□
+Corollary 4.10. Suppose that λ is an infinite regular cardinal, and ν < λ.
+If there exists a cardinal θ < λ such that νθ ≥ λ, then Extract2(νθ, λ, cf(θ), ω)
+holds for the least such θ.
+
+14
+IDO FELDMAN AND ASSAF RINOT
+Proof. Let θ denote the least cardinal such that νθ ≥ λ. Then T := <θν belongs
+to T (θ, λ), so that νθ = |B(T )| is in T (θ, λ). Now, appeal to Lemma 4.9 with
+(κ, µ) := (νθ, λ).
+□
+Corollary 4.11. For every regular uncountable cardinal κ that is not a strong limit,
+Extract2(κ, κ, ω, ω) holds.
+□
+Proposition 4.12. Suppose that λ is a regular uncountable cardinal.
+For every cardinal κ > 2<λ, Extract2(κ, λ, 2, 2) fails.
+Proof. Set ν := 2<λ, and note that νθ = ν for every θ < λ. Towards a contradiction,
+suppose that e : [κ]<ω → [κ]2 is a map witnessing Extract2(κ, λ, 2, 2), and yet κ > ν.
+Without loss of generality, we may assume that e(z) ∈ [z]2 for every z ∈ [κ]3.
+Claim 4.12.1. For every δ < κ, there exist no subset A ⊆ δ of order-type λ such
+that δ ∈ e({α, β, δ}) for every (α, β) ∈ [A]2.
+Proof. Otherwise, fix a counterexample δ and a witnessing A ⊆ δ. Let ⟨αγ | γ < λ⟩
+be the increasing enumeration of A. Now let r := {δ} and, for every γ < λ, put
+xγ := yγ ⊎ r where yγ := {αγ}. Then, for every (γ, γ′) ∈ [λ]2, setting z := xγ ∪ xγ′,
+we get that r ∩ e(z) ̸= ∅. This is a contradiction.
+□
+Denote κ := ν+. It follows from the claim that for every δ ∈ Eκ
+λ , we may fix some
+Aδ ∈ [δ]<λ with the property that for every ordinal β such that sup(Aδ) < β < δ,
+there exists α ∈ Aδ such that e({α, β, δ}) = (α, β). Now, using Fodor’s lemma,
+we may find ε < κ and θ < λ such that {δ ∈ Eκ
+λ | ssup(Aδ) = ε & |Aδ| = θ} is
+stationary. Recalling that νθ = ν < κ, we may then find some A ∈ [ε]θ for which
+S := {δ ∈ Eκ
+λ | Aδ = A} is stationary. Define a coloring c : [S]2 → A by letting for
+every (β, δ) ∈ [S]2:
+c(β, δ) := min{α ∈ A | e({α, β, δ}) = (α, β)}.
+By the Erd˝os-Rado theorem, κ → (λ)2
+θ holds, so we may pick B ⊆ S of order-
+type λ that is c-homogeneous, with value, say, α. Let ⟨βγ | γ < λ⟩ be the increasing
+enumeration of B. Finally, let r := {α} and, for every γ < λ, put xγ := yγ ⊎ r
+where yγ := {βγ}. Then, for every (γ, γ′) ∈ [λ]2, setting z := xγ ∪ xγ′, we get that
+r ∩ e(z) ̸= ∅. This is a contradiction.
+□
+Corollary 4.13. If κ is a strong limit cardinal, then Extract2(κ, λ, 2, 2) fails for
+every infinite cardinal λ < κ.
+□
+Corollary 4.14. Extract2(ℵ2, ℵ1, ℵ0, ℵ0) holds iff CH fails.
+Proof. By Corollary 4.10 and Proposition 4.12.
+□
+Moving on from the case n = 2 to the general case, we consider the following
+two definitions.
+Definition 4.15. Sn(κ, λ, θ) asserts the existence of a coloring d : [κ]<ω → θ such
+that, for every X ⊆ [κ]<ω of size λ and every prescribed color τ < θ, there exist
+{aj | j < n} ∈ [X]n such that d(z) = τ for every z satisfying
+a0 △ (
+�
+0<j<n aj) ⊆ z ⊆
+�
+j<n aj.
+Proposition 4.16 (monotonicity). Suppose that:
+(1) 2 ≤ n ≤ n′ < ω;
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+15
+(2) ω ≤ λ ≤ λ′;
+(3) θ ≤ θ′.
+Then Sn(κ, λ, θ′) implies Sn′(κ, λ′, θ).
+Proof. This is mostly trivial, so we settle for proving that if d : [κ]<ω → θ witnesses
+Sn(κ, λ, θ) for some integer n ≥ 2, then it also witnesses Sn+1(κ, λ, θ).
+To this end, let X ⊆ [κ]<ω be a given family of size λ. Pick x ∈ X, and note that
+X ′ := {a ∪ x | a ∈ X \ {x}} is a λ-sized subset of [κ]<ω. Now, given a prescribed
+color τ < θ, pick {a′
+j | j < n} ∈ [X ′]n such that d(z) = τ for every z satisfying
+a′
+0 △ (
+�
+0<j<n a′
+j) ⊆ z ⊆
+�
+j<n a′
+j.
+For each j < n, pick aj ∈ X \{x} such that a′
+j = aj∪x. As ⟨a′
+j | j < n⟩ is an injective
+sequence, so is ⟨aj | j < n⟩. Set an := x. Altogether, {aj | j < n + 1} ∈ [X]n+1. It
+is clear that �
+j<n a′
+j = �
+j<n+1 aj. In addition,
+a′
+0 △ (�
+0<j<n a′
+j)
+=
+(a′
+0 \ (�
+0<j<n a′
+j))
+∪
+((�
+0<j<n a′
+j) \ a′
+0)
+⊆
+(a0 \ (�
+0<j<n aj))
+∪
+((�
+0<j<n aj) \ a0)
+⊆
+(a0 \ (�
+0<j<n+1 aj))
+∪
+((�
+0<j<n+1 aj) \ a0)
+=
+a0 △ (�
+0<j<n+1 aj).
+Therefore, d(z) = τ for every set z with a0 △ (�
+0<j<n+1 aj) ⊆ z ⊆ �
+j<n+1 aj.
+□
+Definition 4.17. Extractn(κ, λ, θ, χ) asserts the existence of a map e : [κ]<ω → nκ
+such that for every sequence ⟨xγ | γ < λ⟩ of subsets of κ, every r ∈ [κ]<θ, and every
+nonzero σ < χ such that:
+(1) for every (γ, γ′) ∈ [λ]2, xγ ∩ xγ′ ⊆ r;
+(2) for every γ < λ, yγ := xγ \ r has order-type σ,
+there exist j < σ and disjoint cofinal subsets Γ0, Γ1 of λ satisfying the following:
+(a) For every (γ, γ′) ∈ [Γ0 ∪ Γ1]2, yγ(j) < yγ′(j);
+(b) For every strictly increasing sequence ⟨γi | i < n⟩ of ordinals from Γ0 ∪ Γ1
+such that {γi | i < n} /∈ ([Γ0]n ∪ [Γ1]n), for every z ∈ [�
+i<n xγi]<ω that
+covers {yγi(j) | i < n}, we have
+e(z) = ⟨yγi(j) | i < n⟩.
+Remark 4.18. Without loss of generality, we may assume that for every z ∈ [κ]<ω
+of size ≥ n, e(z) consists of ordinals from z.
+The proof of [FBR17, Theorem 4.7] makes it clear that the following holds.
+Proposition 4.19. Suppose that Sn(κ, λ, θ) holds for given cardinals θ ≤ λ ≤ κ
+with λ regular and uncountable. For every map ϕ : G → [G]<ω that is <λ-to-one,
+there exists a corresponding coloring c : G → θ satisfying the following.
+For every binary operation ∗ on G such that, for all x ̸= y in G,
+ϕ(x) △ ϕ(y) ⊆ ϕ(x ∗ y) ⊆ ϕ(x) ∪ ϕ(y),
+for every X ∈ [G]λ and every τ < θ, there is an injective sequence ⟨xj | 1 ≤ j ≤ n⟩
+of elements of X such that c(x1 ∗ · · · ∗ xn) = τ.6
+□
+In order to generalize Lemma 4.8, we now introduce the relation κ
+sup
+�−→ [λ, λ]n
+θ .
+It is a strengthening of κ ↛ [λ]n
+θ , and a weakening of κ ↛ [λ, . . . , λ]n
+θ .
+6As always, we mean that this holds true for all implementations of x1 ∗ · · · ∗ xn.
+
+16
+IDO FELDMAN AND ASSAF RINOT
+Definition 4.20. κ
+sup
+�−→ [λ, λ]n
+θ asserts the existence of a coloring c : [κ]n → θ such
+that for all τ < θ and disjoint A, B ∈ P(κ) satisfying the two:
+(i) otp(A) = otp(B) = λ,
+(ii) sup(A) = sup(B),
+there is ⃗x ∈ [A ∪ B]n \ ([A]n ∪ [B]n) with c(⃗x) = τ.
+Remark 4.21. In the special case of λ = κ, κ
+sup
+�−→ [λ, λ]2
+θ coincides with the classical
+relation κ ↛ [λ, λ]2
+θ.
+Lemma 4.22. Suppose that:
+• 2 ≤ n < ω;
+• θ ≤ λ ≤ κ are cardinals with λ regular and uncountable;
+• κ
+sup
+�−→ [λ, λ]n
+θ holds;
+• Extractn(κ, λ, ω, ω) holds.
+Then Sn(κ, λ, θ) holds.
+Proof. The proof is similar to that of Lemma 4.8, so we settle for a sketch. Fix a
+map c : [κ]n → θ witnessing κ
+sup
+�−→ [λ, λ]n
+θ and a map e : [κ]<ω → nκ witnessing
+Extractn(κ, λ, ω, ω). We claim that d := c ◦ e is a witness for Sn(κ, λ, θ). To this
+end, suppose that we are given a subfamily X ⊆ [κ]<ω of size λ, and a prescribed
+color τ < θ. By the ∆-system lemma, find a sequence ⟨xγ | γ < λ⟩ consisting of
+elements of X, some r ∈ [κ]<ω, and a nonzero σ < χ such that:
+(1) for every (γ, γ′) ∈ [λ]2, xγ ∩ xγ′ = r;
+(2) for every γ < λ, yγ := xγ \ r has order-type σ.
+Now, let j < σ and disjoint cofinal subsets Γ0, Γ1 of λ be given, as in Defini-
+tion 4.17. Put A := {yγ(j) | γ ∈ Γ0} and B := {yγ(j) | γ ∈ Γ1}. By the choice of
+c, pick ⃗x ∈ [A ∪ B]n \ ([A]n ∪ [B]n) such that c(⃗x) = τ. Find a sequence ⃗γ = ⟨γi |
+i < n⟩ of ordinals from Γ0 ∪ Γ1 such that ⃗x = ⟨yγi(j) | i < n⟩. Clearly, ⃗γ is strictly
+increasing, {γi | i < n} ⊈ Γ0 and {γi | i < n} ⊈ Γ1. So, for every set z of interest,
+d(z) = c(e(z)) = c(⃗x) = τ,
+as sought.
+□
+Lemma 4.23. Suppose that κ ∈ T (µ, θ). Then there exists a map e : [κ]<ω → 3κ
+witnessing Extract3(κ, λ, cf(θ), ω) for every regular cardinal λ with max{µ, θ+} ≤
+λ ≤ κ.
+Proof. As κ ∈ T (µ, θ), let us fix T ∈ T (µ, θ) admitting an injective sequence
+⟨bξ | ξ < κ⟩ consisting of elements of B(T ). For notational simplicity, we shall
+write ∆(α, β) for ∆(bα, bβ). For any triplet w ∈ [κ]3, let ∆2(w) := min{∆(α, β) |
+(α, β) ∈ [w]2}. First, given z ∈ [κ]<ω, let:
+• Mz := {(α, β, γ) ∈ [z]3 | ∆2({α, β, γ}) = max{∆2(w) | w ∈ [z]3}}, and
+• M ∗
+z := {(α, β, γ) ∈ Mz | α = min{α′ | (α′, β′, γ′) ∈ Mz}}.
+Then, pick any function e : [κ]<ω → 3κ satisfying that for every z ∈ [κ]<ω:
+• for every z ∈ [κ]<ω, e(z) is a strictly increasing sequence of ordinals in κ.
+If |z| ≥ 3, then e(z) consists of ordinals from z;
+• if Mz∗ is a singleton, then e(z) is its unique element.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+17
+To see that e is a sought, suppose that λ is a regular cardinal satisfying max{µ, θ+} ≤
+λ ≤ κ, and that we are given ⟨xγ | γ < λ⟩, r ∈ [κ]<cf(θ) and σ < ω as in Defini-
+tion 4.17. As in the proof of Lemma 4.9, we may find a cofinal subset Γ ⊆ λ, an
+ordinal δ < θ, and a sequence ⟨tj | j < σ⟩ of nodes in Tδ+1 such that for every
+(γ, γ′) ∈ [Γ]2:
+(I) sup(∆“[r ∪ yγ]2) = δ;
+(II) for every j < σ, byγ(j) ↾ (δ + 1) = tj;
+(III) for every j < σ, yγ(j) < yγ′(j).
+In addition, we may fix Γ0, Γ1 ∈ [Γ]λ+ and a sequence ⟨(sj, ij, i′
+j) | j < m⟩ of triples
+in T × µ × µ such that, for every j < σ:
+• for every γ ∈ Γ0, sj⌢⟨ij⟩ ⊑ byγ(j),
+• for every γ ∈ Γ1, sj⌢⟨i′
+j⟩ ⊑ byγ(j), and
+• ij ̸= i′
+j.
+By possibly passing to cofinal subsets, we may assume that Γ0 ∩ Γ1 = ∅.
+Let
+δ∗ := max{dom(sj) | j < σ} and j∗ := min{j < σ | dom(sj) = δ∗}.
+Claim 4.23.1. Let (γ, γ′, γ′′) ∈ [Γ0∪Γ1]3\([Γ0]3∪[Γ1]3). Let z ∈ [xγ ∪xγ′ ∪xγ′′]<ω
+be such that {yγ(j∗), yγ′(j∗), yγ′′(j∗)} ⊆ z. Then e(z) = (yγ(j∗), yγ′(j∗), yγ′′(j∗)).
+Proof. As made clear by the proof of Claim 4.9.2, for every (ξ, ζ) ∈ [{γ, γ′, γ′′}]2,
+for every α ∈ r ∪ aξ and β ∈ r ∪ yζ, ∆(α, β) > δ iff there is a j < σ such that
+α = yξ(j) and β = yζ(j). In addition, if {ξ, ζ} ⊈ Γ0 and {ξ, ζ} ⊈ Γ1, then for every
+j < m, ∆(yξ(j), yζ(j)) = dom(sj). So, in this case,
+{(α, β) ∈ [r ∪ yξ ∪ yζ]2 | ∆(α, β) = max(∆“[r ∪ yξ ∪ yζ]2)}
+= {(yξ(j), yζ(j)) | dom(sj) = δ∗}.
+Now, since {γ, γ′, γ′′} ⊈ Γ0 and {γ, γ′, γ′′} ⊈ Γ1, it follows that
+∆2(yγ(j∗), yγ′(j∗), yγ′′(j∗)) = δ∗.
+Consequently,
+∅ ⊊ Mz = {(yγ(j), yγ′(j), yγ′′(j)) | dom(sj) = δ∗}.
+So, by Clause (III),
+e(z) = (yγ(j∗), yγ′(j∗), yγ′′(j∗)),
+as sought.
+□
+This completes the proof.
+□
+Proposition 4.24. Suppose that λ ≤ κ is a pair of infinite cardinals.
+If Extract2(κ, λ, 3, 3) holds, then so does κ
+sup
+�−→ [λ, λ]4
+2.
+Proof. Suppose that κ
+sup
+�−→ [λ, λ]4
+2 fails, and we shall prove that Extract2(κ, λ, 3, 3)
+fails, as well. To this end, let e : [κ]<ω → [κ]2 be given. Define a coloring c : [κ]4 → 2
+by letting for all α0 < α1 < α2 < α3 < κ:
+c(α0, α1, α2, α3) := 1 iff e(α0, α1, α2, α3) = (α2, α3).
+Now, since κ
+sup
+−→ [λ, λ]4
+2 holds, we may find τ < 2 and disjoint A, B ∈ P(κ)
+satisfying all of the following:
+(i) otp(A) = otp(B) = λ,
+(ii) sup(A) = sup(B),
+
+18
+IDO FELDMAN AND ASSAF RINOT
+(iii) for every (α0, α1, α2, α3) ∈ [A ∪ B]4 \ ([A]4 ∪ [B]4),
+c(α0, α1, α2, α3) ̸= τ.
+Using Clauses (i) and (ii), fix a sequence ⟨(αi, βi) | i < λ⟩ of pairs in A × B such
+that, for all i < j < λ, αi < βi < αj.
+◮ If τ = 1, then let r := {α0, α1}, and for every γ < λ, let xγ := r ⊎ yγ, where
+yγ := {βγ+1}. Now, for every (γ, γ′) ∈ [λ]2, as z := xγ ∪ xγ′ is in [A ∪ B]4 \ ([A]4 ∪
+[B]4), c(z) = 0, and then e(z) is not disjoint from r.
+◮ If τ = 0, then let r := ∅, and for every γ < λ, let xγ := r ⊎ yγ, where yγ :=
+{αγ, βγ}. Now, for every (γ, γ′) ∈ [λ]2, as z := xγ ∪xγ′ is in [A∪B]4 \ ([A]4 ∪[B]4),
+c(z) = 1, and then e(z) = yγ′ which is disjoint from yγ.
+□
+Corollary 4.25. If λ = ℵ0 or if λ is weakly compact, then Extract2(κ, λ, 3, 3) fails
+for every cardinal κ ≥ λ.
+□
+5. Maximal number of colors
+This section is dedicated to the proof of Theorem C. The main corollary of this
+section reads as follows:
+Corollary 5.1. Suppose that λ = µ+ for an infinite cardinal µ = µ<µ.
+Then the following are equivalent:
+(1) (λ+, λ) ։ (µ+, µ) fails;
+(2) λ+ sup
+�−→ [λ, λ]3
+λ holds.
+Proof. We focus on the nontrivial (that is, forward) implication. As λ = µ+, by
+[Tod07, Lemma 9.2.3], the failure of (λ+, λ) ։ (µ+, µ) is equivalent to the existence
+of a subadditive coloring ̺ : [λ+]2 → λ witnessing U(λ+, λ, λ, λ, ω). In particular,
+̺ ↾ [X]2 witnesses U(λ, λ, λ, 3) for every X ⊆ λ+ of order-type λ. Now, there are
+two cases to consider:
+◮ If 2µ > µ+, then since µ<µ = µ, T := <µµ is a weak µ-Kurepa tree with
+λ+-many branches. In addition, µ<µ = µ implies that µ is regular, so that Eλ
+µ is
+a nonreflecting stationary set. Now the result follows from the upcoming Theo-
+rem 5.2, using κ := λ+.
+◮ If 2µ = µ+, then λ ↛ [µ; λ]2
+λ holds by a theorem of Sierpi´nski (see [IR23,
+Lemma 8.3]). Now the result follows from Theorem 5.3 below.
+□
+When reading the hypotheses of the upcoming theorem, it may worth keeping
+in mind that if λ = µ+ for an infinite regular cardinal µ, then Eλ
+µ is a nonreflecting
+stationary set, and if κ = λ+, then Fact 2.14 provides a subadditive map ρ : [κ]2 →
+λ. The conclusion of the theorem is a conditional form of κ
+sup
+�−→ [λ, λ]3
+λ in which a
+third clause is added to Definition 4.20.
+Theorem 5.2. Suppose that:
+• µ < λ < κ are infinite regular cardinals;
+• Eλ
+µ admits a nonreflecting stationary set;
+• ̺ : [κ]2 → λ is a subadditive coloring of pairs;
+• there exists a weak µ-Kurepa tree with at least κ-many branches.
+Then there exists a corresponding coloring of triples c : [κ]3 → λ such that, for
+all τ < λ and disjoint A, B ∈ P(κ) satisfying the three:
+(i) otp(A) = otp(B) = λ,
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+19
+(ii) sup(A) = sup(B),
+(iii) ̺ ↾ [A ∪ B]2 witnesses U(λ, λ, λ, 3),
+there exists (α, β, γ) ∈ [A ∪ B]3 \ ([A]3 ∪ [B]3) such that c(α, β, γ) = τ.
+Proof. Let T ⊆ <µ2 be a weak µ-Kurepa tree with at least κ-many branches, and
+let ⃗b = ⟨bξ | ξ < κ⟩ be an injective sequence consisting of elements of B(T ). For
+all α ̸= β from κ, we write ∆(α, β) for ∆(bα, bβ). For all B ⊆ κ and t ∈ T , denote
+Bt := {β ∈ B | t ⊑ bβ}. As Eλ
+µ admits a nonreflecting stationary set, by [LHR23,
+Lemma 3.31], we may fix a coloring e : [λ]2 → µ for which the following set is
+stationary:
+∂(e) := {σ ∈ Eλ
+µ | ∀ǫ ∈ λ \ σ ∀δ < µ [sup{ζ < σ | e(ζ, ǫ) ≤ δ} < σ]}.
+Let h : λ → λ be a surjection such that Sτ := {σ ∈ ∂(e) | h(σ) = τ} is stationary
+for every τ < λ.
+For every (α, β, γ) ∈ κ × κ × κ, let
+Z(α,β,γ) := {ζ ∈ λ \ ̺({α, β}) | max(∆“{α, β, γ}2) ≥ e(ζ, ̺({β, γ}))}.
+Now, derive a coloring c : [κ]3 → λ by letting:
+c({α, β, γ}) :=
+
+
+
+
+
+h(min(Z(β,α,γ))),
+if γ = max{α, β, γ} & bα <lex bβ <lex bγ;
+h(min(Z(α,β,γ))),
+if γ = max{α, β, γ} & bγ <lex bα <lex bβ;
+0,
+otherwise.
+Suppose that A, B ∈ P(κ) are disjoint sets satisfying conditions (i)–(iii) above.
+Recalling Lemma 3.4(2), by possibly shrinking A and B, we may assume the exis-
+tence of some χ < µ such that ∆[A × B] = {χ}. By possibly switching the roles of
+A and B, we may also assume the following:
+(iv) for every (α, β) ∈ A × B, bα <lex bβ.
+Note that |T | = µ < λ < κ ≤ 2µ. Now, given a prescribed color τ < λ, let M be
+an elementary submodel of H(2µ)+ containing {A, B, ̺,⃗b, T } such that σ := M ∩λ is
+in Sτ. In particular, |M| ≥ cf(σ) = µ. Denote υ := sup(A) and υM := sup(M ∩ υ).
+The proof is now divided into two cases:
+Case 1: For every β ∈ B, sup(̺β“A) < λ. By Condition (iii), find A′ ∈ [A]λ
+and B′ ∈ [B]λ such that min(̺[A′ ⊛ B′]) > σ. Fix α ∈ A′ \ υM arbitrarily. Appeal
+to Corollary 3.5 with X := B′ and i := 1 to pick γ ∈ B′ \ (α + 1) such that for
+cofinally many δ < µ, the two hold:
+(1) bγ(δ) = 1, and
+(2) {β ∈ B′ | ∆(β, γ) = δ} has size λ.
+Denote D := {δ < µ | Clauses (1) and (2) both hold}.
+As (α, γ) ∈ A′ ⊛ B′, the ordinal ǫ := ̺(α, γ) is bigger than σ. Pick δ ∈ D above
+max{χ, e(σ, ǫ)}. Since σ ∈ ∂(e), the following set is bounded below σ:
+Z := {ζ < σ | e(ζ, ǫ) ≤ δ}.
+Set t := (bγ ↾ δ)⌢⟨0⟩. From δ ∈ D we infer that (B′)t has size λ. As t ∈ T ⊆ M,
+in particular, Bt is a set of size λ lying in M. By Condition (iii) and elementarity
+of M one can find β0 ̸= β1 in Bt ∩ M such that ̺(β0, β1) > sup(Z).
+As ̺ is
+subadditive, we may now find β ∈ {β0, β1} such that ̺(β, α) > sup(Z). As β ∈ B,
+sup(̺β“A) < λ. As {̺, A} ∈ M, it follows that sup(̺β“A) ∈ M. In particular,
+̺(β, α) < σ.
+
+20
+IDO FELDMAN AND ASSAF RINOT
+Claim 5.2.1. All of the following hold:
+(1) (β, α, γ) ∈ B ⊛ A ⊛ B;
+(2) bα <lex bβ <lex bγ;
+(3) max(∆“{α, β, γ}2) = δ;
+(4) min(Z(β,α,γ)) = σ.
+Proof.
+(1) γ was chosen to be in B \ (α + 1). In addition, α ∈ A \ υM, whereas
+β ∈ M ∩ B.
+(2) As α ∈ A and β ∈ B, Condition (iv) entails that bα <lex bβ. In addition,
+as bβ ↾ (δ + 1) = t = (bγ ↾ δ)⌢⟨0⟩ and bγ(δ) = 1, we get that bβ <lex bγ.
+(3) By the previous analysis, ∆(β, γ) = δ.
+In addition, ∆(α, γ) = χ < δ.
+Recalling that |∆“{α, β, γ}2| = 2, we are done.
+(4) By Clause (3) and the fact that ̺(α, γ) = ǫ, we infer that Z(β,α,γ) = {ζ ∈ λ\
+̺(β, α) | δ ≥ e(ζ, ǫ)}. In particular, σ ∈ Z(β,α,γ). Now, if ζ := min(Z(β,α,γ))
+is < σ, then ζ ∈ Z, contradicting the fact that ̺(β, α) > sup(Z).
+□
+By the preceding claim and the definition of c,
+c({α, β, γ}) = h(min(Z(β,α,γ))) = h(σ) = τ,
+as sought.
+Case 2: There is β ∈ B such that sup(̺β“A) = λ. As {̺, A} ∈ M, we may
+pick β ∈ B ∩ M such that sup(̺β“A) = λ. Clearly, |̺β“A| = λ. Define a function
+f : ̺β“A → A via
+f(ξ) := min{α ∈ A | ̺(β, α) = ξ}.
+As {β, A, ̺} ∈ M, we infer that ̺β“A, f and Im(f) are all in M. Note that f is
+injective, so that | Im(f)| = λ. It also follows that
+Γ := {γ ∈ Im(f) \ (β + 1) | ̺(β, γ) ≤ σ}
+is bounded in Im(f).
+Appeal to Corollary 3.5 with X := Im(f) \ (β + 1) and i := 0 to pick γ ∈
+X \ (Γ ∪ υM) such that for cofinally many δ < µ, the two hold:
+(1) bγ(δ) = 0, and
+(2) {α ∈ X | ∆(α, γ) = δ} has size λ.
+Denote D := {δ < µ | Clauses (1) and (2) both hold}.
+As γ /∈ Γ, ǫ := ̺(β, γ) is bigger than σ. Pick δ ∈ D above max{χ, e(σ, ǫ)}. Since
+σ ∈ ∂(e), the following set is bounded below σ:
+Z := {ζ < σ | e(ζ, ǫ) ≤ δ}.
+Set t := (bγ ↾ δ)⌢⟨1⟩.
+From δ ∈ D we infer that Xt is a set of size λ.
+As
+t ∈ T ⊆ M and X ∈ M, Xt is in M. As α �→ ̺(β, α) is injective over X, we
+may find an α ∈ Xt ∩ M such that ̺(β, α) > sup(Z). Because of the fact that
+{β, α} ∈ M, we altogether get that sup(Z) < ̺(β, α) < σ.
+Claim 5.2.2. All of the following hold:
+(1) (β, α, γ) ∈ B ⊛ A ⊛ A;
+(2) bγ <lex bα <lex bβ;
+(3) max(∆“{α, β, γ}2) = δ;
+(4) min(Z(α,β,γ)) = σ.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+21
+Proof.
+(1) β was chosen to be in B ∩M, γ was chosen to be in A\υM, whereas
+α ∈ M ∩ A with α > β.
+(2) As α ∈ A and β ∈ B, Condition (iv) entails that bα <lex bβ. In addition,
+bα ↾ (δ + 1) = t = (bγ ↾ δ)⌢⟨1⟩ and bγ(δ) = 0, so bγ <lex bα.
+(3) By the previous analysis, ∆(α, γ) = δ.
+In addition, ∆(α, β) = χ < δ.
+Recalling that |∆“{α, β, γ}2| = 2, we are done.
+(4) By Clause (3) and the fact that ̺(β, γ) = ǫ, we infer that Z(α,β,γ) = {ζ ∈ λ\
+̺(β, α) | δ ≥ e(ζ, ǫ)}. In particular, σ ∈ Z(α,β,γ). Now, if ζ := min(Z(α,β,γ))
+is < σ, then ζ ∈ Z, contradicting the fact that ̺(β, α) > sup(Z).
+□
+By the preceding claim and the definition of c,
+c({α, β, γ}) = h(min(Z(α,β,γ))) = h(σ) = τ,
+as sought.
+□
+Theorem 5.3. Suppose that:
+• µ = µ<µ is an infinite cardinal, λ = µ+ and κ = λ+;
+• ̺ : [κ]2 → λ is a subadditive coloring of pairs;
+• λ ↛ [µ; λ]2
+λ holds.
+Then, there exists a corresponding coloring of triples c : [κ]3 → λ such that, for
+all τ < λ and disjoint A, B ∈ P(κ) satisfying the three:
+(i) otp(A) = otp(B) = λ,
+(ii) sup(A) = sup(B),
+(iii) ̺ ↾ [A ∪ B]2 witnesses U(λ, λ, λ, 3),
+there exists (α, β, γ) ∈ [A ∪ B]3 \ ([A]3 ∪ [B]3) such that c(α, β, γ) = τ.
+Proof. Let d : [κ]2 → λ be a coloring witnessing λ ↛ [µ; λ]2
+λ. Let T := <λ2. Let
+⃗b = ⟨bξ | ξ < κ⟩ be an injective enumeration of elements of B(T ). For α ̸= β from κ,
+we write ∆(α, β) for ∆(bα, bβ). Likewise, for B ⊆ κ, we write T ⇝B for T ⇝{bβ|β∈B}.
+For all B ⊆ κ and t ∈ T , denote Bt := {β ∈ B | t ⊑ bβ}. Let e : [λ]2 → µ be a
+map with injective fibers. Let h : λ → λ be a surjection such that Sτ := {σ ∈ Eλ
+µ |
+h(σ) = τ} is stationary for every τ < λ. For every (α, β, γ) ∈ κ ⊛ κ ⊛ κ, define:
+Z(α,β,γ) := {ζ ∈ λ \ ̺(α, γ) | e(∆(α, β), ̺(β, γ)) ≥ e(ζ, ̺(β, γ))}.
+We define a coloring c : [κ]3 → λ by letting for all α < β < γ < κ:
+c(α, β, γ) :=
+
+
+
+
+
+
+
+
+
+d(∆(α, γ), ∆(β, γ)),
+if ∆(α, β) < ∆(β, γ);
+d(∆(α, γ), ̺(β, γ)),
+if ∆(α, β) = ∆(β, γ);
+d(∆(α, β), ̺(α, γ)),
+if ∆(α, β) > ∆(β, γ) & bα <lex bβ;
+h(min(Z(α,β,γ))),
+if ∆(α, β) > ∆(β, γ) & bβ <lex bα.
+Suppose that A, B ∈ P(κ) are disjoint sets satisfying conditions (i)–(iii) above
+and let τ < λ be a prescribed color. By possibly passing to a cofinal subset of A, we
+may assume that A = A′ in the sense of Lemma 3.7. In particular, we may assume
+the existence of θA ≤ λ such that T ⇝A ∈ T (λ, θA) and, in addition, if θA < λ,
+then |B(T ⇝A)| = λ. As µ<µ < µ+ = λ, this means that if θA < λ, then θA ∈ Eλ
+µ.
+Likewise, we may assume the existence of θB ∈ Eλ
+µ ∪{λ} such that T ⇝B ∈ T (λ, θB).
+Without loss of generality, we may also assume that θA ≤ θB. The proof is now
+divided into two main cases.
+Case 1: θA = θB = λ. In this case, we shall need the following claim.
+
+22
+IDO FELDMAN AND ASSAF RINOT
+Claim 5.3.1. There exists t ∈ T ⇝B satisfying all of the following:
+• If t /∈ T ⇝A, then D := {∆(bβ, t) | β ∈ B} has size µ, and there exists
+t′ ∈ T ⇝A incompatible with t such that sup(D) > ∆(t, t′);
+• If t ∈ T ⇝A, then D := {∆(bα, t) | α ∈ A} has size µ.
+Proof. There are two cases to consider:
+◮ Suppose that there exists ǫ < λ such that T ⇝A ∩ T ⇝B ∩ ǫ2 = ∅.
+Pick
+t′ ∈ T ⇝A ∩ ǫ2. For each α < λ, pick tα ∈ T ⇝B ∩ α2. Then, by Lemma 3.6, there
+exists α ∈ Eλ
+µ above ǫ such that D := {∆(bβ, tα) | β ∈ B} ∩ α is cofinal in α. To
+see that t := tα is as sought, notice that since t ↾ ǫ /∈ T ⇝A, it must be the case that
+∆(t, t′) < ǫ.
+◮ Otherwise.
+Thus, for each α < λ, we may pick tα ∈ T ⇝A ∩ T ⇝B ∩ α2.
+As ⟨tα | α < λ⟩ ∈ �
+α<λ T ⇝A ∩ α2, Lemma 3.6 provides an α ∈ Eλ
+µ such that
+D := {∆(bβ, tα) | β ∈ A} ∩ α is cofinal in α. So t := tα is as sought.
+□
+Let t ∈ T ⇝B and the corresponding D be as in the claim. There are two subcases
+to consider:
+Subcase 1.1: t ∈ T ⇝A. Pick ¯A ∈ [A]µ such that ¯D := {∆(bα, t) | α ∈ ¯A} is a
+µ-sized subset of D ∩ dom(t). Clearly, B′ := Bt \ sup( ¯A) is a set of size λ. Since
+T ⇝B′ ∈ T (λ, λ), E := ∆[B′ ⊛ B′] is of size λ, as well. By the choice of d, we may
+now find (δ, ǫ) ∈ ¯D ⊛ E such that d(δ, ǫ) = τ. Pick α ∈ ¯A such that ∆(bα, t) = δ.
+Finally, find (β, γ) ∈ B′ ⊛ B′ such that ∆(β, γ) = ǫ. Then
+∆(β, γ) = ǫ > δ = ∆(bα, t) = ∆(α, γ),
+and hence ∆(α, β) = ∆(α, γ) < ∆(β, γ). Altogether, (α, β, γ) ∈ A ⊛ B ⊛ B, and
+c(α, β, γ) = d(∆(α, γ), ∆(β, γ)) = d(δ, ǫ) = τ.
+Subcase 1.2: t /∈ T ⇝A. Pick t′ ∈ T ⇝A incompatible with t such that sup(D) >
+∆(t, t′). As cf(θB) = µ, we may now pick ¯B ∈ [B]µ such that ¯D := {∆(bβ, t) |
+β ∈ ¯B} is a µ-sized subset of D with min( ¯D) > ∆(t, t′).
+As t′ ∈ T ⇝A, At′ has size λ and so does A′ := At′ \ sup( ¯B). As t ∈ T ⇝B, Bt
+has size λ, so since ̺ ↾ [A ∪ B]2 witnesses U(λ, 2, λ, 3), the set E := ̺[A′ ⊛ Bt] has
+size λ, as well. By the choice of d, find (δ, ǫ) ∈ ¯D ⊛ E such that d(δ, ǫ) = τ. Find
+α ∈ ¯B such that ∆(bα, t) = δ. Find (β, γ) ∈ A′ ⊛ Bt such that ̺(β, γ) = ǫ. As
+(β, γ) ∈ At′ ⊛ Bt,
+∆(β, γ) = ∆(t′, t) < min( ¯D) ≤ δ = ∆(bα, t) = ∆(α, γ),
+and hence ∆(α, β) = ∆(β, γ). Altogether, (α, β, γ) ∈ B ⊛ A ⊛ B, and
+c(α, β, γ) = d(∆(α, γ), ̺(β, γ)) = d(δ, ǫ) = τ.
+Case 2: θA < λ. Set θ := θA. We shall need the following claim.
+Claim 5.3.2. There exist χ < θ, A′ ∈ [A]λ and B′ ∈ [B]λ such that ∆[A′ × B′] =
+{χ}.
+Proof. Denote ¯A := B(T ⇝A) and ¯B := B(T ⇝B). Recall that by our application of
+Lemma 3.7, | ¯A| = λ, and if θB < λ, then | ¯B| = λ, as well. We shall prove the claim
+by showing that there exist χ < θ and a pair (t, t′) ∈ (T ⇝A)χ+1 × (T ⇝B)χ+1 such
+that ∆(t, t′) = χ. Indeed, once we have such a pair (t, t′), the sets A′ := At and
+B′ := Bt′ would be as sought.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+23
+There are two cases to consider:
+◮ If θB = θ, then set ¯T := T ∩ <θ2. In this case, ¯B and ¯A are λ-sized subsets
+of B( ¯T). So Lemma 3.4(2) yields an s ∈ ¯T together with i ̸= i′ such that s⌢⟨i⟩ ∈
+¯T ⇝ ¯
+A ⊆ T ⇝A and s⌢⟨i′⟩ ∈ ¯T ⇝ ¯
+B ⊆ T ⇝B. Evidently, χ := dom(s), t := s⌢⟨i⟩ and
+t′ := s⌢⟨i′⟩ are as sought.
+◮ If θB > θ, then pick r ∈ (T ⇝B)θ. For every a ∈ ¯A \ {r}, χa := ∆(a, r) is
+smaller than θ. As | ¯A| = λ, we can find χ < θ such that λ-many a’s in ¯A\{r} satisfy
+χa = χ. As the χth level of T ⇝A has size < λ, we may then find t ∈ (T ⇝A)χ+1
+such that that λ-many a’s in ¯A \ {r} satisfy χa = χ and a ↾ (χ + 1) = t. Clearly,
+χ, t and t′ := r ↾ (χ + 1) are as sought.
+□
+Let χ be given by the claim. For notational simplicity, we shall assume that
+∆[A× B] = {χ}. Let M be an elementary submodel of H(2λ)+ containing {χ, A, B,
+T ⇝A, T ⇝B, ̺, d} such that σ := M ∩ λ is in Sτ. In particular, |M| = µ. Denote
+υ := sup(A) and υM := sup(M ∩ υ).
+Note that since T ⇝A ∈ T (λ, θ) and as
+λ = µ+ > |θ|, it follows that T ⇝A has size ≤ µ. So, T ⇝A ⊆ M.
+Consider the following sets:
+• A0 := {α ∈ A | |̺α[B]| = λ},
+• A1 := {α ∈ A | |̺α[B]| ≤ µ}.
+Observe that A0, A1 ∈ M. We examine two subcases.
+Subcase 2.1: A0 has size λ. Appeal to Corollary 3.5 with X := A0 and i := 0
+to pick α ∈ A0 such that for cofinally many δ < θ, the two hold:
+(1) bα(δ) = 0, and
+(2) {β ∈ A | ∆(α, β) = δ} has size λ.
+Since θ ∈ Eλ
+µ, D := {δ < θ | Clauses (1) and (2) both hold} has size µ. For each
+δ ∈ D, use Clause (2) to fix βδ ∈ A above α such that ∆(α, βδ) = δ.
+Consider ς := sup{βδ | δ < µ}. As |B ∩ ς| ≤ µ, the fact that α ∈ A0 implies that
+E := ̺α[B \ ς] has size λ. By the choice of d, then, we may pick δ ∈ D \ (χ+ 1) and
+ǫ ∈ E above δ such that d(δ, ǫ) = τ. Pick γ ∈ B \ ς such that ǫ = ̺(α, γ). Clearly,
+α < βδ < γ.
+Recall that ∆(α, βδ) = δ > χ = ∆(βδ, γ). Since bα(δ) = 0, we conclude that
+bβδ(δ) = 1 and bα <lex bβδ. Altogether, (α, βδ, γ) ∈ A ⊛ A ⊛ B, and
+c(α, βδ, γ) = d(∆(α, βδ), ̺(α, γ)) = d(δ, ǫ) = τ,
+as sought.
+Subcase 2.2: A1 has size λ. By Condition (iii), find A′ ∈ [A1]λ and B′ ∈ [B]λ
+such that min(̺[A′ ⊛ B′]) > σ. Appeal to Corollary 3.5 with X := A′ and i := 0
+to pick β ∈ A′ \ υM such that for cofinally many δ < θ, the two hold:
+(1) bβ(δ) = 0, and
+(2) {α ∈ A′ | ∆(α, β) = δ} has size λ.
+Since θ ∈ Eλ
+µ, D := {δ < θ | Clauses (1) and (2) both hold} has size µ.
+Pick γ ∈ B′ \ (β + 1) arbitrarily. As (β, γ) ∈ A′ ⊛ B′, the ordinal ǫ := ̺(β, γ)
+is bigger than σ. Since eǫ is an injection to µ = |D|, we may pick δ ∈ D such
+that e(δ, ǫ) > max{e(χ, ǫ), e(σ, ǫ)}. In addition, since µ = cf(σ), the following set
+is bounded below σ:
+Z := {ζ < σ | e(ζ, ǫ) ≤ e(δ, ǫ)}.
+
+24
+IDO FELDMAN AND ASSAF RINOT
+Set t := (bβ ↾ δ)⌢⟨1⟩. From δ ∈ D we infer that (A′)t has size λ. As t ∈ T ⇝A ⊆ M,
+in particular, (A1)t is a set of size λ lying in M. By Condition (iii) and elementarity
+of M one can find α0 ̸= α1 in (A1)t ∩ M such that ̺(α0, α1) > sup(Z). As ̺ is
+subadditive, we may now find α ∈ {α0, α1} such that ̺(α, γ) > sup(Z). Note that,
+as {α, B, ̺} ∈ M, and as α ∈ A1, ̺α[B] is a set of size no more than µ lying in M,
+so that sup(̺α[B]) ∈ M. In particular, ̺(α, γ) < σ.
+Claim 5.3.3. All of the following hold:
+(1) (α, β, γ) ∈ A ⊛ A ⊛ B;
+(2) ∆(α, β) > ∆(β, γ);
+(3) bβ <lex bα;
+(4) min(Z(α,β,γ)) = σ.
+Proof.
+(1) γ was chosen to be in B \ (β + 1). In addition, β ∈ A′ \ υM, whereas
+α ∈ M ∩ A.
+(2) Since ∆(α, β) = ∆(t, bβ) = δ > χ = ∆(β, γ).
+(3) By the definition of t and since δ ∈ D.
+(4) As δ = ∆(α, β) and ̺(β, γ) = ǫ, we infer that Z(α,β,γ) = {ζ ∈ λ \ ̺(α, γ) |
+e(δ, ǫ) ≥ e(ζ, ǫ)}. In particular, σ ∈ Z(α,β,γ). Now, if ζ := min(Z(α,β,γ)) is
+below σ, then ζ ∈ Z, contradicting the fact ̺(α, γ) > sup(Z).
+□
+By the preceding claim and the definition of c,
+c(α, β, γ) = h(min(Z(α,β,γ)) = h(σ) = τ,
+as sought.
+□
+6. Countably many colors
+The main result of this section asserts that
+λ+ sup
+�−→ [λ, λ]3
+ω
+holds, provided that λ = µ+ for an infinite cardinal µ = µ<µ. The idea of the
+proof is to build on the colorings c : [λ+]3 → λ given by Theorems 5.2 and 5.3 with
+respect to the subadditive coloring ρ : [λ+]2 → λ given by Fact 2.14. By Clause (iii)
+of these theorems, we must address the problematic case in which the two sets A, B
+of Definition 4.20 do not satisfy that ρ ↾ [A ∪ B]2 witnesses U(λ, λ, λ, 3). Anyone
+that is familiar with [Tod07, §10] would probably suggest to use the oscillation
+of [Tod07, §8] in this problematic case, and this indeed works. Unfortunately, to
+verify that this works in the rectangular context, we had to reopen and tweak the
+proofs. The experts may want to skip directly to Corollary 6.17. The newcomers
+may benefit from the modular exposition.
+Setup 6.1. For the rest of this section, κ stands for a regular uncountable cardinal,
+Υ is a large enough regular cardinal (e.g., (2κ)+), and we fix some C-sequence
+⃗C = ⟨Cβ | β < κ⟩. We shall also assume that otp(Cβ) = cf(β) for all β < κ, though
+this will only play a role in the proof of Lemma 6.16 below.
+The items of the next definition correspond to Definitions 8.1.4, 6.3.1 and 8.1.1
+of [Tod07], where the last item is a non-essential strengthening of the latter.
+Definition 6.2. A subset Γ ⊆ κ with cf(otp(Γ)) > ω is said to be:
+• ⃗C-stationary iff �
+β∈Γ(acc(Cβ) ∪ {β}) is stationary in sup(Γ);
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+25
+• ⃗C-nontrivial iff for every club D ⊆ sup(Γ), there exists α ∈ Γ such that
+D ∩ α ⊈ Cβ for all β ∈ Γ;
+• ⃗C-oscillating iff for every club D ⊆ sup(Γ), there exist β ∈ Γ and an
+increasing sequence ⟨δj | j < ω⟩ of ordinals in D \ Cβ such that (δj, δj+1) ∩
+Cβ ̸= ∅ for all j < ω.
+Lemma 6.3. Suppose that Γ ⊆ κ is such that cf(otp(Γ)) > ω.
+If Γ is ⃗C-nontrivial and ⃗C-stationary, then Γ is ⃗C-oscillating.
+Proof. Suppose that Γ is ⃗C-nontrivial and ⃗C-stationary.
+By the latter, ∆ :=
+�
+β∈Γ(acc(Cβ) ∪ {β}) is a stationary subset of θ. For each δ ∈ ∆, pick βδ ∈ Γ
+such that sup(Cβδ ∩ δ) = δ.
+Next, to verify that Γ is ⃗C-oscillating, let D ⊆ sup(Γ) be a given club.
+Claim 6.3.1. There exists δ ∈ ∆ such that sup(D ∩ δ \ Cβδ) = δ.
+Proof. Suppose not. Then, for every δ ∈ ∆, ǫδ := sup(D ∩ δ \ Cβδ) is smaller than
+δ. Fix ǫ < sup(Γ) for which S := {δ ∈ ∆ | ǫδ < ǫ < δ} is stationary. Now, consider
+the club D′ := D \ ǫ. Then, for every α ∈ Γ, letting δ := min(S \ α), it is the case
+that D′ ∩α ⊆ D ∩[ǫ, δ) ⊆ Cβδ. This contradicts the fact that Γ is ⃗C-nontrivial.
+□
+Let δ be given by the claim. As sup(D ∩δ \Cβδ) = δ = sup(Cβδ ∩δ), it is easy to
+recursively construct an increasing sequence ⟨δj | j < ω⟩ of ordinals in D ∩ δ \ Cβδ
+such that (δj, δj+1) ∩ Cβδ ̸= ∅ for all j < ω.
+□
+Definition 6.4. For two disjoint sets of ordinals y and z, we say that P is a
+y-convex subset of z iff one of the following occurs:
+• P = {ζ ∈ z | ζ < α} and α = min(y);
+• P = {ζ ∈ z | β < ζ} and β = max(y);
+• P = {ζ ∈ z | α < ζ < β} and α < β are two consecutive elements of y.
+Note that if P and Q are nonempty y-convex subsets of z, then either P < Q or
+Q < P.
+Definition 6.5 (Todorˇcevi´c, [Tod07, §8]). For an ordinal ε < κ, define a function
+Oscε : [P(κ)]2 → P(P(κ)) via
+Oscε(x, y) :=
+�
+{P | P is a nonempty y-convex subset of x \ ε},
+if y ∩ x ⊆ ε;
+∅,
+otherwise.
+Then the oscillation mapping oscε : [P(κ)]2 → CARD(κ + 1) is defined via
+oscε(x, y) := | Oscε(x, y)|.
+Remark 6.6.
+(1) If we omit the subscript ε, then Osc(x, y) and osc(x, y) are
+understood to be Oscε(x, y) and oscε(x, y) for ε := ssup(x ∩ y).
+(2) For all ε < α < β < κ such that Cα∩Cβ ⊆ ε, Cα\ε and Cβ have no common
+accumulation points, and hence Oscε(Cα, Cβ) is finite. In this case, we shall
+identify Oscε(Cα, Cβ) with its increasing enumeration ⟨P0, . . . , Pn⟩.
+The next lemma makes explicit some of the features that are present in the proof
+of [Tod07, Lemma 8.1.2].
+Lemma 6.7 (Todorˇcevi´c). Suppose that Γ is a cofinal subset of some θ ≤ κ of
+uncountable cofinality, and that Γ is ⃗C-oscillating. For every cofinal E ⊆ θ, there
+
+26
+IDO FELDMAN AND ASSAF RINOT
+exists β ∈ Γ such that for every positive integer n, there are α ∈ Γ∩β and ε ∈ E ∩α
+such that all of the following hold:
+• oscε(Cα, Cβ) = n;
+• for every j < n, there is a pair ǫ < ǫ′ of ordinals in E \ Cα for which
+Oscε(Cα, Cβ)(j) = Cα ∩ (ǫ, ǫ′).
+Proof. Set µ := cf(θ), and fix a map ψ : µ → θ whose image is cofinal in θ. Let M
+be a continuous ∈-chain of length µ consisting of elementary submodels M ≺ HΥ
+with M ∩µ ∈ µ and {ψ, ⃗C, Γ, E} ∈ M. It follows that D := {sup(M ∩θ) | M ∈ M}
+constitutes a club in θ.
+Recalling that Γ is ⃗C-oscillating, pick β ∈ Γ and an
+increasing sequence ⟨δj | j < ω⟩ of ordinals in D \ Cβ such that (δj, δj+1) ∩ Cβ ̸= ∅
+for all j < ω.
+By possibly replacing δj by δj+1, we may assume that Cβ ∩ δ0
+is nonempty.
+For every j < ω, since δj ∈ D \ Cβ, pick Mj ∈ M such that
+sup(Mj ∩ θ) = δj, and note that γj := sup(Cβ ∩ δj) and Ωj := min(Mj ∩ θ \ γj) are
+both smaller than δj. So, for every j < ω:
+0 < sup(Cβ ∩ δj) = γj ≤ Ωj < δj < γj+1 < β.
+For each k < ω, let Ik denote the collection of all increasing sequences ⃗I = ⟨Ij |
+j ≤ k⟩ of closed intervals in θ. Now, let n be a positive integer and we shall find
+α ∈ Γ ∩ β and ε ∈ E ∩ α as in the conclusion of the lemma.
+Define a sequence of collections ⟨Fn−i | i ≤ n⟩ by recursion on i ≤ n, as follows:
+◮ For i = 0, let Fn be the set of all ⟨Ij | j ≤ n⟩ ∈ In such that the following
+two hold:
+(1) I0 = [0, Ω0];
+(2) α := max(In) belongs to Γ, Cα ⊆ I0 ∪ · · · ∪ In, and Cα ∩ Ij ̸= ∅ for every
+j ≤ n.
+◮ For every i < n such that Fn−i has already been defined, let Fn−i−1 be the
+collection of all ⃗I ∈ In−i−1 with the property that for every ǫ < θ there exists a
+closed interval I ⊆ (ǫ, θ) such that ⃗I⌢⟨I⟩ ∈ Fn−i.
+Claim 6.7.1. ⟨[0, Ω0]⟩ ∈ F0.
+Proof. For every j ≤ n, define:
+Ij :=
+
+
+
+
+
+[0, Ω0],
+if j = 0;
+[δj−1, Ωj],
+if 0 < j < n;
+[δn−1, β],
+otherwise.
+We shall prove by induction on i ≤ n that ⟨Ij | j ≤ n− i⟩ ∈ Fn−i. The base case
+is immediate, since ⟨Ij | j ≤ n⟩ satisfies requirements (1) and (2), with β playing
+the role of α.
+Next, suppose that we are given i < n for which ⟨Ij | j ≤ n− i⟩ ∈ Fn−i has been
+established. Note:
+• ⟨Fk | k ≤ n⟩ ∈ M0 ⊆ Mn−i−1;
+• ⟨Ij | j ≤ n − i − 1⟩ ∈ Mn−i−1 ∩ Fn−i;
+• In−i ∈ Mn−i \ Mn−i−1.
+So, by elementarity of Mn−i−1, ⟨Ij | j ≤ n − i − 1⟩ ∈ Fn−i−1.
+□
+It follows that we may recursively construct a sequence ⟨Ij | j ≤ n⟩ such that:
+(3) I0 = [0, Ω0], so that ⟨I0⟩ ∈ F0 ∩ M0;
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+27
+(4) ⟨Ij | j ≤ k + 1⟩ ∈ Fk+1 ∩ Mk for every k < n;
+(5) Ij+1 ⊆ (min(E \ Ωj + 1), θ) for every j < n.
+For each j < n, denote ǫj := min(E \Ωj +1), and note that since Ij+1 and E are
+in Mj, ǫ′
+j := min(E \ max(Ij+1) + 1) is < δj. Denote γ′
+j := min(Cβ \ γj + 1) so that
+γj < γ′
+j are two consecutive elements of Cβ. Since sup(Cβ ∩ δj) = γj, altogether,
+Ij+1 ⊆ (ǫj, ǫ′
+j) ⊆ (Ωj, δj) ⊆ (γj, γ′
+j) ⊆ (γj, γj+1).
+Now, put α := max(In). Then α ∈ Γ ∩ δn−1 ⊆ Γ ∩ β, Cα ⊆ I0 ∪ · · · ∪ In and
+Cα ∩ Ij ̸= ∅ for every j ≤ n. So (Cα \ I0) ⊆ �
+j<n(Ωj, δj). On the other hand,
+(Cβ \ I0) ∩ (�
+j<n(Ωj, δj)) = ∅. Therefore, for ε := ǫ0, we get that
+Cα ∩ Cβ ⊆ (Ω0 + 1) ⊆ ε.
+Claim 6.7.2. {ǫj, ǫ′
+j | j < n} ∩ Cα = ∅.
+Proof. Suppose not, and fix j < n such that {ǫj, ǫ′
+j} ∩ Cα ̸= ∅. As max(I0) = Ω0 ≤
+Ωj < ǫj < ǫ′
+j, we may fix some i < n such that {ǫj, ǫ′
+j} ∩ Ii+1 ̸= ∅. Recalling that
+Ij+1 ⊆ (ǫj, ǫ′
+j), it must be the case that i ̸= j. Note:
+◮ If i < j, then γi+1 ≤ Ωi+1 ≤ Ωj < ǫj < ǫ′
+j.
+◮ If i > j, then ǫj < ǫ′
+j < δj < γj+1 ≤ γi.
+So, both options contradict the fact that Ii+1 ⊆ (γi, γi+1).
+□
+By Clause (2), for every j ≤ n, Cα ∩ Ij+1 ̸= ∅. Altogether, for every ς ∈ (Ω0, ε]:
+Oscς(Cα, Cβ) = ⟨Cα ∩ Ij+1 | j < n⟩
+= ⟨Cα ∩ (γj, γ′
+j) | j < n⟩
+= ⟨Cα ∩ (ǫj, ǫ′
+j) | j < n⟩.
+In particular, oscε(Cα, Cβ) = n.
+□
+Remark 6.8. In the preceding proof, in the special case that κ = θ or κ = (cf(θ))+,
+one can secure that Ω0 be equal to γ0. So, in this case, we would get that max(Cα ∩
+Cβ) = Ω0, meaning that the conclusion of the lemma remains valid also after
+omitting the subscript ε.
+Definition 6.9. Define χ : [κ]3 → ω by letting for all α < β < κ:
+χ(α, β, γ) := max{k < ω | Tr(α, γ)(k) = Tr(β, γ)(k)}.
+Definition 6.10 ([Tod07, Definition 10.3.1]). A subset A ⊆ κ is said to be stable
+if χ“[A]3 is finite. Otherwise, we say that A is unstable.
+Similar to [Tod07, Definition 10.3.3], we use χ to derive the following stepping-up
+of the two-dimensional oscillation.
+Definition 6.11. The three-dimensional oscillation mapping, osc : [κ]3 → ω is
+defined on the basis of the two-dimensional oscillation defined in Definition 6.5 via:
+osc(α, β, γ) := oscα(CTr(α,β)(χ(α,β,γ)), CTr(α,γ)(χ(α,β,γ))).
+We now verify a rectangular version of [Tod07, Lemma 10.3.4]:
+Lemma 6.12 (Todorˇcevi´c). Suppose that B is a cofinal subset of some θ ≤ κ of
+uncountable cofinality, and that every cofinal subset of B is unstable.
+Then, for every cofinal A ⊆ θ and every positive integer n, there exists (α, β, γ) ∈
+A ⊛ B ⊛ B such that osc(α, β, γ) = n.
+
+28
+IDO FELDMAN AND ASSAF RINOT
+Proof. For each δ < θ, let βδ := min(B \ (δ + 1)) and Λδ := λ2(δ, βδ). By Fodor’s
+lemma, fix Λ < θ, k < ω and a stationary S ⊆ acc(θ) such that, for all δ ∈ S:
+(1) Λδ ≤ Λ;
+(2) ρ2(ðδ,βδ, βδ) = k;
+(3) for every ¯δ < δ, β¯δ < δ.
+Claim 6.12.1. For every δ ∈ S and every ordinal α with Λ < α < ðδ,βδ:
+• Tr(α, βδ) ↾ (k + 1) = Tr(δ, βδ) ↾ (k + 1), and
+• Tr(α, βδ)(k) = ðδ,βδ.
+Proof. By Remark 2.13.
+□
+Let Γ := {ðδ,βδ | δ ∈ S}. For each ξ ∈ Γ, pick δ(ξ) ∈ S such that ξ = ðδ(ξ),βδ(ξ).
+Note that δ(ξ) ≤ ξ ≤ βδ(ξ).
+Claim 6.12.2. Γ is ⃗C-oscillating.
+Proof. As δ ∈ acc(ðδ,βδ)∪{ðδ,βδ} for every δ ∈ S, we have S ⊆ �
+ξ∈Γ acc(Cξ)∪{ξ}.
+So Γ is ⃗C-stationary. By Lemma 6.3, it thus suffices to prove that Γ is ⃗C-nontrivial.
+Towards a contradiction, suppose this is not so, and fix a club D ⊆ θ such that,
+for every α ∈ Γ there exists β ∈ Γ with D ∩ α ⊆ Cβ. As sup(Γ) = θ, we may then
+recursively construct a sparse enough cofinal subset X ⊆ Γ with the property that
+for every pair ξ < ξ′ of ordinals from X, all of the following hold:
+• Λ < δ(ξ);
+• D ∩ (βδ(ξ), δ(ξ′)) ̸= ∅;
+• D ∩ βδ(ξ) ⊆ Cξ′.
+As B′ := {βδ(ξ) | ξ ∈ X} is a cofinal subset of B, it must be unstable. We shall
+reach a contradiction by showing that χ“[B′]3 = {k}. To this end, let α < β < γ
+be a triple of ordinals from B′. Fix a triple ξ < ξ′ < ξ′′ of ordinals from X such
+that α = βδ(ξ), β = βδ(ξ′), and γ = βδ(ξ′′). Then:
+• Λ < δ(ξ) < α < δ(ξ′) < β < δ(ξ′′) ≤ ξ′′ ≤ γ;
+• D ∩ (α, δ(ξ′)) ̸= ∅;
+• D ∩ β ⊆ Cξ′′.
+Pick ι ∈ D ∩ (α, δ(ξ′)) ̸= ∅, so that ι ∈ D ∩ (α, β) ⊆ Cξ′′.
+Appealing to
+Claim 6.12.1 with δ′′ := δ(ξ′′), we infer that:
+• Tr(α, γ) ↾ (k + 1) = Tr(δ′′, γ) ↾ (k + 1) = Tr(β, γ) ↾ (k + 1), and
+• Tr(α, γ)(k) = ξ′′ = Tr(β, γ)(k).
+Therefore
+Tr(α, γ)(k + 1) = min(Cξ′′ \ α) ≤ ι < β ≤ min(Cξ′′ \ β) = Tr(β, ξ)(k + 1).
+Recalling Definition 6.9, this indeed means that χ(α, β, γ) = k.
+□
+Now, given a cofinal A ⊆ θ and a positive integer n, appeal to Lemma 6.7 with
+E := acc+(A \ Λ) to find a pair (ξ, ζ) ∈ Γ ⊛ Γ and an ordinal ε ∈ E ∩ ξ such that:
+• oscε(Cξ, Cζ) = n + 1, and
+• for every j < n + 1, there is a pair ǫ < ǫ′ of ordinals in E \ Cξ for which
+Oscε(Cξ, Cζ)(j) = Cξ ∩ (ǫ, ǫ′).
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+29
+Let ǫ < ǫ′ be a pair of ordinals witnessing the case j = 0 of the preceding.
+Clearly,
+Oscε(Cξ, Cζ)(0) = Cξ ∩ [ǫ, ǫ′].
+Since osc(Cξ, Cζ) > 1 and ξ < ζ, we may fix two consecutive elements ¯α < ¯β of Cζ
+such that
+Oscε(Cξ, Cζ)(1) = Cξ ∩ (¯α, ¯β).
+So, ǫ < ǫ′ ≤ ¯α < ξ.
+Since Cξ is a closed subset of ξ, and ǫ′ ∈ acc+(A \ Λ) ∩ (ξ \ Cξ), we may pick a
+large enough α ∈ A ∩ (Λ, ǫ′) such that
+Oscε(Cξ, Cζ)(0) ⊆ (ǫ, α).
+In particular, Cξ ∩ Cζ ⊆ ε ⊆ α, and
+oscα(Cξ, Cζ) = oscε(Cξ, Cζ) − 1 = n.
+Denote ¯δ := δ(ξ) and δ := δ(ζ). Then (¯δ, δ) ∈ [S]2, ξ = ð¯δ,β¯δ and ζ = ðδ,βδ. Set
+β := β¯δ and γ := βδ, so that (β, γ) ∈ [B]2. Note that
+Λ < α < ǫ′ ≤ ¯α < ξ ≤ β < δ ≤ ζ ≤ γ.
+By Claim 6.12.1, then:
+• Tr(α, γ) ↾ (k + 1) = Tr(δ, γ) ↾ (k + 1) = Tr(β, γ) ↾ (k + 1);
+• Tr(α, γ)(k) = ζ = Tr(β, γ)(k);
+• Tr(α, β)(k) = ξ.
+Therefore,
+Tr(α, γ)(k + 1) = min(Cζ \ α) ≤ ¯α < β ≤ min(Cζ \ β) = Tr(β, γ)(k + 1),
+and χ(α, β, γ) = k.
+Summing all up, (α, β, γ) ∈ A ⊛ B ⊛ B, and
+osc(α, β, γ) = oscα(CTr(α,β)(χ(α,β,γ)), CTr(α,γ)(χ(α,β,γ)))
+= oscα(CTr(α,β)(k), CTr(α,γ)(k))
+= oscα(Cξ, Cζ) = n,
+as sought.
+□
+The ending of the proof of Claim 6.12.2 makes it clear that the following hold.
+Observation 6.13. Suppose:
+• λ2(δ, γ) < α < β < δ < γ < κ;
+• Cðδ,γ ∩ [α, β) is nonempty.
+Then χ(α, β, γ) = ρ2(ðδ,γ, γ).
+□
+The next lemma extracts features present in the proof of [Tod07, Lemma 10.3.2].
+Lemma 6.14. Suppose that X is a stable cofinal subset of some θ ≤ κ of uncount-
+able cofinality. Then there exist a cofinal X′ ⊆ X, a club D ⊆ θ, and a positive
+integer k satisfying all of the following:
+(1) for every (δ, γ) ∈ D ⊛ X′, D ∩ δ ⊆ Cðδ,γ;
+(2) for every (δ, α, δ′, β, δ′′, γ) ∈ D ⊛ θ ⊛ D ⊛ θ ⊛ D ⊛ X′:
+• χ(α, β, γ) = k, and
+• D ∩ δ′′ ⊆ CTr(α,γ)(k).
+
+30
+IDO FELDMAN AND ASSAF RINOT
+Proof. Set µ := cf(θ), and fix a map ψ : µ → θ whose image is cofinal in θ. Let M
+be a continuous ∈-chain of length µ consisting of elementary submodels M ≺ HΥ
+with M ∩ µ ∈ µ and {ψ, ⃗C, X} ∈ M. For each M ∈ M, denote θM := sup(M ∩ θ),
+so that E := {θM | M ∈ M} is a club in θ.
+Claim 6.14.1. Let N ≺ HΥ be such that {ψ, ⃗C, X, M} ∈ N and N ∩ µ ∈ µ.
+Denote θN := sup(N ∩ θ) and let γ ∈ X \ (θN + 1).7
+If sup(E∩θN\CðθN ,γ) = θN, then there exists γ′ ∈ X\(θN+1) with ρ2(ðθN ,γ′, γ′) >
+ρ2(ðθN,γ, γ).
+Proof. Denote ¯γ := ðθN,γ, n := ρ2(¯γ, γ), and
+Mγ := {M ∈ M ∩ N | θM /∈ C¯γ}.
+Now, assuming that sup(E ∩ θN \ C¯γ) = θN, we infer that
+sup{θM | M ∈ Mγ} = θN.
+Pick M ∈ Mγ with θM > λ2(θN, γ).
+Since θM /∈ C¯γ, it is the case that
+ρ2(θM, ¯γ) > 1. So, by Remark 2.13,
+tr(θM, γ) = tr(¯γ, γ)⌢ tr(θM, ¯γ),
+and
+ρ2(θM, γ) > ρ2(¯γ, γ) + 1.
+Thus, Im(tr(θM, γ)) contains not only Im(tr(¯γ, γ)) but also min(C¯γ\θM) which is
+strictly above θM, therefore, ρ2(ðθM,γ, γ) > n. Pick Λ ∈ M ∩ θ above λ2(ðθM,γ, γ).
+Then, the following set belongs to M and γ witnesses that it has θM as an element:
+S := {δ < θ | ∃γ′ ∈ X \ (θ + 1) [ρ2(ðδ,γ′, γ′) > n & λ2(δ, γ′) ≤ Λ]},
+so that S is stationary in θ. Pick δ ∈ S above θN, along with a witnessing γ′. As
+λ2(δ, γ′) ≤ Λ < θM < θN < δ, we get from Remark 2.13 that
+tr(θN, γ′) = tr(ðδ,γ′, γ′)⌢ tr(θN, ðδ,γ′),
+and hence ρ2(θN, γ′) ≥ ρ2(ðδ,γ′, γ′) + 1 > n + 1. Therefore ρ2(ðθN,γ′, γ′) > n, as
+sought.
+□
+Claim 6.14.2. ∆ := {δ < θ | ∃γ ∈ X \ (δ + 1) [E ∩ δ ⊆∗ Cðδ,γ]} covers a club in θ.
+Proof. Let S be a stationary subset of θ, and we shall prove that S ∩ ∆ ̸= ∅. Let
+N ≺ HΥ be such that {ψ, ⃗C, X, M} ∈ N, N ∩ µ ∈ µ, with θN := sup(N ∩ θ)
+in S. Using the fact that X is stable, fix m < ω such that χ“[X]3 ⊆ m. Now,
+if θN /∈ ∆, then by iterating Claim 6.14.1 finitely many times, we may find a
+γ ∈ X \ (θN + 1) such that ρ2(ðθN,γ, γ) > m. Pick α, β ∈ X with λ2(θN, γ) < α <
+β < θN such that CðθN ,γ ∩ (α, β) ̸= ∅. By Observation 6.13, χ(α, β, γ) > m. This
+is a contradiction.
+□
+For each δ ∈ ∆, fix γδ ∈ X \ (δ + 1), ǫδ < δ and kδ < ω such that:
+• E ∩ δ \ ǫδ ⊆ Cðδ,γδ ,γδ;
+• λ2(δ, γδ) ≤ ǫδ;
+• ρ2(ðδ,γδ, γδ) = kδ.
+7As cf(θ) = µ > |N|, X \ θN is co-bounded in X.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+31
+Find ǫ < θ and k < ω for which the following set is stationary:
+S := {δ ∈ ∆ | ǫδ = ǫ & kδ = k}.
+Consider the club D := {δ ∈ acc(E \ ǫ) | ∀¯δ ∈ ∆ ∩ δ (γ¯δ < δ)}, and the set
+X′ := {γδ | δ ∈ S ∩ D}. We shall verify that X′, D and k satisfy the requirements
+of the two clauses.
+(1) Let δ ∈ D and γ ∈ X′ \ (δ + 1). Pick δ∗ ∈ D such that γ = γδ∗. If δ∗ = δ,
+then D ∩ δ∗ ⊆ E ∩ δ \ ǫ ⊆ Cðδ∗,γ. Otherwise, λ2(δ∗, γ) ≤ ǫ < δ < δ∗ < γ = γδ∗. So,
+by Remark 2.13,
+tr(δ, γ) = tr(ðδ∗,γ, γ)⌢ tr(δ, ðδ∗,γ),
+with E∩δ∗\ǫ ⊆ Cðδ∗,γ. Since δ ∈ acc(E)∩(ǫ, δ∗), it is the case that δ ∈ acc(Cðδ∗,γ).
+Altogether,
+tr(ðδ,γ, γ) = tr(ðδ∗,γ, γ).
+In particular, D ∩ δ ⊆ E ∩ (ǫ, δ∗) ⊆ Cðδ∗,γ.
+(2) Let (δ, α, δ′, β, δ′′, γ) ∈ D ⊛ θ ⊛ D ⊛ θ ⊛ D ⊛ X′. Fix δ∗ ∈ S ∩ D such that
+γ = γδ∗. As δ∗ ∈ ∆ and γ¯δ < δ′′ < γ for every ¯δ ∈ ∆ ∩ δ′′, it must be the case that
+δ∗ ≥ δ′′. So λ2(δ∗, γ) ≤ ǫ < α < β < δ∗ < γ and δ′ ∈ D ∩ (α, β) ⊆ Cðδ∗,γ ∩ (α, β).
+Then, Observation 6.13 implies that χ(α, β, γ) = ρ2(ðδ∗,γ, γ) = k.
+□
+Remark 6.15. While we will not be needing this fact, we point out that the proofs of
+Lemmas 6.12 and 6.14 together show that for every θ ≤ κ of uncountable cofinality,
+and every cofinal subset X ⊆ θ, the following are equivalent:
+• Every cofinal subset of X is unstable;
+• For every stationary S ⊆ θ and every ⟨βδ | δ ∈ S⟩ in �
+δ∈S(X \ (δ + 1)),
+the set Γ := {ðδ,βδ | δ ∈ S} is ⃗C-nontrivial.
+The following is an easy strengthening of [Tod07, Lemma 10.3.2]:
+Lemma 6.16 (Todorˇcevi´c). Suppose that κ = λ+ for some regular uncountable
+cardinal λ, and let ρ : [κ]2 → λ be the corresponding map given by Fact 2.14.
+Suppose also that X is a stable subset of κ of order-type λ. Then there exists a
+cofinal subset X′ ⊆ X such that ρ ↾ [X′]2 witnesses U(λ, λ, λ, ω).
+Proof. Denote θ := sup(X). Let X′ ⊆ X and D ⊆ θ be given by Lemma 6.14. To
+see that ρ ↾ [X′]2 witnesses U(λ, λ, λ, ω), let A ∈ [X′]σ be some λ-sized pairwise
+disjoint, with σ < ω, and let τ < λ.
+As A consists of λ-many pairwise disjoint subsets of X′ and otp(X′) = λ, for
+every δ ∈ D, we may pick bδ ∈ A with min(bδ) > δ. Then put Λδ := max{λ2(δ, β) |
+β ∈ bδ}. Recalling that X′ and D were given by Lemma 6.14, for all δ ∈ D and
+β ∈ bδ,
+D ∩ δ ⊆ Cðδ,β.
+Next, fix some Λ < θ and a stationary set S ⊆ D such that for every δ ∈ S:
+(1) Λδ ≤ Λ < δ,
+(2) otp(D ∩ δ) > τ, and
+(3) For every γ ∈ D ∩ δ, sup(bγ) < δ.
+Evidently, B := {bδ | δ ∈ S} is a λ-sized subset of A. Now, given a ̸= b in B, we
+may find a pair γ ̸= δ of ordinals from S such that a = bγ and b = bδ. Without loss
+of generality, γ < δ. Let (α, β) ∈ a × b. Then
+Λδ < γ < α < δ < β,
+
+32
+IDO FELDMAN AND ASSAF RINOT
+so by Remark 2.13, ðδ,β ∈ Im(tr(α, β)). As the map ρ was given by Fact 2.14,
+ρ(α, β) ≥ otp(Cðδ,β ∩ α) ≥ otp(Cðδ,β ∩ γ) ≥ otp(D ∩ γ) > τ,
+as sought.
+□
+It is clear that every subset of a stable set is stable. The next corollary addresses
+the question of closure under unions.
+Corollary 6.17. Suppose that A, B are cofinal stable subsets of some θ ≤ κ of
+uncountable cofinality. Then, there exist cofinal subsets A′ ⊆ A and B′ ⊆ B such
+that:
+• A′ ∪ B′ is stable, and
+• for every (α, β, γ) ∈ [A′∪B′]3, if (β, γ) ∈ [A′]2∪[B′]2, then osc(α, β, γ) = 0.
+Proof. Appeal to Lemma 6.14 with A to get a cofinal A′ ⊆ A, a club D1 ⊆ θ and
+an integer k1. Likewise, appeal to Lemma 6.14 with B to get a cofinal B′ ⊆ B,
+a club D2 ⊆ θ and an integer k2. Consider the club D := D1 ∩ D2. Pick sparse
+enough cofinal subsets A′′ ⊆ A′ and B′′ ⊆ B′ such that, letting X := A′′ ∪ B′′, for
+every (α, β) ∈ [X]2, there are ι < θ and δ, δ′, δ′′ ∈ D with δ < α < δ′ < ι < δ′′ < β.
+Claim 6.17.1. X is stable. Furthermore, max(χ“[X]3) = k2.
+Proof. Let (α, β, γ) ∈ [X]3. Pick δ, δ′, δ′′ such that
+(δ, α, δ′, β, δ′′, γ) ∈ D ⊛ X ⊛ D ⊛ X ⊛ D ⊛ X.
+Then χ(α, β, γ) = k1 if γ ∈ A′, and χ(α, β, γ) = k2 otherwise.
+□
+Claim 6.17.2. Let Y ∈ {A′′, B′′} and (α, β, γ) ∈ X⊛Y ⊛Y . Then osc(α, β, γ) = 0.
+Proof. If Y = A′′, then denote k := k1. Otherwise, denote k := k2. Now, pick
+ι, δ, δ′, δ′′, δ′′′ such that
+(δ, α, δ′, ι, δ′′, β, δ′′′, γ) ∈ D ⊛ X ⊛ D ⊛ θ ⊛ D ⊛ Y ⊛ D ⊛ Y.
+In particular,
+{(δ, α, δ′, ι, δ′′, β), (δ, α, δ′′, β, δ′′′, γ)} ⊆ D ⊛ θ ⊛ D ⊛ θ ⊛ D ⊛ X.
+Then D ∩ δ′′ ⊆ CTr(α,β)(k) and D ∩ δ′′′ ⊆ CTr(α,γ)(k). In addition, χ(α, β, γ) = k,
+so that
+δ′ ∈ D ∩ (α, δ′′) ⊆ CTr(α,β)(χ(α,β,γ)) ∩ CTr(α,γ)(χ(α,β,γ)).
+Recalling Definition 6.5, this means that
+Oscα(CTr(α,β)(χ(α,β,γ)), CTr(α,γ)(χ(α,β,γ))) = ∅,
+and hence osc(α, β, γ) = 0.
+□
+So A′′ and B′′ are as sought.
+□
+Corollary 6.18. Suppose that:
+(1) µ < λ < λ+ = κ are infinite regular cardinals;
+(2) Eλ
+µ admits a nonreflecting stationary set;
+(3) there exists a weak µ-Kurepa tree with at least κ-many branches.
+Then κ
+sup
+�−→ [λ, λ]3
+ω holds.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+33
+Proof. Let c : [κ]3 → λ be the map given by Theorem 5.2 with respect to the
+subadditive coloring ρ : [κ]2 → λ of Fact 2.14. Define cω : [κ]3 → ω via
+cω(α, β, γ) :=
+�
+c(α, β, γ),
+if c(α, β, γ) < ω;
+0,
+otherwise.
+Let T ⊆ <µ2 be a weak µ-Kurepa tree with at least κ-many branches, and let
+⟨bξ | ξ < κ⟩ be an injective sequence consisting of elements of B(T ). For notational
+simplicity, we shall write ∆(α, β) for ∆(bα, bβ). Define a coloring d : [κ]3 → ω by
+letting for all α < β < γ < κ:
+d(α, β, γ) :=
+�
+osc(α, β, γ) − 1,
+if osc(α, β, γ) > 0 and ∆(α, β) < ∆(β, γ);
+cω(α, β, γ),
+otherwise.
+To see that d witnesses κ
+sup
+�−→ [λ, λ]3
+ω, let A, B be disjoint subsets of κ of order-
+type λ with sup(A) = sup(B), and let n < ω. Using Lemma 3.4 and by possibly
+passing to cofinal subsets, we may assume the existence of s ∈ T and i ̸= i′ such
+that s⌢⟨i⟩ ⊑ bα for all α ∈ A, and s⌢⟨i′⟩ ⊑ bβ for all β ∈ B.
+Claim 6.18.1. Let (α, β, γ) ∈ [A ∪ B]3 \ ([A]3 ∪ [B]3).
+Then (β, γ) ∈ ([A]2 ∪ [B]2) iff ∆(α, β) < ∆(β, γ).
+Proof. For every (ǫ, δ) ∈ (A ⊛ B) ∪ (B ⊛ A), ∆(ǫ, δ) = ∆(s⌢⟨i⟩, s⌢⟨i′⟩) = dom(s).
+For every (ǫ, δ) ∈ (A ⊛ A) ∪ (B ⊛ B), ∆(ǫ, δ) ≥ dom(s + 1) > dom(s).
+By the hypothesis on (α, β, γ), there are three cases to consider:
+◮ If (α, β, γ) ∈ A ⊛ B ⊛ B, then ∆(α, β) = dom(s) < ∆(β, γ).
+◮ If (α, β, γ) ∈ B ⊛ A ⊛ A, then ∆(α, β) = dom(s) < ∆(β, γ).
+◮ If (β, γ) ∈ (A ⊛ B) ∪ (B ⊛ A), then ∆(β, γ) = dom(s) ≤ ∆(α, β).
+□
+There are three cases to consider:
+◮ Suppose that there exist A′ ∈ [A]λ and B′ ∈ [B]λ such that A′ and B′ are
+stable. Then, by Corollary 6.17, we may moreover assume that A′ ∪ B′ is
+stable, and that for every (α, β, γ) ∈ [A′ ∪ B′]3 with (β, γ) ∈ [A′]2 ∪ [B′]2,
+osc(α, β, γ) = 0. By Lemma 6.16, then, ρ ↾ [A′∪B′]2 witnesses U(λ, λ, λ, 3).
+As c : [κ]3 → λ was given by Theorem 5.2, we may find (α, β, γ) ∈ [A′ ∪
+B′]3 \ ([A′]3 ∪ [B′]3) such that c(α, β, γ) = n. Now, if (β, γ) ∈ [A′]2 ∪ [B′]2,
+then osc(α, β, γ) = 0, and if (β, γ) /∈ [A′]2 ∪ [B′]2, then ∆(α, β) ≥ ∆(β, γ).
+It thus follows that d(α, β, γ) = cω(α, β, γ) = c(α, β, γ) = n, as sought.
+◮ Suppose that every cofinal subset of B is unstable. By appealing to Lemma 6.12
+with A and B, we may find (α, β, γ) ∈ A ⊛ B ⊛ B such that osc(α, β, γ) =
+n + 1.
+As (β, γ) ∈ [B]2, it is the case that ∆(α, β) < ∆(β, γ).
+So
+d(α, β, γ) = osc(α, β, γ) − 1 = n.
+◮ Otherwise. So, every cofinal subset of A is unstable, and then the argument
+is similar to that of the previous case.
+□
+We are finally in conditions to prove the rectangular extension of [Tod07, Theo-
+rem 10.3.6].
+Corollary 6.19. Suppose that λ = µ+ for an infinite cardinal µ = µ<µ.
+Then λ+ sup
+�−→ [λ, λ]3
+ω holds.
+
+34
+IDO FELDMAN AND ASSAF RINOT
+Proof. Denote κ := λ+. If 2µ > µ+, then since µ<µ = µ, there is a weak µ-Kurepa
+tree with κ-many branches, and Eλ
+µ constitutes a nonreflecting stationary set. So,
+by Corollary 6.18, we may assume here that 2µ = µ+. In particular, λ ↛ [µ; λ]2
+λ
+holds by a theorem of Sierpi´nski. Let c : [κ]3 → λ be the map given by Theorem 5.3
+with respect to the subadditive coloring ρ : [κ]2 → λ of Fact 2.14. Derive cω from
+c as in the proof of Theorem 6.18. Also, denote T := <λ2, and let ⟨bξ | ξ < κ⟩
+be the injective sequence of elements of B(T ) used in the proof of Theorem 5.3 to
+define the coloring c. For notational simplicity, we shall write ∆(α, β) for ∆(bα, bβ).
+Likewise, for B ⊆ κ, we write T ⇝B for T ⇝{bβ|β∈B}.
+Finally, define a coloring d : [κ]3 → ω by letting for all α < β < γ < κ:8
+d(α, β, γ) :=
+�
+osc(α, β, γ) − 1,
+if osc(α, β, γ) > 0 and ∆(α, β) < ∆(β, γ);
+cω(α, β, γ),
+otherwise.
+To see that d witnesses κ
+sup
+�−→ [λ, λ]3
+ω, let A, B be disjoint subsets of κ of order-
+type λ with sup(A) = sup(B), and let n < ω. Using Corollary 6.17 and by possibly
+passing to cofinal subsets, we may assume that one of the following holds:
+(I) A ∪ B is stable, and for every (α, β, γ) ∈ [A ∪ B]3 with (β, γ) ∈ [A]2 ∪ [B]2,
+osc(α, β, γ) = 0;
+(II) every cofinal subset of B is unstable;
+(III) every cofinal subset of A is unstable.
+Let us dispose of Case (I) right away. In this case, by Lemma 6.16, ρ ↾ [A ∪ B]2
+witnesses U(λ, λ, λ, 3). So by the choice of c, we may find (α, β, γ) ∈ [A ∪ B]3 \
+([A]3∪[B]3) such that c(α, β, γ) = n. Going over the division into cases in the proof
+of Theorem 5.3, we see that ∆(α, β) ≥ ∆(β, γ) in all subcases but to Subcase 1.1.
+So, in all of these cases, d(α, β, γ) = cω(α, β, γ) = c(α, β, γ) = n. Finally, looking
+at Subcase 1.1, we see that the provided triple (α, β, γ) is an element of A ⊛ B ⊛ B
+(or an element of B ⊛ A ⊛ A, once lifting the initial “without loss of generality”
+assumption). So (β, γ) ∈ [A]2 ∪ [B]2, and hence osc(α, β, γ) = 0. Therefore, again
+d(α, β, γ) = cω(α, β, γ) = n.
+Moving on to handling Cases (II) and (III), we shall need the following claim.
+Claim 6.19.1. There are cofinal subsets A′ ⊆ A and B′ ⊆ B such that, for every
+(α, β, γ) ∈ (A ⊛ B ⊛ B) ∪ (B ⊛ A ⊛ A), ∆(α, β) < ∆(β, γ).
+Proof. By possibly passing to a cofinal subset of B, we may assume that B = B′
+in the sense of Lemma 3.7. So let θB ≤ λ be such that T ⇝B′ is a normal tree in
+T (λ, θB) for every B′ ∈ [B]λ. Likewise, we may assume that A = A′ in the sense
+of Lemma 3.7, and let θA ≤ λ be such that T ⇝A′ is a normal tree in T (λ, θA) for
+every A′ ∈ [A]λ.
+If min{θA, θB} < λ, then the proof of Claim 5.3.2 provides χ < min{θA, θB} and
+a pair (t, t′) ∈ (T ⇝A)χ+1 ×(T ⇝B)χ+1 such that ∆(t, t′) = χ. Thus letting A′ := At
+and B′ := Bt′, we see that ∆(α, β) = χ whenever (α, β) ∈ (A′ ⊛ B′) ∪ (B′ ⊛ A′),
+and ∆(α, β) > χ whenever (α, β) ∈ (A′ ⊛ A′) ∪ (B′ ⊛ B′). Thus, we may assume
+that θA = θB = λ, so that T ⇝A, T ⇝B ∈ T (λ, λ). Now, if (T ⇝A) ⊈ (T ⇝B) and
+(T ⇝A) ⊈ (T ⇝B), then by normality of the two trees there must exist χ < λ,
+8It may appear that this is the same map from the proof of Corollary 6.18. Note, however,
+that there ∆ was a map from [κ]2 to µ, whereas here ∆ is a map from [κ]2 to λ.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+35
+t ∈ (T ⇝A)χ \ (T ⇝B)χ and t′ ∈ (T ⇝B)χ \ (T ⇝A)χ. Clearly, A′ := At and B′ := Bt′
+are as sought.
+Thus, the only nontrivial case is in which θA = θB = λ and T ⇝A ⊆ T ⇝B or
+T ⇝A ⊆ T ⇝B. Without loss of generality, assume that T ⇝A ⊆ T ⇝B. Now, there
+are two options:
+◮ If B(T ⇝A) is nonempty, then there is b : λ → 2 that constitutes a branch
+through both T ⇝A and T ⇝B. In this case, it is easy to recursively simultaneously
+construct cofinal subsets A′ ⊆ A and B′ ⊆ B such that for all triple α < β < γ of
+ordinals from A′ ∪ B′, it is the case that ∆(α, β) < ∆(β, γ).
+◮ Otherwise. So T ⇝A is a λ-Aronszajn tree. In particular, we may find χ < λ
+and t ̸= t′ in (T ⇝A)χ. Altogether, t ∈ (T ⇝A)χ, t′ ∈ (T ⇝B)χ and ∆(t, t′) < χ.
+Then A′ := At and B′ := Bt′ are as sought.
+□
+At this point, the proof is similar to that of Theorem 6.18.
+Succinctly, in
+Case (II), we appeal to Lemma 6.12 with A and B, to find (α, β, γ) ∈ A ⊛ B ⊛ B
+such that osc(α, β, γ) = n+1. By the preceding claim, it is the case that ∆(α, β) <
+∆(β, γ). So d(α, β, γ) = osc(α, β, γ) − 1 = n. The handling of Case (III) is simi-
+lar.
+□
+7. Connecting the dots
+The next result implies Theorem A′.
+Theorem 7.1. For every infinite cardinal µ satisfying µ<µ < µ+ < 2µ:
+(1) S3(µ++, µ+, ω) holds;
+(2) G ↛ [µ+]FS3
+ω
+holds for every Abelian group (G, +) of size µ++.
+Proof. Suppose that µ is an infinite cardinal satisfying µ<µ < µ+ < 2µ. Denote
+λ := µ+ and κ := λ+. Then, T := <µ2 is a weak µ-Kurepa tree with at least
+κ-many branches. So, by Corollary 6.18, κ
+sup
+�−→ [λ, λ]3
+ω holds. As T witnesses that
+κ ∈ T (λ, µ), by Lemma 4.23, Extract3(κ, λ, µ, ω) holds, as well.
+Together with
+Lemma 4.22, this yields Clause (1). Then, Clause (2) follows from Proposition 4.19
+and the fact (see [FBR17, Lemma 2.2]) that every Abelian group is a well-behaved
+magma.
+□
+By [FBR17, Theorem 3.8], for every infinite cardinal µ = 2<µ, S2(2µ, 2µ, ω)
+holds. By the upcoming corollary, for every infinite cardinal µ = 2<µ, S2(2µ, µ+, 2)
+holds. While it is easy to get S2(2µ, µ+, µ) from 2µ = µ+, Remark 4.4 shows that
+S2(2µ, µ+, µ) is also compatible with 2µ > µ+. Note, however, that by a theorem
+of Shelah [She88, Theorem 2.1], one cannot prove S2(2µ, µ+, 3) in ZFC. Thus, in
+view of the number of colors, the following corollary is optimal.
+Corollary 7.2. For every infinite cardinal µ, S2(µθ, µ+, 2) holds, for θ := logµ(µ+).
+In particular, S2(2µ, µ+, 2) holds for every strong limit cardinal µ.
+Proof. Appeal to the upcoming theorem with λ := µ+ and κ := µθ.
+□
+Lemma 7.3. Suppose that θ < λ ≤ κ are infinite cardinals, with λ being regular.
+If κ ∈ T (λ, θ), then:
+(1) κ
+sup
+�−→ [λ, λ]2
+2 holds;
+(2) S2(κ, λ, 2) holds.
+
+36
+IDO FELDMAN AND ASSAF RINOT
+Proof. Suppose that κ ∈ T (λ, θ), and fix T ∈ T (λ, θ) admitting an injective se-
+quence ⟨bξ | ξ < κ⟩ consisting of elements of B(T ).
+(1) Consider the Sierpi´nski map c : [κ]2 → 2 defined by letting, for all α < β < κ:
+c(α, β) := 1 iff bα <lex bβ.
+Claim 7.3.1. c witnesses κ
+sup
+�−→ [λ, λ]2
+2.
+Proof. Suppose that we are given two disjoint subsets A, B of κ with otp(A) =
+otp(B) = λ and sup(A) = sup(B). By Lemma 3.4(2) (using µ := λ), we may find
+s ∈ T and i ̸= i′ such that A′ := {α ∈ A | s⌢⟨i⟩ ⊑ bα} and B′ := {β ∈ B |
+s⌢⟨i′⟩ ⊑ bβ} are both of size λ. As sup(A′) = sup(B′), we may now fix (α, β, γ) ∈
+A′ ⊛ B′ ⊛ A′. To see that {c(α, β), c(β, γ)} = 2, consider the following cases:
+◮ If i < i′, then bα, bγ <lex bβ and hence c(α, β) = 1 > 0 = c(β, γ);
+◮ If i′ < i, then bβ < bα, bγ and hence c(α, β) = 0 < 1 = c(β, γ).
+□
+(2) By Lemma 4.9 (again, using µ := λ), in particular, Extract2(κ, λ, ω, ω) holds.
+So, by Lemma 4.22, Clause (1) implies Clause (2).
+□
+Corollary 7.4. If there exists a weak µ-Kurepa tree with κ-many branches, then
+S2(κ, µ+, 2) holds.
+□
+Theorem B now follows (using µ := 2):
+Corollary 7.5. For every infinite set G, for every map ϕ : G → [G]<ω, and for
+every pair of cardinals µ, θ such that µ<θ < |G| ≤ µθ, there exists a corresponding
+coloring c : G → 2 satisfying the following.
+For every binary operation ∗ on G, if ϕ witnesses that (G, ∗) is well-behaved,
+then for every X ⊆ G of size (µ<θ)+ and every i ∈ {0, 1}, there are x ̸= y in X
+such that c(x ∗ y) = i.
+Proof. Given G, ϕ, µ and θ as above, denote κ := |G| and λ := (µ<θ)+, so that
+λ ≤ κ ≤ µθ. Evidently, T := <θµ witnesses that κ ∈ T (λ, θ), so S2(κ, λ, 2) holds
+by Lemma 7.3. Now, appeal to Proposition 4.19.
+□
+8. Acknowledgments
+The results of this paper stemmed from the first author’s M.Sc. thesis written
+under the supervision of the second author at Bar-Ilan University, and supported
+by the Israel Science Foundation (grant agreement 2066/18).
+The first author was partially supported by the Israel Science Foundation (grant
+agreement 203/22). The second author was partially supported by the Israel Science
+Foundation (grant agreement 203/22) and by the European Research Council (grant
+agreement ERC-2018-StG 802756).
+References
+[BR21]
+Ari Meir Brodsky and Assaf Rinot. A microscopic approach to Souslin-tree constructions.
+Part II. Ann. Pure Appl. Logic, 172(5):Paper No. 102904, 65, 2021.
+[EHR65] P. Erd˝os, A. Hajnal, and R. Rado. Partition relations for cardinal numbers. Acta Math.
+Acad. Sci. Hungar., 16:93–196, 1965.
+[FBR17] David Fernandez-Breton and Assaf Rinot. Strong failures of higher analogs of Hindman’s
+theorem. Trans. Amer. Math. Soc., 369(12):8939–8966, 2017.
+[Fle78]
+William G. Fleissner. Some spaces related to topological inequalities proven by the
+Erd˝os-Rado theorem. Proc. Amer. Math. Soc., 71(2):313–320, 1978.
+
+SUMS OF TRIPLES IN ABELIAN GROUPS
+37
+[HLS17] Neil Hindman, Imre Leader, and Dona Strauss. Pairwise sums in colourings of the reals.
+Abh. Math. Semin. Univ. Hambg., 87(2):275–287, 2017.
+[HS12]
+Neil Hindman and Dona Strauss. Algebra in the Stone- ˇCech compactification. de Gruyter
+Textbook. Walter de Gruyter & Co., Berlin, 2012.
+[IR23]
+Tanmay Inamdar and Assaf Rinot. Was Ulam right? I: Basic theory and subnormal
+ideals. Topology Appl., 323(C):Paper No. 108287, 53pp, 2023.
+[Kom16] P´eter Komj´ath. A certain 2-coloring of the reals. Real Anal. Exchange, 41(1):227–231,
+2016.
+[Kom20] P´eter Komj´ath. Weiss’s question. Mathematika, 66(4):954–958, 2020.
+[Kun80]
+Kenneth Kunen. Set theory, volume 102 of Studies in Logic and the Foundations of
+Mathematics. North-Holland Publishing Co., Amsterdam, 1980. An introduction to in-
+dependence proofs.
+[LHR18] Chris Lambie-Hanson and Assaf Rinot. Knaster and friends I: Closed colorings and
+precalibers. Algebra Universalis, 79(4):Art. 90, 39, 2018.
+[LHR23] Chris Lambie-Hanson and Assaf Rinot. Knaster and friends III: Subadditive colorings.
+J. Symbolic Logic, 51pp, to appear 2023. https://doi.org/10.1017/jsl.2022.50.
+[Mil78]
+Keith R. Milliken. Hindman’s theorem and groups. J. Combin. Theory Ser. A, 25(2):174–
+180, 1978.
+[Po´o21]
+M´ark Po´or. On the spectra of cardinalities of branches of Kurepa trees. Arch. Math.
+Logic, 60(7-8):927–966, 2021.
+[Rin14]
+Assaf Rinot. Chain conditions of products, and weakly compact cardinals. Bull. Symb.
+Log., 20(3):293–314, 2014.
+[RZ21]
+Assaf Rinot and Jing Zhang. Transformations of the transfinite plane. Forum Math.
+Sigma, 9(e16):1–25, 2021.
+[She88]
+Saharon Shelah. Was Sierpi´nski right? I. Israel J. Math., 62(3):355–380, 1988.
+[SW16]
+Daniel Soukup and William Weiss. Pairwise sums in colourings of the reals. Unpublished
+note, 2016. https://danieltsoukup.github.io/academic/finset_colouring.pdf.
+[Tod94]
+Stevo Todorcevic. Some partitions of three-dimensional combinatorial cubes. J. Combin.
+Theory Ser. A, 68(2):410–437, 1994.
+[Tod07]
+Stevo Todorcevic. Walks on ordinals and their characteristics, volume 263 of Progress
+in Mathematics. Birkh¨auser Verlag, Basel, 2007.
+Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel.
+Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel.
+URL: http://www.assafrinot.com
+
diff --git a/yNAzT4oBgHgl3EQftP0T/content/tmp_files/load_file.txt b/yNAzT4oBgHgl3EQftP0T/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1729 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf,len=1728
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='01671v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='LO]  4 Jan 2023  SUMS OF TRIPLES IN ABELIAN GROUPS  IDO FELDMAN AND ASSAF RINOT  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Motivated by a problem in additive Ramsey theory, we extend  Todorˇcevi´c’s partitions of three-dimensional combinatorial cubes to handle ad-  ditional three-dimensional objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As a corollary, we get that if the continuum  hypothesis fails, then for every Abelian group G of size ℵ2, there exists a col-  oring c : G → Z such that for every uncountable X ⊆ G and every integer k,  there are three distinct elements x, y, z of X such that c(x + y + z) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Introduction  By Hindman’s celebrated theorem (see [HS12, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9]), for every partition  of an infinite commutative cancellative semigroup (G, +) into two cells A and B,  there exists an infinite subset X ⊆ G such that the set of its finite sums  FS(X) := {x1 + · · · + xn | x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , xn are distinct elements of X & n ∈ N \\ 2}  is completely contained in A or completely contained in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Equivalently, for every  coloring c : G → 2, there exists an infinite X ⊆ G such that c ↾ FS(X) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Hindman’s theorem does not generalize to the uncountable, as it follows from a  theorem of Milliken (see [Mil78, Theorem 9]) that the following assertion is consis-  tent with the usual axioms of set theory: for every (not necessarily Abelian) group  (G, ∗) whose size is a regular uncountable cardinal, there is a coloring c : G → G  such that c ↾ FS2(X) is onto G for every X ⊆ G of size |G|, where this time  FSn(X) := {x1 ∗ · · · ∗ xn | x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , xn are distinct elements of X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  A few years ago, starting with a paper by Hindman, Leader and Strauss [HLS17],  the study of higher analogs of Hindman’s theorem regained interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We mention  only a few results that are relevant to this paper:  (1) Improving upon a theorem from [HLS17], Komj´ath [Kom16], and inde-  pendently Soukup and Weiss [SW16], proved that there exists a coloring  c : R → 2 such that for every uncountable X ⊆ R and every i ∈ {0, 1},  there are x ̸= y in X such that c(x + y) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) Solving a problem of Weiss, Komj´ath [Kom20] proved that there exists a  coloring c : R → 2 such that for every uncountable X ⊆ R and every  i ∈ {0, 1}, there are x ̸= y in X such that c(|x − y|) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As for dimension  d > 1, assuming the continuum hypothesis, there exists a coloring c : R → 2  such that for every uncountable X ⊆ Rd and every i ∈ {0, 1}, there are  x ̸= y in X such that c(∥x − y∥) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) In [FBR17], Fern´andez-Bret´on and Rinot proved that there exists a coloring  c : R → N such that for every X ⊆ R of size |R| and every i ∈ N, there are  x ̸= y in X such that c(x + y) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Date: Preprint as of January 4, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For the latest version, visit http://p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='assafrinot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='com/57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Primary 03E02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Secondary 03E75, 03E35, 05A17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  1  2  IDO FELDMAN AND ASSAF RINOT  (4) By [FBR17], for class many cardinals κ (including κ = ℵn for every positive  integer n), for every commutative cancellative semigroup G of size κ, there  exists a coloring c : G → G such that for all X, Y ⊆ G of size κ and every  g ∈ G, there are x ∈ X and y ∈ Y such that c(x + y) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1  (5) By [FBR17], for every commutative cancellative semigroup G, there exists  a coloring c : G → N such that c ↾ FS(X) is onto N for every uncountable  X ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It is also consistent that the same holds after replacing N by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Note that in the results listed in (1), (2) and (5), the triggering set X may have  cardinality smaller than that of G, whereas in (3) and (4), |X| coincides with |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Another important difference is that unlike the results of (1)–(4), in (5), no bound  is asserted on the length of the sums needed to generate all the infinite colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This  raises a natural question whose simplest instance reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that (G, +) is an Abelian group of size ℵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Must there exist a positive integer n and a coloring c : G → N such that c ↾  FSn(X) is onto N for every uncountable X ⊆ G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  A moment’s reflection makes it clear that an affirmative answer (even for one  particular group G) immediately implies the relation ℵ2 ↛ [ℵ1]n  ℵ0 from the classical  study of partition relations for cardinal numbers [EHR65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By a theorem of Erd˝os  and Rado, the above relation may consistently fail for n = 2, and it is a remarkable  theorem of Todorˇcevi´c [Tod94] that it does hold for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The first main result of  this paper gives a consistent extension of Todorˇcevi´c’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If the continuum hypothesis fails, then for every Abelian group (G, +)  of size ℵ2, there exists a coloring c : G → N such that for every uncountable  X ⊆ G and every i ∈ N, there are three distinct elements x, y, z of X such that  c(x + y + z) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Theorem A is not limited to Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In fact, it works for all so-called  well-behaved magmas, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' A magma is a structure (G, ∗), where ∗ is a binary operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We say  that it is well-behaved iff there exists a map ϕ : G → [G]<ω such that:2  G is countable-to-one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  for all x ̸= y in G, ϕ(x) △ ϕ(y) ⊆ ϕ(x ∗ y) ⊆ ϕ(x) ∪ ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Every infinite commutative cancellative semigroup (G, +) is well-behaved (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=', [FBR17, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Also, every free group (G, ∗) is well-behaved, as wit-  nessed by the map that sends a word to the set of its letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As a third ex-  ample, consider the magma appearing in result (2) above, namely, (R, d) where  d(x, y) := |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Indeed, viewing R as a Q-vector space over some Hamel basis  B, any x ∈ R \\ {0} is the unique linear combination �  i≤n qivi of nonzero rational  numbers q0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , qn, and an injective sequence ⟨vi | i ≤ n⟩ of elements of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So  ϕ : R → [R]<ω sending x to the unique {vi | i ≤ n} (and sending 0 to the emptyset)  is countable-to-one, and for all x ̸= y, ϕ(x) △ ϕ(y) ⊆ ϕ(|x − y|) ⊆ ϕ(x) ∪ ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The full statement of Theorem A reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Theorem A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every infinite cardinal µ such that µ<µ < µ+ < 2µ, for every  well-behaved magma (G, ∗) of size µ++, there is a coloring c : G → N such that for  1More is true, see [FBR17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  2Here, [G]<ω denotes the collection of all finite subsets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  3  every X ⊆ G of size µ+ and every i ∈ N, there are three distinct elements x, y, z of  X such that c(x ∗ y ∗ z) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3  While not so explicit, the approach of going through well-behaved magmas is  already present in [FBR17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In particular, the coloring of result (4) attains all  possible colors not only over evaluations of the form x + y, but also over any  nontrivial Q-combination of x and y, such as |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  This suggests that it is  possible to obtain a coloring simultaneously witnessing result (1) together with  the first half of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Indeed, Komj´ath’s theorems follow from the following finding  (using θ := ℵ0):  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every infinite cardinal θ such that 2<θ = θ, for every set G with  θ < |G| ≤ 2θ, and every map ϕ : G → [G]<ω, there exists a corresponding coloring  c : G → 2 satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For every binary operation ∗ on G, if ϕ witnesses that (G, ∗) is well-behaved,  then for every X ⊆ G of size θ+ and every i ∈ {0, 1}, there are x ̸= y in X such  that c(x ∗ y) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The proofs of Theorems A′ and B are obtained in a few steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As a first step,  we consider a coloring principle Sn(κ, λ, θ) that is sufficient to imply that any well-  behaved magma (G, ∗) of size κ admits a coloring c : G → θ that takes on every  possible color on FSn(X) for every set X ⊆ G of size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The next step is the  introduction of an extraction principle Extractn(κ, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=') that is sufficient for the  reduction of Sn(κ, λ, θ) into a rectangular-type strengthening κ  sup  �−→ [λ, λ]n  θ of the  classical partition relation κ ↛ [λ]n  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This leaves us with two independent tasks:  proving instances of Extractn(κ, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' ), and proving instances of κ  sup  �−→ [λ, λ]n  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The  harder task is the latter, and the second main result of this paper is an extension of  Todorˇcevi´c’s theorem [Tod94] that Chang’s conjecture fails iff ω2 ↛ [ω1]3  ω1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Here ω2 ↛ [ω1]3  ω1 is improved to ω2  sup  �−→ [ω1, ω1]3  ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Specifically:  Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The following are equivalent:  (1) (ℵ2, ℵ1) ։ (ℵ1, ℵ0) fails;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) There exists a coloring c : [ω2]3 → ω1 with the property that for all disjoint  A, B ⊆ ω2 of order-type ω1 such that sup(A) = sup(B), for every color  τ < ω1, there is (α, β, γ) ∈ [A ∪ B]3 \\ ([A]3 ∪ [B]3) such that c(α, β, γ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Organization of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In Section 2, we provide some necessary pre-  liminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In Section 3, we recall the definition of a weak Kurepa tree and study related  objects such as the branch spectrum T (µ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This will play a role in both getting  instances of Extractn(κ, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=') and of κ  sup  �−→ [λ, λ]n  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In Section 4, we prove that Sn(κ, λ, θ) implies that any well-behaved magma  (G, ∗) of size κ admits a coloring with the strong properties mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  It is proved that in the special case of λ = κ, S2(κ, λ, θ) already follows from  κ ↛ [λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  θ, and that, in general, Sn(κ, λ, θ) follows from κ  sup  �−→ [λ, λ]n  θ together with  Extractn(κ, λ, ω, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We then use tree combinatorics to obtain sufficient conditions  for Extractn(κ, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=') to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The definitions of Extractn(κ, λ, θ, χ) and κ  sup  �−→  [λ, λ]n  θ will be found in this section as Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  3As ∗ is not assumed to be associative, the claim is that we get c(x ∗ y ∗ z) = i for both  implementations of x ∗ y ∗ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  4  IDO FELDMAN AND ASSAF RINOT  In Section 5, we prove the general case of Theorem C in which ℵ2 is substituted  by the double successor of a cardinal µ satisfying µ<µ = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The proof is a bit long,  since the analysis goes through a division into a total of six cases and subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In Section 6, we verify that Todorˇcevi´c’s theorems on the correspondence between  unstable sets and oscillation remains valid in the rectangular context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We then  combine it with the results of Section 5 and get that λ+ sup  �−→ [λ, λ]3  ω holds for every  successor λ = µ+ of an infinite cardinal µ = µ<µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In Section 7, we obtain the intended applications in additive Ramsey theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Theorem A′ is gotten as a corollary of the results of Sections 4 and 6, and Theorem B  is gotten as a corollary of a theorem asserting that S2(κ, µ+, 2) holds whenever there  exists a weak µ-Kurepa tree with κ-many branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Preliminaries  In this section, κ, λ, µ, θ, χ stand for nonzero cardinals, and n stand for a positive  integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We let Hκ denote the collection of all sets of hereditary cardinality less  than κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We write [κ]λ := {A ⊆ κ | |A| = λ} and [κ]<λ := {A ⊆ κ | |A| < λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let  Eκ  χ := {α < κ | cf(α) = χ}, and define Eκ  ≤χ, Eκ  <χ, Eκ  ≥χ, Eκ  >χ, Eκ  ̸=χ analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For two distinct functions f, g ∈ θµ, write f <lex g to mean that f(δ) < g(δ) for  the least δ < λ such that f(δ) ̸= g(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For functions f, g ∈ ≤θµ, we write f ⊑ g to  mean that dom(f) ≤ dom(g) and g ↾ dom(f) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For sets of ordinals A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , An, we define  A1 ⊛ · · · ⊛ An := {(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , αn) ∈ A1 × · · · × An | α1 < · · · < αn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By convention, whenever we write (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , αn) ∈ [A]n (as opposed to {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , αn} ∈  [A]n), we mean that (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , αn) ∈ A ⊛ · · · ⊛ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For a set of ordinals A, we write ssup(A) := sup{α + 1 | α ∈ A}, acc+(A) :=  {α < ssup(A) | sup(a ∩ α) = α > 0}, and acc(A) := A ∩ acc+(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For two sets of  ordinals A and B, we write A < B to mean that A × B coincides with A ⊛ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1 (Positive round-bracket relations, [EHR65, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' κ → (λ)n  θ asserts  that for every coloring c : [κ]n → θ, there exists A ⊆ κ of order-type λ such that c  is constant over [A]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2 (Negative square-bracket relations, [EHR65, §18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' A coloring c :  [κ]n → θ is said to witness:  κ ↛ [λ]n  θ iff c[[A]n] = θ for every A ∈ [κ]λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  κ ↛ [λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , λn]n  θ iff c[A1 × · · · × An] = θ for every ⟨Ai | 1 ≤ i ≤ n⟩ ∈  �n  i=1[κ]λi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  κ ↛ [λ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λn]n  θ iff c[A1 ⊛ · · · ⊛ An] = θ for every ⟨Ai | 1 ≤ i ≤ n⟩ ∈  �n  i=1[κ]λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Note that (κ ↛ [λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]n  θ ) =⇒ (κ ↛ [λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , λ]n  θ ) =⇒ (κ ↛ [λ]n  θ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3 (Fiber maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Given a coloring of pairs c : [κ]2 → θ and some  β < κ, we sometimes write cβ for the βth-fiber map of c, that is, for the unique  map cβ : β → θ to satisfy cβ(α) = c(α, β) for every α < β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  We say that c has injective fibers iff cβ is injective of every β < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4 ([LHR18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 →  θ such that for every σ < χ, every pairwise disjoint subfamily A ⊆ [κ]σ of size κ,  SUMS OF TRIPLES IN ABELIAN GROUPS  5  for every τ < θ, there exists B ⊆ A of size µ such that min(c[a × b]) > τ for all  a ̸= b from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Of special interest are witnesses c : [κ]2 → θ to U(κ, µ, θ, χ) that are  moreover subadditive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=', satisfying that for all α < β < γ < κ, the following hold:  c(α, γ) ≤ max{c(α, β), c(β, γ)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  c(α, β) ≤ max{c(α, γ), c(β, γ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  These colorings are studied in [LHR23], and they will show up here in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Given a coloring c : [κ]2 → θ and a subset X ⊆ κ of order-type λ, we say that  “c ↾ [X]2 witnesses U(λ, µ, θ, χ)” if for the order-preserving bijection π : λ ↔ X,  the coloring d : [λ]2 → θ defined via d(α, β) := c(π(α), π(β)) is a witness for  U(λ, µ, θ, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To be able to express that this happens globally, we introduce the  following 5-cardinal extension of the principle of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' U(κ, λ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such  that for every σ < χ, every pairwise disjoint subfamily A ⊆ [κ]σ of size λ, for every  τ < θ, there exists B ⊆ A of size µ such that min(c[a × b]) > τ for all a ̸= b from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7 ([Tod07, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every regular uncountable cardinal λ, if  U(λ+, λ, 2, λ, 2) holds, then there exists a subadditive witness to U(λ+, λ, λ, λ, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Finally, we arrive at the notion motivating this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For a magma (G, ∗), we write G ↛ [λ]FSn  θ  to assert that there  exists a coloring c : G → θ with the property that for every subset A ⊆ G of size λ  and every prescribed color τ < θ, there is an injective sequence ⟨ai | 1 ≤ i ≤ n⟩ of  elements of A such that c(a1 ∗ · · · ∗ an) = τ for all implementations of a1 ∗ · · ·∗ an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4  In the special case of n = 2, [FBR17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5] and [RZ21, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='20]  provide sufficient conditions for G ↛ [λ]FSn  θ  to follow from |G| ↛ [λ]n  θ for all values  of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Higher dimensional reductions are out of reach at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Walks on ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In this subsection, we provide a minimal background  on walks on ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This background is only necessary for Section 6, hence the  exposition here is quite succinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' A thorough treatment may be found in [Tod07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For the rest of this subsection, κ denotes a regular uncountable cardinal, and we  fix some C-sequence over κ, that is, a sequence ⃗C = ⟨Cβ | β < κ⟩ such that, for  every β < κ, Cβ is closed subset of β with sup(Cβ) = sup(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9 (Todorˇcevi´c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' From ⃗C, derive maps Tr : [κ]2 → ωκ, ρ2 : [κ]2 → ω,  and tr : [κ]2 → <ωκ, by letting for all α < β < κ:  Tr(α, β) : ω → κ is defined by recursion on n < ω:  Tr(α, β)(n) :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  β,  n = 0  min(CTr(α,β)(n−1) \\ α),  n > 0 & Tr(α, β)(n − 1) > α  α,  otherwise  ρ2(α, β) := min{l < ω | Tr(α, β)(l) = α};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  tr(α, β) := Tr(α, β) ↾ ρ2(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  4The issue of implementation arises from the fact that we do not assume ∗ to be associative,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=', it is possible that (a1 ∗ a2) ∗ a3 ̸= a1 ∗ (a2 ∗ a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  6  IDO FELDMAN AND ASSAF RINOT  To explain: Given a pair of ordinals α < β below κ, one would like to walk from  β down to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This is done by recursion, letting β0 := β, and βn+1 := min(Cβn \\ α),  thus, obtaining an ordinal βn+1 such that α ≤ βn+1 ≤ βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since the ordinals are  well-founded, there must exist some integer k such that βk+1 = α, so that, the  walk is β = β0 > β1 > · · · > βk+1 = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This walk is recorded by Tr(α, β), since,  for every n ≤ k, we have that Tr(α, β) = βn, and for every n > k, we have that  Tr(α, β) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The length of the walk is recorded by the positive integer ρ2(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Now, since Tr(α, β) is eventually constant with value α, its nontrivial part is those  ordinals greater than α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=', β0 > β1 > · · · > βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' this is recorded by tr(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='10 ([Rin14, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Define a function λ2 : [κ]2 → κ via  λ2(α, β) := sup(α ∩ {sup(Cη ∩ α) | η ∈ Im(tr(α, β))}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Note that λ2(α, β) < α whenever 0 < α < β < κ, since tr(α, β) is a finite  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='11 ([LHR18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that λ2(α, β) < ǫ < α < β < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then tr(ǫ, β) end-extends tr(α, β), and one of the following cases holds:  (1) α ∈ Im(tr(ǫ, β));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' or  (2) α ∈ acc(Cð) for ð := min(Im(tr(α, β))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12 ([RZ21, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every (α, β) ∈ [κ]2, we define an  ordinal ðα,β ∈ [α, β] via:  ðα,β :=  �  min(Im(tr(α, β))),  α ∈ acc(Cmin(Im(tr(α,β))));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  α,  otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It is easy to see that sup(Cðα,β) = sup(α) for all α < β < κ, and it  follows from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='11 that  tr(ǫ, β) = tr(ðα,β, β)⌢ tr(ǫ, ðα,β),  whenever λ2(α, β) < ǫ < α < β < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14 (Todorˇcevi´c, [Tod07, §9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If κ = λ+ for a regular cardinal λ and  otp(Cβ) ≤ λ for all β < κ, then there exists a subadditive coloring ρ : [κ]2 → λ with  the property that ρ(α, β) ≥ otp(Cη ∩ α) for all α < β < κ and η ∈ Im(tr(α, β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Weak Kurepa trees and the branch spectrum  In this section, µ denotes a cardinal and θ denotes an ordinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' T (µ, θ) denotes the collection of all subsets T ⊆ <θµ such that  the following two hold:  (1) T is downward-closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='e, for every t ∈ T , {t ↾ α | α < θ} ⊆ T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) for every α < θ, the set Tα := T ∩ αµ is nonempty and has size < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  We say that T is a tree of height θ if there exists a cardinal µ such that T ∈  T (µ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5 Note that θ is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For such a tree T , we shall refer to  Tα as the αth-level of T , and the set {b ∈ θµ | ∀α < θ (b ↾ α ∈ Tα)} of all branches  through T is denoted by B(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Also, for all f, g ∈ ≤µθ, we let  ∆(f, g) :=  �  min{δ ∈ dom(f) ∩ dom(g) | f(δ) ̸= g(δ)},  if f ⊈ g & g ⊈ f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  min{dom(f), dom(g)},  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  5There is no loss of generality here, see [BR21, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  7  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' T ∈ T (µ, θ) is said to be normal iff for all α < β < θ and t ∈ Tα,  there exists t′ ∈ Tβ with t ⊑ t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Given a tree T and a subset B ⊆ B(T ), we consider the subtree:  T ⇝B := {t ∈ T | |{b ∈ B | t ⊑ b}| = |B|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that T ∈ T (µ, θ), and λ is an infinite regular cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (1) If λ ≥ µ, then for every B ∈ [B(T )]λ, T ⇝B is in T (µ, θ) and is normal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) If λ ≥ max{µ, |θ|+}, then for all A, B ∈ [B(T )]λ, there are s ∈ T and i ̸= i′  such that s⌢⟨i⟩ ∈ T ⇝A and s⌢⟨i′⟩ ∈ T ⇝B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' (1) Suppose that B ∈ [B(T )]λ and λ ≥ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It is clear that ∅ ∈ T ⇝B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Thus,  to prove that T ⇝B has height θ and is normal, let α < β < θ and t ∈ (T ⇝B)α, and  we shall show that there exists t′ ∈ (T ⇝B)β extending t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By the choice of t, B′ := {b ∈ B | t ⊑ b}| has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since T ∈ T (µ, θ), it is  the case that 0 < |Tβ| < |B′| = cf(|B′|), and then the pigeonhole principle provides  t′ ∈ Tβ such that {b ∈ B′ | t′ ⊑ b} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Evidently, t′ is as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) Suppose that A, B ∈ [B(T )]λ and λ ≥ max{µ, |θ|+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By possibly passing to  λ-sized subsets of A and B, we may assume that A ∩ B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let ⟨aj | j < λ⟩ be  some injective enumeration of A, and likewise let ⟨bj | j < λ⟩ be some injective  enumeration of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For each j < λ, as aj ̸= bj, we may let δj := ∆(aj, bj) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As λ  is a regular cardinal greater than |θ|, we may fix some J ∈ [λ]λ on which the map  j �→ δj is constant with value, say, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As Tδ+1 has size < µ ≤ λ, we may moreover  assume that the map j �→ ((aj ↾ δ + 1), (bj ↾ δ + 1)) is constant over J, with value,  say, (s⌢⟨i⟩, s⌢⟨i′⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that T ∈ T (λ, θ) ∩ P(<θ2), where λ = cf(λ) > cf(θ) ≥ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Suppose that we are given i < 2 and X ∈ [B(T )]λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, for λ-many x ∈ X, there  are cofinally many δ < θ such that the following two hold:  (1) x(δ) = i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) {y ∈ X | ∆(x, y) = δ} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In particular, the set Y of all x ∈ X for which there are  boundedly many δ < θ satisfying Clauses (1) and (2) has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So, for each  x ∈ Y , the following ordinal is smaller than θ:  ǫx := sup{δ < θ | x(δ) = i & |{y ∈ X | ∆(x, y) = δ}| = λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As |Y | = cf(λ) > cf(θ), we may find some ǫ < θ such that Z := {x ∈ Y | ǫx = ǫ}  has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As |Tǫ+1| < λ, we may also find some t ∈ Tǫ+1 such that Zt := {x ∈ Z |  t ⊑ x} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, by appealing to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4(2) with µ := λ, A := Zt and  B := Zt, we may find s ∈ T and j < 2 such that ˆA := {a ∈ A | s⌢⟨j⟩ ⊑ a} and  ˆB := {b ∈ B | s⌢⟨1 − j⟩ ⊑ b} are both of size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As A = B and by possibly  switching the roles of ˆA and ˆB, we may assume that j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote δ := dom(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For all a ∈ ˆA and b ∈ ˆB, since a, b ∈ Zt, both s⌢⟨i⟩ and s⌢⟨1 − i⟩ are compatible  with t, so that δ = dom(s) ≥ dom(t) > ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, for every x ∈ ˆA, it is the case that  x(δ) = i and {y ∈ X | ∆(x, y) = δ} covers ˆB, but | ˆB| = λ, so we got a contradiction  to the fact that δ > ǫ = ǫx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that T ∈ T (µ, µ), and µ is a regular uncountable cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Suppose also that ⟨bξ | ξ < µ⟩ is an injective enumeration of some B ∈ [B(T )]µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  8  IDO FELDMAN AND ASSAF RINOT  For every ⟨tα | α < µ⟩ ∈ �  α<µ(T ⇝B ∩ αµ), for club many α < µ,  sup({∆(bβ, tα) | β < α} ∩ α) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In particular, for club many α < µ,  sup{γ < µ | α ∈ acc+({∆(bβ, bγ) | β < α})} = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The ‘In particular’ part follows the main claim together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4(1),  using λ := µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Next, to prove the main claim, let ⟨tα | α < µ⟩ ∈ �  α<µ(T ⇝B ∩ αµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Denote Γα := {γ < µ | tα ⊑ bγ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Consider the club  C := {α ∈ acc(µ) | ∀¯α < α [min(Γ¯α \\ ¯α) < α]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Note that for every α < µ, Dα := {∆(bβ, tα) | β ∈ α \\ Γα} is a subset of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The following set covers a club in µ:  A := {α < µ | sup(Dα) = α}  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Fix an ordinal ǫ < µ for which the following set is stationary:  S := {α ∈ C | sup(Dα) = ǫ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  There are two cases to consider:  ◮ Suppose that there exists a function b : µ → µ such that Sb := {α ∈ S |  b ↾ α = tα} is cofinal in µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick ¯α ∈ Sb \\ (ǫ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since Γ¯α has more than one  element, we may now find β ∈ Γ¯α such b ̸= bβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then ǫ < ¯α ≤ ∆(bβ, b) < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick  α ∈ Sb above max{∆(bβ, b), β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then ǫ < ∆(bβ, tα) < α, contradicting the fact  that sup(Dα) = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ Suppose the first case fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' First, since |Tǫ+1| < µ, pick a node t ∈ Tǫ+1 such  that S′ := {α ∈ S | tα ↾ (ǫ+1) = t} is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since for every function b : µ → µ,  the set Sb := {α ∈ S | b ↾ α = tα} is bounded in µ, we may now pick a pair  (¯α, α) ∈ S′ such that t¯α ̸⊑ tα, so that ǫ < ∆(t¯α, tα) < ¯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let β := min(Γ¯α\\¯α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then  bβ ↾ ¯α = t¯α and hence ∆(bβ, tα) = ∆(t¯α, tα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' That is, ǫ < ∆(bβ, tα) < ¯α ≤ β < α,  contradicting the fact that sup(Dα) = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Let α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Recall that Γα has size µ, and note that, for every γ ∈ Γα,  {∆(bβ, bγ) | β < α} ∩ α = {∆(bβ, tα) | β < α} ∩ α = Dα,  so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that T ∈ T (2µ, µ) ∩ P(<µµ), where µ is an infinite regular  cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every B ∈ [B(T )]µ, there exist B′ ∈ [B]µ and θ ≤ µ such that:  (1) For every B′′ ∈ [B′]µ, T ⇝B′′ is in T (µ, θ) and is normal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) If θ < µ or if T contains no µ-Aronszajn subtrees, then |B(T ⇝B′′)| = µ for  every B′′ ∈ [B′]µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) If θ < µ, then |{b ∈ B′ | ssup{∆(t, b ↾ θ) | t ∈ T ⇝B′ & t ̸⊑ b} < θ}| < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let B ∈ [B(T )]µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If T ⇝B ∈ T (µ, µ), then the pair (B′, θ) := (B, µ) is as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that T ⇝B is in T (µ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every B′′ ∈ [B]µ, T ⇝B′′ = (T ⇝B)⇝B′′,  and hence Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4(1) implies that T ⇝B′′ ∈ T (µ, µ) and is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition,  if there exists some B′′ ∈ [B]µ such that |B(T ⇝B′′)| < µ, then looking at B′′′ :=  B′′ \\ B(T ⇝B′′), we get that T ⇝B′′′ is a µ-Aronszajn subtree of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  SUMS OF TRIPLES IN ABELIAN GROUPS  9  From now on, suppose that T ⇝B ∈ T (2µ, µ) \\ T (µ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let θ < µ be the least  such that (T ⇝B)θ has size ≥ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let ⟨ti | i < µ⟩ be an injective sequence of elements  of (T ⇝B)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For each i < µ, pick bi ∈ B such that ti ⊑ bi, and set B′ := {bi | i < µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  To see that the pair (B′, θ) is as sought, let B′′ ∈ [B′]µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' T ⇝B′′ is in T (µ, θ) and is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every α ∈ [θ, µ), it is the case that for every t ∈ (T ⇝B′′)α, there exists  a unique i < µ such that ti ⊑ t, and hence {b ∈ B′′ | t ⊑ b} ⊆ {bi} is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Therefore, (T ⇝B′′)α is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition, as B′′ ⊆ B′ ⊆ B, it is the case that  |(T ⇝B′′)α| ≤ |(T ⇝B)α| < µ for all α < θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Clearly, ∅ ∈ T ⇝B′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Finally, let α < β < µ with t ∈ (T ⇝B′′)α, and we shall find  t′ ∈ (T ⇝B′′)β extending t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the choice of t, B∗ := {b ∈ B′′ | t ⊑ b}| has size µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By the minimality of θ, the map b �→ b ↾ β from B∗ to Tβ cannot have an image of  size µ, and hence there exists B∗∗ ∈ [B∗]µ on which the said map is constant, with  some value, say t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clearly, t′ is as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' |B(T ⇝B′′)| = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, I := {i < µ | bi ∈ B′′ & ti /∈ B(T ⇝B′′)} has size  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It follows that there exists an α < θ such that Iα := {i ∈ I | (ti ↾ α) /∈ T ⇝B′′}  has size µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' However, µ is a regular cardinal greater than |Tα| ≥ |(T ⇝B′′)α|, and  hence there must exist some s ∈ (T ⇝B′′)α such that {i ∈ Iα | (ti ↾ α) = s} has size  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' |{i < µ | ssup{∆(t, ti) | t ∈ T ⇝B′ & t ̸⊑ ti} < θ}| < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose not, and pick ǫ < θ such that the following set has size µ:  I := {i < µ | ssup{∆(t, ti) | t ∈ T ⇝B′ & t ̸⊑ ti} = ǫ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then pick s ∈ Tǫ such that {i ∈ I | ti ↾ ǫ = s} has size µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Finally, as in the proof  of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4(2), we may find some s′ ∈ T ⇝B′ extending s and j ̸= j′ such that  {i ∈ I | s′⌢⟨j⟩ ⊑ ti} and {i ∈ I | s′⌢⟨j′⟩ ⊑ ti} are both of size µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular,  there exist i ̸= i′ in I such that ti ∩ ti′ = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So ∆(s′⌢⟨i′⟩, ti) ≥ ǫ, contradicting the  fact that i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let µ denote an infinite cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (1) A weak µ-Kurepa tree is a tree T of height µ, of size µ, satisfying |B(T )| > µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) A µ-Kurepa tree is a tree T of height µ for which {α < µ | |Tα| > |α|} is  nonstationary, and |B(T )| > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As in Exercise 34 of [Kun80, §II], if there exists a µ-Kurepa tree  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' weak µ-Kurepa tree), then there exists one which is a subset of <µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='10 (Branch spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' T (µ, θ) stands for the collection of all cardi-  nals κ for which there exists T ∈ T (µ, θ) with κ ≤ |B(T )|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let µ and θ denote infinite cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then:  (1) sup(T (µ, θ)) ≤ µθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) If θ is the least cardinal to satisfy µθ > µ, then max(T (µ+, θ)) = (µ+)θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) If there exists a weak µ-Kurepa tree, then µ+ ∈ T (µ+, µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  10  IDO FELDMAN AND ASSAF RINOT  (4) If there exists a µ-Kurepa tree, then µ+ ∈ T (µ, µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (5) If µ is a strong limit, then 2µ ∈ T (µ, cf(µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' T (µ, θ) need not have a maximal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By [Po´o21], it is consistent  for T (ω1, ω1) to have ℵω2 has a supremum that is not attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note, however, that  T (µ, θ) is closed under unions of length < cf(µ), and hence T (µ, θ) is < cf(µ)-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every κ ∈ T (µ, θ), there exists a coloring c : [κ]2 → θ  witnessing U(κ, λ, λ, θ, 2) for every regular cardinal λ ∈ [µ, κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Given κ ∈ T (µ, θ), let us fix T ∈ T (µ, θ) admitting an injective sequence ⟨bξ |  ξ < κ⟩ consisting of elements of B(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Define c : [κ]2 → θ via c(α, β) := ∆(bα, bβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Now, given τ < θ and A ∈ [κ]λ for a regular cardinal λ ∈ [µ, κ], since |Tτ+1| < µ, it  is possible to find x ∈ Tτ+1 for which B := {α ∈ A | x ⊑ bα} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Evidently,  c(α, β) > τ for all α ̸= β from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Coloring well-behaved magmas  In this section, we obtain sufficient conditions for G ↛ [λ]FSn  θ  to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To ease  on the reader, we start with the special case of n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The upcoming Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2  reduces this case to the following simple combinatorial principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' S2(κ, λ, θ) asserts the existence of a coloring d : [κ]<ω → θ such  that, for every X ⊆ [κ]<ω of size λ and every prescribed color τ < θ, there exist  two distinct x, y ∈ X such that d(z) = τ whenever (x △ y) ⊆ z ⊆ (x ∪ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2 ([FBR17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that S2(κ, λ, θ) holds, for given  cardinals θ ≤ λ ≤ κ with λ regular and uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then G ↛ [λ]FS2  θ  holds for every well-behaved magma (G, ∗) with |G| = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let d be coloring witnessing S2(κ, λ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Suppose that (G, ∗) is a well-  behaved magma with |G| = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By identifying [G]<ω with [κ]<ω, we may thus  fix a map ϕ : G → [G]<ω such that:  ϕ is <λ-to-one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  for all x ̸= y in G, ϕ(x) △ ϕ(y) ⊆ ϕ(x ∗ y) ⊆ ϕ(x) ∪ ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Define a coloring c : G → θ by letting c := d ◦ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To see that c is as sought,  let X ∈ [G]λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As ϕ is <λ-to-one, X := {ϕ(x) | x ∈ X} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Thus, given  a prescribed color τ < θ, we may find x, y ∈ X with ϕ(x) ̸= ϕ(y) such that  d(z) = τ whenever (ϕ(x) △ ϕ(y)) ⊆ z ⊆ (ϕ(x) ∪ ϕ(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, x ̸= y and  c(x ∗ y) = d(ϕ(x ∗ y)) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  The question arises: How do one obtain instances of S2(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' )?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The proof of  [FBR17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4] makes it clear that the following holds:  Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that λ is a regular uncountable cardinal and that θ is an infinite  cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then Pr1(κ, λ, θ, ω) implies S2(κ, λ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The principle Pr1(κ, λ, θ, ω) is a particular strengthening of κ ↛  [λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Since it will not play a role in this paper, we omit its definition, and  settle for pointing out the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By a theorem of Fleissner [Fle78,  §5], for every regular uncountable cardinal κ, in the forcing extension for adding  κ-many Cohen reals, Pr1(κ, ω1, ω, ω) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It thus follows that if κ is a regular  cardinal ≥ c, then after adding κ-many Cohen reals, S2(2ℵ0, ℵ1, ℵ0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  11  In case that λ = κ, we can now improve Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that κ is a regular uncountable cardinal and that θ is an  infinite cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then κ ↛ [κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' κ]2  θ implies S2(κ, κ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that c : [κ]2 → θ is a coloring witnessing κ ↛ [κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' κ]2  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Fix a  bijection π : θ ↔ θ × ω, and then find c0 : [κ]2 → θ and c1 : [κ]2 → ω such that  π(c(α, β)) = (c0(α, β), c1(α, β)) for every (α, β) ∈ [κ]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Define a coloring d : [κ]<ω → θ, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For z ∈ [κ]<2, just let d(z) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Next,  for z ∈ [κ]<ω of size ≥ 2, first let ⟨αi | i < |z|⟩ denote the increasing enumeration  of z, and then let d(z) := c0(αjz, αjz+1), for  jz := min{j < |z| − 1 | c1(αj, αj+1) = max{c1(αi, αi+1) | i < |z| − 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  To see this works, suppose that we are given a κ-sized family X ⊆ [κ]<ω, and a  prescribed color τ < θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By thinning out, we may assume that X forms an head-  tail-tail ∆-system with some root r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By further thinning out, we may assume the  existence of some n < ω such that c1“[x]2 ⊆ n for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Split X into two κ-sized  sets X = X0 ∪ X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set A := {min(x \\ r) | x ∈ X0} and B := {max(x \\ r) | x ∈ X1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As c witnesses κ ↛ [κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' κ]2  θ, fix (α, β) ∈ A ⊛ B such that c(α, β) = π−1(τ, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Pick the unique x, y ∈ X such that α = min(x \\ r) and β = max(x \\ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As  X0 ∩ X1 = ∅, x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Consequently, x \\ r < y \\ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now fix an arbitrary set z such  that (x △ y) ⊆ z ⊆ (x ∪ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clearly |z| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let ⟨αi | i < |z|⟩ denote the increasing  enumeration of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every i < |z|, if {αi, αi+1} ⊆ x then c1(αi, αi+1) < n, and  likewise, if {αi, αi+1} ⊆ y then c1(αi, αi+1) < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As x \\ r < y \\ r and {α, β} ⊆ z,  it follows that there exists j < |z| such that aj = α and αj+1 = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For this j,  we would have c1(αj, αj+1) = c1(α, β) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Altogether, d(z) = c0(α, β) = τ, as  sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  As for the general case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=', λ ≤ κ), we now present an extraction principle that  is sufficient to derive S2(κ, λ, θ) from κ ↛ [λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Roughly speaking, the upcoming  principle asserts the existence of a map e : [κ]<ω → [κ]2 that, in some scenarios,  manages to extract two distinguished points e(z) from any given set z ∈ [κ]<ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Extract2(κ, λ, θ, χ) asserts the existence of a map e : [κ]<ω → [κ]2  satisfying that for every sequence ⟨xγ | γ < λ⟩ of subsets of κ, every r ∈ [κ]<θ, and  every nonzero σ < χ such that:  (1) for every (γ, γ′) ∈ [λ]2, xγ ∩ xγ′ ⊆ r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) for every γ < λ, yγ := xγ \\ r has order-type σ,  there exist j < σ and disjoint cofinal subsets Γ0, Γ1 of λ satisfying the following:  (a) For every (γ, γ′) ∈ [Γ0 ∪ Γ1]2, yγ(j) < yγ′(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (b) For every (γ, γ′) ∈ (Γ0 ⊛Γ1)∪(Γ1 ⊛Γ0), for every z ∈ [xγ ∪xγ′]<ω covering  {yγ(j), yγ′(j)}, we have  e(z) = (yγ(j), yγ′(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Without loss of generality, we may assume that e(z) ∈ [z]2 for every  z ∈ [κ]<ω of size ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Also note that Extract2(κ, κ, cf(κ), 2) is a theorem of ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that λ is a regular uncountable cardinal and that θ is an  arbitrary cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If κ ↛ [λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  θ and Extract2(κ, λ, ω, ω) both hold, then so does  S2(κ, λ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  12  IDO FELDMAN AND ASSAF RINOT  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that c : [κ]2 → θ is a witness for κ ↛ [λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  θ, and that e : [κ]<ω →  [κ]2 is a witness for Extract2(κ, λ, ω, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We claim that d := c ◦ e is a witness for  S2(κ, λ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To this end, suppose that we are given a subfamily X ⊆ [κ]<ω of size  λ, and a prescribed color τ < θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As λ is regular and uncountable, by the ∆-system  lemma, we may find a sequence ⟨xγ | γ < λ⟩ consisting of elements of X, some  r ∈ [κ]<ω, and a nonzero σ < χ such that:  (1) for every (γ, γ′) ∈ [λ]2, xγ ∩ xγ′ = r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) for every γ < λ, yγ := xγ \\ r has order-type σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  It thus follows from the choice of e that we may pick some integer j < σ and cofinal  subsets Γ0, Γ1 of λ satisfying the following:  (a) For every (γ, γ′) ∈ [Γ0 ∪ Γ1]2, yγ(j) < yγ′(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (b) For every (γ, γ′) ∈ (Γ0 ⊛Γ1)∪(Γ1 ⊛Γ0), for every z ∈ [xγ ∪xγ′]<ω covering  {yγ(j), yγ′(j)}, we have  e(z) = (yγ(j), yγ′(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Put A := {yγ(j) | γ ∈ Γ0} and B := {yγ(j) | γ ∈ Γ1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the choice of c, we may  find (α, β) ∈ A⊛B such that c(α, β) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick γ ∈ Γ0 such that yγ(j) = α, and pick  γ′ ∈ Γ1 such that yγ′(j) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As α < β, Clause (a) implies that (γ, γ′) ∈ (Γ0 ⊛ Γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As xγ ∩ xγ′ = r, we infer that {α, β} ⊆ xγ △ xγ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, for every set z such that  (xγ △ xγ′) ⊆ z ⊆ (xγ ∪ xγ′), we get that  d(z) = c(e(z)) = c(α, β) = τ,  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Motivated by the preceding reduction, one would like to see how to get Extract2(κ, λ,  ω, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The next lemma provides a sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that κ ∈ T (µ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then there exists a map e : [κ]<ω → [κ]2  witnessing Extract2(κ, λ, cf(θ), ω) for every regular cardinal λ with max{µ, θ+} ≤  λ ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As κ ∈ T (µ, θ), let us fix T ∈ T (µ, θ) admitting an injective sequence ⟨bξ |  ξ < κ⟩ consisting of elements of B(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For notational simplicity, we shall write  ∆(α, β) for ∆(bα, bβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' First, for every z ∈ [κ]<ω, let:  Mz := {(α, β) ∈ [z]2 | ∆(α, β) = max(∆“[z]2)}, and  M ∗  z := {(α, β) ∈ Mz | α = min{α′ | (α′, β′) ∈ Mz}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then, pick any function e : [κ]<ω → [κ]2 satisfying that for every z ∈ [κ]<ω:  for every z ∈ [κ]<ω of size ≥ 2, e(z) ∈ [z]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  if Mz∗ is a singleton, then e(z) is its unique element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  To see that e is a sought, suppose that λ is a regular cardinal satisfying max{µ, θ+} ≤  λ ≤ κ, and that we are given ⟨xγ | γ < λ⟩, r ∈ [κ]<cf(θ) and σ < ω as in Defini-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the pigeonhole principle and the Dushnik-Miller theorem, we may find  a cofinal subset Γ ⊆ λ, an ordinal δ < θ, and a sequence ⟨tj | j < σ⟩ of nodes in  Tδ+1 such that for every (γ, γ′) ∈ [Γ]2:  (I) sup(∆“[r ∪ yγ]2) = δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (II) for every j < σ, byγ(j) ↾ (δ + 1) = tj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (III) for every j < σ, yγ(j) < yγ′(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' There exist Γ0, Γ1 ∈ [Γ]λ and a sequence ⟨(sj, ij, i′  j) | j < σ⟩ of  triples in T × µ × µ such that, for every j < σ:  SUMS OF TRIPLES IN ABELIAN GROUPS  13  for every γ ∈ Γ0, sj⌢⟨ij⟩ ⊑ byγ(j),  for every γ ∈ Γ1, sj⌢⟨i′  j⟩ ⊑ byγ(j), and  ij ̸= i′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We shall define by recursion a sequence of pairs ⟨(Aj, Bj) | j ≤ σ⟩ such that,  for all j < σ, Aj+1 ∈ [Aj ∩ Γ]λ and Bj+1 ∈ [Bj ∩ Γ]λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  We commence by letting both A0 and B0 be Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, for every j < σ for which  the pair (Aj, Bj) has already been defined, we do the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set Aj := {yγ(j) |  γ ∈ Aj} and Bj := {yγ(j) | γ ∈ Bj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Clause (III), Aj and Bj have size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, by  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4(2), we may find sj ∈ T and ij ̸= i′  j such that {α ∈ Aj | sj⌢⟨ij⟩ ⊑ bα}  and {β ∈ Bj | sj⌢⟨i′  j⟩ ⊑ bβ} are both of size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, let Aj+1 := {γ ∈ Aj |  sj⌢⟨ij⟩ ⊆ byγ(j)} and Bj+1 := {γ ∈ Bj | sj⌢⟨1 − ij⟩ ⊆ byγ(j)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Clearly, Γ0 := Aσ and Γ1 := Bσ are as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Let Γ0, Γ1 and ⟨(sj, ij, i′  j) | j < σ⟩ be given by the preceding claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note that  Γ0 is disjoint from Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set δ∗ := max{dom(sj) | j < σ} and j∗ := min{j < σ |  dom(sj) = δ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let (γ, γ′) ∈ (Γ0 ⊛ Γ1) ∪ (Γ1 ⊛ Γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let z ∈ [xγ ∪ xγ′]<ω be such that  {yγ(j∗), yγ′(j∗)} ⊆ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then e(z) = (yγ(j∗), yγ′(j∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It is clear that 2 ≤ otp(z) < ω, so that Mz is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let (α, β) ∈ [z]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By the choice of z, we must analyze the following cases:  (1) Suppose that α ∈ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As β ∈ r ∪ yγ ∪ yγ′, it follows from Clause (I) that ∆(α, β) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) Suppose that α ∈ yγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (a) If β ∈ r ∪ yγ, then it follows from Clause (I) that ∆(α, β) ≤ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (b) If β ∈ yγ′, then let jα, jβ < σ be such that, α = yγ(jα) and β = yγ′(jβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  There are two possible options:  (i) If jα = jβ = j, then by Clause (II), fα ↾ (δ + 1) = tj = fβ ↾  (δ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So ∆(α, β) > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (ii) If jα ̸= jβ, then by Clauses (I) and (II),  ∆(yγ′(jα), β) ≤ δ < ∆(α, yγ′(jα)),  and hence ∆(α, β) = ∆(yγ′(jα), β) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) If α ∈ yγ′, then the analysis is analogous to that of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Altogether, so far we have shown that  ∅ ⊊ Mz ⊆ {(yγ(j), yγ′(j)) | j < σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Recalling that (γ, γ′) ∈ (Γ0 ⊛ Γ1) ∪ (Γ1 ⊛ Γ0), we infer from the choice of δ∗ that  ∅ ⊊ Mz ⊆ {(yγ(j), yγ′(j)) | j < σ, dom(sj) = δ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So, since {yγ(j∗), yγ′(j∗)} ⊆ z, it is the case that M ∗  z = {(yγ(j∗), yγ′(j∗))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In  particular, e(z) = (yγ(j∗), yγ′(j∗)), as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that λ is an infinite regular cardinal, and ν < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  If there exists a cardinal θ < λ such that νθ ≥ λ, then Extract2(νθ, λ, cf(θ), ω)  holds for the least such θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  14  IDO FELDMAN AND ASSAF RINOT  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let θ denote the least cardinal such that νθ ≥ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then T := <θν belongs  to T (θ, λ), so that νθ = |B(T )| is in T (θ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, appeal to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9 with  (κ, µ) := (νθ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every regular uncountable cardinal κ that is not a strong limit,  Extract2(κ, κ, ω, ω) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that λ is a regular uncountable cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For every cardinal κ > 2<λ, Extract2(κ, λ, 2, 2) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set ν := 2<λ, and note that νθ = ν for every θ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Towards a contradiction,  suppose that e : [κ]<ω → [κ]2 is a map witnessing Extract2(κ, λ, 2, 2), and yet κ > ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Without loss of generality, we may assume that e(z) ∈ [z]2 for every z ∈ [κ]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every δ < κ, there exist no subset A ⊆ δ of order-type λ such  that δ ∈ e({α, β, δ}) for every (α, β) ∈ [A]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Otherwise, fix a counterexample δ and a witnessing A ⊆ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let ⟨αγ | γ < λ⟩  be the increasing enumeration of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now let r := {δ} and, for every γ < λ, put  xγ := yγ ⊎ r where yγ := {αγ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, for every (γ, γ′) ∈ [λ]2, setting z := xγ ∪ xγ′,  we get that r ∩ e(z) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Denote κ := ν+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It follows from the claim that for every δ ∈ Eκ  λ , we may fix some  Aδ ∈ [δ]<λ with the property that for every ordinal β such that sup(Aδ) < β < δ,  there exists α ∈ Aδ such that e({α, β, δ}) = (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, using Fodor’s lemma,  we may find ε < κ and θ < λ such that {δ ∈ Eκ  λ | ssup(Aδ) = ε & |Aδ| = θ} is  stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Recalling that νθ = ν < κ, we may then find some A ∈ [ε]θ for which  S := {δ ∈ Eκ  λ | Aδ = A} is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Define a coloring c : [S]2 → A by letting for  every (β, δ) ∈ [S]2:  c(β, δ) := min{α ∈ A | e({α, β, δ}) = (α, β)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By the Erd˝os-Rado theorem, κ → (λ)2  θ holds, so we may pick B ⊆ S of order-  type λ that is c-homogeneous, with value, say, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let ⟨βγ | γ < λ⟩ be the increasing  enumeration of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Finally, let r := {α} and, for every γ < λ, put xγ := yγ ⊎ r  where yγ := {βγ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, for every (γ, γ′) ∈ [λ]2, setting z := xγ ∪ xγ′, we get that  r ∩ e(z) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If κ is a strong limit cardinal, then Extract2(κ, λ, 2, 2) fails for  every infinite cardinal λ < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Extract2(ℵ2, ℵ1, ℵ0, ℵ0) holds iff CH fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='10 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Moving on from the case n = 2 to the general case, we consider the following  two definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Sn(κ, λ, θ) asserts the existence of a coloring d : [κ]<ω → θ such  that, for every X ⊆ [κ]<ω of size λ and every prescribed color τ < θ, there exist  {aj | j < n} ∈ [X]n such that d(z) = τ for every z satisfying  a0 △ (  �  0<j<n aj) ⊆ z ⊆  �  j<n aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='16 (monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that:  (1) 2 ≤ n ≤ n′ < ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  15  (2) ω ≤ λ ≤ λ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) θ ≤ θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then Sn(κ, λ, θ′) implies Sn′(κ, λ′, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This is mostly trivial, so we settle for proving that if d : [κ]<ω → θ witnesses  Sn(κ, λ, θ) for some integer n ≥ 2, then it also witnesses Sn+1(κ, λ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  To this end, let X ⊆ [κ]<ω be a given family of size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick x ∈ X, and note that  X ′ := {a ∪ x | a ∈ X \\ {x}} is a λ-sized subset of [κ]<ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, given a prescribed  color τ < θ, pick {a′  j | j < n} ∈ [X ′]n such that d(z) = τ for every z satisfying  a′  0 △ (  �  0<j<n a′  j) ⊆ z ⊆  �  j<n a′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For each j < n, pick aj ∈ X \\{x} such that a′  j = aj∪x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As ⟨a′  j | j < n⟩ is an injective  sequence, so is ⟨aj | j < n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set an := x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Altogether, {aj | j < n + 1} ∈ [X]n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It  is clear that �  j<n a′  j = �  j<n+1 aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition,  a′  0 △ (�  0<j<n a′  j)  =  (a′  0 \\ (�  0<j<n a′  j))  ∪  ((�  0<j<n a′  j) \\ a′  0)  ⊆  (a0 \\ (�  0<j<n aj))  ∪  ((�  0<j<n aj) \\ a0)  ⊆  (a0 \\ (�  0<j<n+1 aj))  ∪  ((�  0<j<n+1 aj) \\ a0)  =  a0 △ (�  0<j<n+1 aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Therefore, d(z) = τ for every set z with a0 △ (�  0<j<n+1 aj) ⊆ z ⊆ �  j<n+1 aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Extractn(κ, λ, θ, χ) asserts the existence of a map e : [κ]<ω → nκ  such that for every sequence ⟨xγ | γ < λ⟩ of subsets of κ, every r ∈ [κ]<θ, and every  nonzero σ < χ such that:  (1) for every (γ, γ′) ∈ [λ]2, xγ ∩ xγ′ ⊆ r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) for every γ < λ, yγ := xγ \\ r has order-type σ,  there exist j < σ and disjoint cofinal subsets Γ0, Γ1 of λ satisfying the following:  (a) For every (γ, γ′) ∈ [Γ0 ∪ Γ1]2, yγ(j) < yγ′(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (b) For every strictly increasing sequence ⟨γi | i < n⟩ of ordinals from Γ0 ∪ Γ1  such that {γi | i < n} /∈ ([Γ0]n ∪ [Γ1]n), for every z ∈ [�  i<n xγi]<ω that  covers {yγi(j) | i < n}, we have  e(z) = ⟨yγi(j) | i < n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Without loss of generality, we may assume that for every z ∈ [κ]<ω  of size ≥ n, e(z) consists of ordinals from z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The proof of [FBR17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7] makes it clear that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that Sn(κ, λ, θ) holds for given cardinals θ ≤ λ ≤ κ  with λ regular and uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every map ϕ : G → [G]<ω that is <λ-to-one,  there exists a corresponding coloring c : G → θ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For every binary operation ∗ on G such that, for all x ̸= y in G,  ϕ(x) △ ϕ(y) ⊆ ϕ(x ∗ y) ⊆ ϕ(x) ∪ ϕ(y),  for every X ∈ [G]λ and every τ < θ, there is an injective sequence ⟨xj | 1 ≤ j ≤ n⟩  of elements of X such that c(x1 ∗ · · · ∗ xn) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6  □  In order to generalize Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='8, we now introduce the relation κ  sup  �−→ [λ, λ]n  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  It is a strengthening of κ ↛ [λ]n  θ , and a weakening of κ ↛ [λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , λ]n  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  6As always, we mean that this holds true for all implementations of x1 ∗ · · · ∗ xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  16  IDO FELDMAN AND ASSAF RINOT  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' κ  sup  �−→ [λ, λ]n  θ asserts the existence of a coloring c : [κ]n → θ such  that for all τ < θ and disjoint A, B ∈ P(κ) satisfying the two:  (i) otp(A) = otp(B) = λ,  (ii) sup(A) = sup(B),  there is ⃗x ∈ [A ∪ B]n \\ ([A]n ∪ [B]n) with c(⃗x) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In the special case of λ = κ, κ  sup  �−→ [λ, λ]2  θ coincides with the classical  relation κ ↛ [λ, λ]2  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that:  2 ≤ n < ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  θ ≤ λ ≤ κ are cardinals with λ regular and uncountable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  κ  sup  �−→ [λ, λ]n  θ holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Extractn(κ, λ, ω, ω) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then Sn(κ, λ, θ) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The proof is similar to that of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='8, so we settle for a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Fix a  map c : [κ]n → θ witnessing κ  sup  �−→ [λ, λ]n  θ and a map e : [κ]<ω → nκ witnessing  Extractn(κ, λ, ω, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We claim that d := c ◦ e is a witness for Sn(κ, λ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To this  end, suppose that we are given a subfamily X ⊆ [κ]<ω of size λ, and a prescribed  color τ < θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the ∆-system lemma, find a sequence ⟨xγ | γ < λ⟩ consisting of  elements of X, some r ∈ [κ]<ω, and a nonzero σ < χ such that:  (1) for every (γ, γ′) ∈ [λ]2, xγ ∩ xγ′ = r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) for every γ < λ, yγ := xγ \\ r has order-type σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Now, let j < σ and disjoint cofinal subsets Γ0, Γ1 of λ be given, as in Defini-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Put A := {yγ(j) | γ ∈ Γ0} and B := {yγ(j) | γ ∈ Γ1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the choice of  c, pick ⃗x ∈ [A ∪ B]n \\ ([A]n ∪ [B]n) such that c(⃗x) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Find a sequence ⃗γ = ⟨γi |  i < n⟩ of ordinals from Γ0 ∪ Γ1 such that ⃗x = ⟨yγi(j) | i < n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clearly, ⃗γ is strictly  increasing, {γi | i < n} ⊈ Γ0 and {γi | i < n} ⊈ Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, for every set z of interest,  d(z) = c(e(z)) = c(⃗x) = τ,  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that κ ∈ T (µ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then there exists a map e : [κ]<ω → 3κ  witnessing Extract3(κ, λ, cf(θ), ω) for every regular cardinal λ with max{µ, θ+} ≤  λ ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As κ ∈ T (µ, θ), let us fix T ∈ T (µ, θ) admitting an injective sequence  ⟨bξ | ξ < κ⟩ consisting of elements of B(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For notational simplicity, we shall  write ∆(α, β) for ∆(bα, bβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For any triplet w ∈ [κ]3, let ∆2(w) := min{∆(α, β) |  (α, β) ∈ [w]2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' First, given z ∈ [κ]<ω, let:  Mz := {(α, β, γ) ∈ [z]3 | ∆2({α, β, γ}) = max{∆2(w) | w ∈ [z]3}}, and  M ∗  z := {(α, β, γ) ∈ Mz | α = min{α′ | (α′, β′, γ′) ∈ Mz}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then, pick any function e : [κ]<ω → 3κ satisfying that for every z ∈ [κ]<ω:  for every z ∈ [κ]<ω, e(z) is a strictly increasing sequence of ordinals in κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  If |z| ≥ 3, then e(z) consists of ordinals from z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  if Mz∗ is a singleton, then e(z) is its unique element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  17  To see that e is a sought, suppose that λ is a regular cardinal satisfying max{µ, θ+} ≤  λ ≤ κ, and that we are given ⟨xγ | γ < λ⟩, r ∈ [κ]<cf(θ) and σ < ω as in Defini-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9, we may find a cofinal subset Γ ⊆ λ, an  ordinal δ < θ, and a sequence ⟨tj | j < σ⟩ of nodes in Tδ+1 such that for every  (γ, γ′) ∈ [Γ]2:  (I) sup(∆“[r ∪ yγ]2) = δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (II) for every j < σ, byγ(j) ↾ (δ + 1) = tj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (III) for every j < σ, yγ(j) < yγ′(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In addition, we may fix Γ0, Γ1 ∈ [Γ]λ+ and a sequence ⟨(sj, ij, i′  j) | j < m⟩ of triples  in T × µ × µ such that, for every j < σ:  for every γ ∈ Γ0, sj⌢⟨ij⟩ ⊑ byγ(j),  for every γ ∈ Γ1, sj⌢⟨i′  j⟩ ⊑ byγ(j), and  ij ̸= i′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By possibly passing to cofinal subsets, we may assume that Γ0 ∩ Γ1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Let  δ∗ := max{dom(sj) | j < σ} and j∗ := min{j < σ | dom(sj) = δ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let (γ, γ′, γ′′) ∈ [Γ0∪Γ1]3\\([Γ0]3∪[Γ1]3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let z ∈ [xγ ∪xγ′ ∪xγ′′]<ω  be such that {yγ(j∗), yγ′(j∗), yγ′′(j∗)} ⊆ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then e(z) = (yγ(j∗), yγ′(j∗), yγ′′(j∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As made clear by the proof of Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2, for every (ξ, ζ) ∈ [{γ, γ′, γ′′}]2,  for every α ∈ r ∪ aξ and β ∈ r ∪ yζ, ∆(α, β) > δ iff there is a j < σ such that  α = yξ(j) and β = yζ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition, if {ξ, ζ} ⊈ Γ0 and {ξ, ζ} ⊈ Γ1, then for every  j < m, ∆(yξ(j), yζ(j)) = dom(sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, in this case,  {(α, β) ∈ [r ∪ yξ ∪ yζ]2 | ∆(α, β) = max(∆“[r ∪ yξ ∪ yζ]2)}  = {(yξ(j), yζ(j)) | dom(sj) = δ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Now, since {γ, γ′, γ′′} ⊈ Γ0 and {γ, γ′, γ′′} ⊈ Γ1, it follows that  ∆2(yγ(j∗), yγ′(j∗), yγ′′(j∗)) = δ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Consequently,  ∅ ⊊ Mz = {(yγ(j), yγ′(j), yγ′′(j)) | dom(sj) = δ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So, by Clause (III),  e(z) = (yγ(j∗), yγ′(j∗), yγ′′(j∗)),  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that λ ≤ κ is a pair of infinite cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  If Extract2(κ, λ, 3, 3) holds, then so does κ  sup  �−→ [λ, λ]4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that κ  sup  �−→ [λ, λ]4  2 fails, and we shall prove that Extract2(κ, λ, 3, 3)  fails, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To this end, let e : [κ]<ω → [κ]2 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Define a coloring c : [κ]4 → 2  by letting for all α0 < α1 < α2 < α3 < κ:  c(α0, α1, α2, α3) := 1 iff e(α0, α1, α2, α3) = (α2, α3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Now, since κ  sup  −→ [λ, λ]4  2 holds, we may find τ < 2 and disjoint A, B ∈ P(κ)  satisfying all of the following:  (i) otp(A) = otp(B) = λ,  (ii) sup(A) = sup(B),  18  IDO FELDMAN AND ASSAF RINOT  (iii) for every (α0, α1, α2, α3) ∈ [A ∪ B]4 \\ ([A]4 ∪ [B]4),  c(α0, α1, α2, α3) ̸= τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Using Clauses (i) and (ii), fix a sequence ⟨(αi, βi) | i < λ⟩ of pairs in A × B such  that, for all i < j < λ, αi < βi < αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ If τ = 1, then let r := {α0, α1}, and for every γ < λ, let xγ := r ⊎ yγ, where  yγ := {βγ+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, for every (γ, γ′) ∈ [λ]2, as z := xγ ∪ xγ′ is in [A ∪ B]4 \\ ([A]4 ∪  [B]4), c(z) = 0, and then e(z) is not disjoint from r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ If τ = 0, then let r := ∅, and for every γ < λ, let xγ := r ⊎ yγ, where yγ :=  {αγ, βγ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, for every (γ, γ′) ∈ [λ]2, as z := xγ ∪xγ′ is in [A∪B]4 \\ ([A]4 ∪[B]4),  c(z) = 1, and then e(z) = yγ′ which is disjoint from yγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If λ = ℵ0 or if λ is weakly compact, then Extract2(κ, λ, 3, 3) fails  for every cardinal κ ≥ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Maximal number of colors  This section is dedicated to the proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The main corollary of this  section reads as follows:  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that λ = µ+ for an infinite cardinal µ = µ<µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then the following are equivalent:  (1) (λ+, λ) ։ (µ+, µ) fails;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) λ+ sup  �−→ [λ, λ]3  λ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We focus on the nontrivial (that is, forward) implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As λ = µ+, by  [Tod07, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3], the failure of (λ+, λ) ։ (µ+, µ) is equivalent to the existence  of a subadditive coloring ̺ : [λ+]2 → λ witnessing U(λ+, λ, λ, λ, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular,  ̺ ↾ [X]2 witnesses U(λ, λ, λ, 3) for every X ⊆ λ+ of order-type λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, there are  two cases to consider:  ◮ If 2µ > µ+, then since µ<µ = µ, T := <µµ is a weak µ-Kurepa tree with  λ+-many branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition, µ<µ = µ implies that µ is regular, so that Eλ  µ is  a nonreflecting stationary set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now the result follows from the upcoming Theo-  rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2, using κ := λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ If 2µ = µ+, then λ ↛ [µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  λ holds by a theorem of Sierpi´nski (see [IR23,  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now the result follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  When reading the hypotheses of the upcoming theorem, it may worth keeping  in mind that if λ = µ+ for an infinite regular cardinal µ, then Eλ  µ is a nonreflecting  stationary set, and if κ = λ+, then Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14 provides a subadditive map ρ : [κ]2 →  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The conclusion of the theorem is a conditional form of κ  sup  �−→ [λ, λ]3  λ in which a  third clause is added to Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that:  µ < λ < κ are infinite regular cardinals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Eλ  µ admits a nonreflecting stationary set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ̺ : [κ]2 → λ is a subadditive coloring of pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  there exists a weak µ-Kurepa tree with at least κ-many branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then there exists a corresponding coloring of triples c : [κ]3 → λ such that, for  all τ < λ and disjoint A, B ∈ P(κ) satisfying the three:  (i) otp(A) = otp(B) = λ,  SUMS OF TRIPLES IN ABELIAN GROUPS  19  (ii) sup(A) = sup(B),  (iii) ̺ ↾ [A ∪ B]2 witnesses U(λ, λ, λ, 3),  there exists (α, β, γ) ∈ [A ∪ B]3 \\ ([A]3 ∪ [B]3) such that c(α, β, γ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let T ⊆ <µ2 be a weak µ-Kurepa tree with at least κ-many branches, and  let ⃗b = ⟨bξ | ξ < κ⟩ be an injective sequence consisting of elements of B(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For  all α ̸= β from κ, we write ∆(α, β) for ∆(bα, bβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For all B ⊆ κ and t ∈ T , denote  Bt := {β ∈ B | t ⊑ bβ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As Eλ  µ admits a nonreflecting stationary set, by [LHR23,  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='31], we may fix a coloring e : [λ]2 → µ for which the following set is  stationary:  ∂(e) := {σ ∈ Eλ  µ | ∀ǫ ∈ λ \\ σ ∀δ < µ [sup{ζ < σ | e(ζ, ǫ) ≤ δ} < σ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Let h : λ → λ be a surjection such that Sτ := {σ ∈ ∂(e) | h(σ) = τ} is stationary  for every τ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For every (α, β, γ) ∈ κ × κ × κ, let  Z(α,β,γ) := {ζ ∈ λ \\ ̺({α, β}) | max(∆“{α, β, γ}2) ≥ e(ζ, ̺({β, γ}))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Now, derive a coloring c : [κ]3 → λ by letting:  c({α, β, γ}) :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  h(min(Z(β,α,γ))),  if γ = max{α, β, γ} & bα <lex bβ <lex bγ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  h(min(Z(α,β,γ))),  if γ = max{α, β, γ} & bγ <lex bα <lex bβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Suppose that A, B ∈ P(κ) are disjoint sets satisfying conditions (i)–(iii) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Recalling Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4(2), by possibly shrinking A and B, we may assume the exis-  tence of some χ < µ such that ∆[A × B] = {χ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By possibly switching the roles of  A and B, we may also assume the following:  (iv) for every (α, β) ∈ A × B, bα <lex bβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Note that |T | = µ < λ < κ ≤ 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, given a prescribed color τ < λ, let M be  an elementary submodel of H(2µ)+ containing {A, B, ̺,⃗b, T } such that σ := M ∩λ is  in Sτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, |M| ≥ cf(σ) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote υ := sup(A) and υM := sup(M ∩ υ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The proof is now divided into two cases:  Case 1: For every β ∈ B, sup(̺β“A) < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Condition (iii), find A′ ∈ [A]λ  and B′ ∈ [B]λ such that min(̺[A′ ⊛ B′]) > σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Fix α ∈ A′ \\ υM arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Appeal  to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5 with X := B′ and i := 1 to pick γ ∈ B′ \\ (α + 1) such that for  cofinally many δ < µ, the two hold:  (1) bγ(δ) = 1, and  (2) {β ∈ B′ | ∆(β, γ) = δ} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Denote D := {δ < µ | Clauses (1) and (2) both hold}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As (α, γ) ∈ A′ ⊛ B′, the ordinal ǫ := ̺(α, γ) is bigger than σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick δ ∈ D above  max{χ, e(σ, ǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since σ ∈ ∂(e), the following set is bounded below σ:  Z := {ζ < σ | e(ζ, ǫ) ≤ δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Set t := (bγ ↾ δ)⌢⟨0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' From δ ∈ D we infer that (B′)t has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As t ∈ T ⊆ M,  in particular, Bt is a set of size λ lying in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Condition (iii) and elementarity  of M one can find β0 ̸= β1 in Bt ∩ M such that ̺(β0, β1) > sup(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As ̺ is  subadditive, we may now find β ∈ {β0, β1} such that ̺(β, α) > sup(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As β ∈ B,  sup(̺β“A) < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As {̺, A} ∈ M, it follows that sup(̺β“A) ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular,  ̺(β, α) < σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  20  IDO FELDMAN AND ASSAF RINOT  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' All of the following hold:  (1) (β, α, γ) ∈ B ⊛ A ⊛ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) bα <lex bβ <lex bγ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) max(∆“{α, β, γ}2) = δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (4) min(Z(β,α,γ)) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (1) γ was chosen to be in B \\ (α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition, α ∈ A \\ υM, whereas  β ∈ M ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) As α ∈ A and β ∈ B, Condition (iv) entails that bα <lex bβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition,  as bβ ↾ (δ + 1) = t = (bγ ↾ δ)⌢⟨0⟩ and bγ(δ) = 1, we get that bβ <lex bγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) By the previous analysis, ∆(β, γ) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In addition, ∆(α, γ) = χ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Recalling that |∆“{α, β, γ}2| = 2, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (4) By Clause (3) and the fact that ̺(α, γ) = ǫ, we infer that Z(β,α,γ) = {ζ ∈ λ\\  ̺(β, α) | δ ≥ e(ζ, ǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, σ ∈ Z(β,α,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, if ζ := min(Z(β,α,γ))  is < σ, then ζ ∈ Z, contradicting the fact that ̺(β, α) > sup(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  By the preceding claim and the definition of c,  c({α, β, γ}) = h(min(Z(β,α,γ))) = h(σ) = τ,  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Case 2: There is β ∈ B such that sup(̺β“A) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As {̺, A} ∈ M, we may  pick β ∈ B ∩ M such that sup(̺β“A) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clearly, |̺β“A| = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Define a function  f : ̺β“A → A via  f(ξ) := min{α ∈ A | ̺(β, α) = ξ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As {β, A, ̺} ∈ M, we infer that ̺β“A, f and Im(f) are all in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note that f is  injective, so that | Im(f)| = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It also follows that  Γ := {γ ∈ Im(f) \\ (β + 1) | ̺(β, γ) ≤ σ}  is bounded in Im(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Appeal to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5 with X := Im(f) \\ (β + 1) and i := 0 to pick γ ∈  X \\ (Γ ∪ υM) such that for cofinally many δ < µ, the two hold:  (1) bγ(δ) = 0, and  (2) {α ∈ X | ∆(α, γ) = δ} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Denote D := {δ < µ | Clauses (1) and (2) both hold}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As γ /∈ Γ, ǫ := ̺(β, γ) is bigger than σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick δ ∈ D above max{χ, e(σ, ǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since  σ ∈ ∂(e), the following set is bounded below σ:  Z := {ζ < σ | e(ζ, ǫ) ≤ δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Set t := (bγ ↾ δ)⌢⟨1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  From δ ∈ D we infer that Xt is a set of size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As  t ∈ T ⊆ M and X ∈ M, Xt is in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As α �→ ̺(β, α) is injective over X, we  may find an α ∈ Xt ∩ M such that ̺(β, α) > sup(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Because of the fact that  {β, α} ∈ M, we altogether get that sup(Z) < ̺(β, α) < σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' All of the following hold:  (1) (β, α, γ) ∈ B ⊛ A ⊛ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) bγ <lex bα <lex bβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) max(∆“{α, β, γ}2) = δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (4) min(Z(α,β,γ)) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  21  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (1) β was chosen to be in B ∩M, γ was chosen to be in A\\υM, whereas  α ∈ M ∩ A with α > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) As α ∈ A and β ∈ B, Condition (iv) entails that bα <lex bβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition,  bα ↾ (δ + 1) = t = (bγ ↾ δ)⌢⟨1⟩ and bγ(δ) = 0, so bγ <lex bα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) By the previous analysis, ∆(α, γ) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In addition, ∆(α, β) = χ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Recalling that |∆“{α, β, γ}2| = 2, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (4) By Clause (3) and the fact that ̺(β, γ) = ǫ, we infer that Z(α,β,γ) = {ζ ∈ λ\\  ̺(β, α) | δ ≥ e(ζ, ǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, σ ∈ Z(α,β,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, if ζ := min(Z(α,β,γ))  is < σ, then ζ ∈ Z, contradicting the fact that ̺(β, α) > sup(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  By the preceding claim and the definition of c,  c({α, β, γ}) = h(min(Z(α,β,γ))) = h(σ) = τ,  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that:  µ = µ<µ is an infinite cardinal, λ = µ+ and κ = λ+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ̺ : [κ]2 → λ is a subadditive coloring of pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  λ ↛ [µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  λ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then, there exists a corresponding coloring of triples c : [κ]3 → λ such that, for  all τ < λ and disjoint A, B ∈ P(κ) satisfying the three:  (i) otp(A) = otp(B) = λ,  (ii) sup(A) = sup(B),  (iii) ̺ ↾ [A ∪ B]2 witnesses U(λ, λ, λ, 3),  there exists (α, β, γ) ∈ [A ∪ B]3 \\ ([A]3 ∪ [B]3) such that c(α, β, γ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let d : [κ]2 → λ be a coloring witnessing λ ↛ [µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let T := <λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let  ⃗b = ⟨bξ | ξ < κ⟩ be an injective enumeration of elements of B(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For α ̸= β from κ,  we write ∆(α, β) for ∆(bα, bβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Likewise, for B ⊆ κ, we write T ⇝B for T ⇝{bβ|β∈B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For all B ⊆ κ and t ∈ T , denote Bt := {β ∈ B | t ⊑ bβ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let e : [λ]2 → µ be a  map with injective fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let h : λ → λ be a surjection such that Sτ := {σ ∈ Eλ  µ |  h(σ) = τ} is stationary for every τ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every (α, β, γ) ∈ κ ⊛ κ ⊛ κ, define:  Z(α,β,γ) := {ζ ∈ λ \\ ̺(α, γ) | e(∆(α, β), ̺(β, γ)) ≥ e(ζ, ̺(β, γ))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  We define a coloring c : [κ]3 → λ by letting for all α < β < γ < κ:  c(α, β, γ) :=  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  d(∆(α, γ), ∆(β, γ)),  if ∆(α, β) < ∆(β, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  d(∆(α, γ), ̺(β, γ)),  if ∆(α, β) = ∆(β, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  d(∆(α, β), ̺(α, γ)),  if ∆(α, β) > ∆(β, γ) & bα <lex bβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  h(min(Z(α,β,γ))),  if ∆(α, β) > ∆(β, γ) & bβ <lex bα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Suppose that A, B ∈ P(κ) are disjoint sets satisfying conditions (i)–(iii) above  and let τ < λ be a prescribed color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By possibly passing to a cofinal subset of A, we  may assume that A = A′ in the sense of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, we may assume  the existence of θA ≤ λ such that T ⇝A ∈ T (λ, θA) and, in addition, if θA < λ,  then |B(T ⇝A)| = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As µ<µ < µ+ = λ, this means that if θA < λ, then θA ∈ Eλ  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Likewise, we may assume the existence of θB ∈ Eλ  µ ∪{λ} such that T ⇝B ∈ T (λ, θB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Without loss of generality, we may also assume that θA ≤ θB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The proof is now  divided into two main cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Case 1: θA = θB = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In this case, we shall need the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  22  IDO FELDMAN AND ASSAF RINOT  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' There exists t ∈ T ⇝B satisfying all of the following:  If t /∈ T ⇝A, then D := {∆(bβ, t) | β ∈ B} has size µ, and there exists  t′ ∈ T ⇝A incompatible with t such that sup(D) > ∆(t, t′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  If t ∈ T ⇝A, then D := {∆(bα, t) | α ∈ A} has size µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' There are two cases to consider:  ◮ Suppose that there exists ǫ < λ such that T ⇝A ∩ T ⇝B ∩ ǫ2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Pick  t′ ∈ T ⇝A ∩ ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For each α < λ, pick tα ∈ T ⇝B ∩ α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6, there  exists α ∈ Eλ  µ above ǫ such that D := {∆(bβ, tα) | β ∈ B} ∩ α is cofinal in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To  see that t := tα is as sought, notice that since t ↾ ǫ /∈ T ⇝A, it must be the case that  ∆(t, t′) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ Otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Thus, for each α < λ, we may pick tα ∈ T ⇝A ∩ T ⇝B ∩ α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As ⟨tα | α < λ⟩ ∈ �  α<λ T ⇝A ∩ α2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6 provides an α ∈ Eλ  µ such that  D := {∆(bβ, tα) | β ∈ A} ∩ α is cofinal in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So t := tα is as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Let t ∈ T ⇝B and the corresponding D be as in the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' There are two subcases  to consider:  Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1: t ∈ T ⇝A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick ¯A ∈ [A]µ such that ¯D := {∆(bα, t) | α ∈ ¯A} is a  µ-sized subset of D ∩ dom(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clearly, B′ := Bt \\ sup( ¯A) is a set of size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since  T ⇝B′ ∈ T (λ, λ), E := ∆[B′ ⊛ B′] is of size λ, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the choice of d, we may  now find (δ, ǫ) ∈ ¯D ⊛ E such that d(δ, ǫ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick α ∈ ¯A such that ∆(bα, t) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Finally, find (β, γ) ∈ B′ ⊛ B′ such that ∆(β, γ) = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then  ∆(β, γ) = ǫ > δ = ∆(bα, t) = ∆(α, γ),  and hence ∆(α, β) = ∆(α, γ) < ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Altogether, (α, β, γ) ∈ A ⊛ B ⊛ B, and  c(α, β, γ) = d(∆(α, γ), ∆(β, γ)) = d(δ, ǫ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2: t /∈ T ⇝A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick t′ ∈ T ⇝A incompatible with t such that sup(D) >  ∆(t, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As cf(θB) = µ, we may now pick ¯B ∈ [B]µ such that ¯D := {∆(bβ, t) |  β ∈ ¯B} is a µ-sized subset of D with min( ¯D) > ∆(t, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As t′ ∈ T ⇝A, At′ has size λ and so does A′ := At′ \\ sup( ¯B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As t ∈ T ⇝B, Bt  has size λ, so since ̺ ↾ [A ∪ B]2 witnesses U(λ, 2, λ, 3), the set E := ̺[A′ ⊛ Bt] has  size λ, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the choice of d, find (δ, ǫ) ∈ ¯D ⊛ E such that d(δ, ǫ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Find  α ∈ ¯B such that ∆(bα, t) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Find (β, γ) ∈ A′ ⊛ Bt such that ̺(β, γ) = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As  (β, γ) ∈ At′ ⊛ Bt,  ∆(β, γ) = ∆(t′, t) < min( ¯D) ≤ δ = ∆(bα, t) = ∆(α, γ),  and hence ∆(α, β) = ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Altogether, (α, β, γ) ∈ B ⊛ A ⊛ B, and  c(α, β, γ) = d(∆(α, γ), ̺(β, γ)) = d(δ, ǫ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Case 2: θA < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set θ := θA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We shall need the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' There exist χ < θ, A′ ∈ [A]λ and B′ ∈ [B]λ such that ∆[A′ × B′] =  {χ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote ¯A := B(T ⇝A) and ¯B := B(T ⇝B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Recall that by our application of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7, | ¯A| = λ, and if θB < λ, then | ¯B| = λ, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We shall prove the claim  by showing that there exist χ < θ and a pair (t, t′) ∈ (T ⇝A)χ+1 × (T ⇝B)χ+1 such  that ∆(t, t′) = χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Indeed, once we have such a pair (t, t′), the sets A′ := At and  B′ := Bt′ would be as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  23  There are two cases to consider:  ◮ If θB = θ, then set ¯T := T ∩ <θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In this case, ¯B and ¯A are λ-sized subsets  of B( ¯T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4(2) yields an s ∈ ¯T together with i ̸= i′ such that s⌢⟨i⟩ ∈  ¯T ⇝ ¯  A ⊆ T ⇝A and s⌢⟨i′⟩ ∈ ¯T ⇝ ¯  B ⊆ T ⇝B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Evidently, χ := dom(s), t := s⌢⟨i⟩ and  t′ := s⌢⟨i′⟩ are as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ If θB > θ, then pick r ∈ (T ⇝B)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every a ∈ ¯A \\ {r}, χa := ∆(a, r) is  smaller than θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As | ¯A| = λ, we can find χ < θ such that λ-many a’s in ¯A\\{r} satisfy  χa = χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As the χth level of T ⇝A has size < λ, we may then find t ∈ (T ⇝A)χ+1  such that that λ-many a’s in ¯A \\ {r} satisfy χa = χ and a ↾ (χ + 1) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clearly,  χ, t and t′ := r ↾ (χ + 1) are as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Let χ be given by the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For notational simplicity, we shall assume that  ∆[A× B] = {χ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let M be an elementary submodel of H(2λ)+ containing {χ, A, B,  T ⇝A, T ⇝B, ̺, d} such that σ := M ∩ λ is in Sτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, |M| = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote  υ := sup(A) and υM := sup(M ∩ υ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Note that since T ⇝A ∈ T (λ, θ) and as  λ = µ+ > |θ|, it follows that T ⇝A has size ≤ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, T ⇝A ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Consider the following sets:  A0 := {α ∈ A | |̺α[B]| = λ},  A1 := {α ∈ A | |̺α[B]| ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Observe that A0, A1 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We examine two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1: A0 has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Appeal to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5 with X := A0 and i := 0  to pick α ∈ A0 such that for cofinally many δ < θ, the two hold:  (1) bα(δ) = 0, and  (2) {β ∈ A | ∆(α, β) = δ} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Since θ ∈ Eλ  µ, D := {δ < θ | Clauses (1) and (2) both hold} has size µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For each  δ ∈ D, use Clause (2) to fix βδ ∈ A above α such that ∆(α, βδ) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Consider ς := sup{βδ | δ < µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As |B ∩ ς| ≤ µ, the fact that α ∈ A0 implies that  E := ̺α[B \\ ς] has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the choice of d, then, we may pick δ ∈ D \\ (χ+ 1) and  ǫ ∈ E above δ such that d(δ, ǫ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick γ ∈ B \\ ς such that ǫ = ̺(α, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clearly,  α < βδ < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Recall that ∆(α, βδ) = δ > χ = ∆(βδ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since bα(δ) = 0, we conclude that  bβδ(δ) = 1 and bα <lex bβδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Altogether, (α, βδ, γ) ∈ A ⊛ A ⊛ B, and  c(α, βδ, γ) = d(∆(α, βδ), ̺(α, γ)) = d(δ, ǫ) = τ,  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2: A1 has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Condition (iii), find A′ ∈ [A1]λ and B′ ∈ [B]λ  such that min(̺[A′ ⊛ B′]) > σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Appeal to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5 with X := A′ and i := 0  to pick β ∈ A′ \\ υM such that for cofinally many δ < θ, the two hold:  (1) bβ(δ) = 0, and  (2) {α ∈ A′ | ∆(α, β) = δ} has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Since θ ∈ Eλ  µ, D := {δ < θ | Clauses (1) and (2) both hold} has size µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Pick γ ∈ B′ \\ (β + 1) arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As (β, γ) ∈ A′ ⊛ B′, the ordinal ǫ := ̺(β, γ)  is bigger than σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since eǫ is an injection to µ = |D|, we may pick δ ∈ D such  that e(δ, ǫ) > max{e(χ, ǫ), e(σ, ǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition, since µ = cf(σ), the following set  is bounded below σ:  Z := {ζ < σ | e(ζ, ǫ) ≤ e(δ, ǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  24  IDO FELDMAN AND ASSAF RINOT  Set t := (bβ ↾ δ)⌢⟨1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' From δ ∈ D we infer that (A′)t has size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As t ∈ T ⇝A ⊆ M,  in particular, (A1)t is a set of size λ lying in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Condition (iii) and elementarity  of M one can find α0 ̸= α1 in (A1)t ∩ M such that ̺(α0, α1) > sup(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As ̺ is  subadditive, we may now find α ∈ {α0, α1} such that ̺(α, γ) > sup(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note that,  as {α, B, ̺} ∈ M, and as α ∈ A1, ̺α[B] is a set of size no more than µ lying in M,  so that sup(̺α[B]) ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, ̺(α, γ) < σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' All of the following hold:  (1) (α, β, γ) ∈ A ⊛ A ⊛ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) ∆(α, β) > ∆(β, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) bβ <lex bα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (4) min(Z(α,β,γ)) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (1) γ was chosen to be in B \\ (β + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition, β ∈ A′ \\ υM, whereas  α ∈ M ∩ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) Since ∆(α, β) = ∆(t, bβ) = δ > χ = ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) By the definition of t and since δ ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (4) As δ = ∆(α, β) and ̺(β, γ) = ǫ, we infer that Z(α,β,γ) = {ζ ∈ λ \\ ̺(α, γ) |  e(δ, ǫ) ≥ e(ζ, ǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, σ ∈ Z(α,β,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, if ζ := min(Z(α,β,γ)) is  below σ, then ζ ∈ Z, contradicting the fact ̺(α, γ) > sup(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  By the preceding claim and the definition of c,  c(α, β, γ) = h(min(Z(α,β,γ)) = h(σ) = τ,  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Countably many colors  The main result of this section asserts that  λ+ sup  �−→ [λ, λ]3  ω  holds, provided that λ = µ+ for an infinite cardinal µ = µ<µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The idea of the  proof is to build on the colorings c : [λ+]3 → λ given by Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3 with  respect to the subadditive coloring ρ : [λ+]2 → λ given by Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Clause (iii)  of these theorems, we must address the problematic case in which the two sets A, B  of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='20 do not satisfy that ρ ↾ [A ∪ B]2 witnesses U(λ, λ, λ, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Anyone  that is familiar with [Tod07, §10] would probably suggest to use the oscillation  of [Tod07, §8] in this problematic case, and this indeed works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Unfortunately, to  verify that this works in the rectangular context, we had to reopen and tweak the  proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The experts may want to skip directly to Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The newcomers  may benefit from the modular exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Setup 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For the rest of this section, κ stands for a regular uncountable cardinal,  Υ is a large enough regular cardinal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=', (2κ)+), and we fix some C-sequence  ⃗C = ⟨Cβ | β < κ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We shall also assume that otp(Cβ) = cf(β) for all β < κ, though  this will only play a role in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='16 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The items of the next definition correspond to Definitions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1  of [Tod07], where the last item is a non-essential strengthening of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' A subset Γ ⊆ κ with cf(otp(Γ)) > ω is said to be:  ⃗C-stationary iff �  β∈Γ(acc(Cβ) ∪ {β}) is stationary in sup(Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  25  ⃗C-nontrivial iff for every club D ⊆ sup(Γ), there exists α ∈ Γ such that  D ∩ α ⊈ Cβ for all β ∈ Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ⃗C-oscillating iff for every club D ⊆ sup(Γ), there exist β ∈ Γ and an  increasing sequence ⟨δj | j < ω⟩ of ordinals in D \\ Cβ such that (δj, δj+1) ∩  Cβ ̸= ∅ for all j < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that Γ ⊆ κ is such that cf(otp(Γ)) > ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  If Γ is ⃗C-nontrivial and ⃗C-stationary, then Γ is ⃗C-oscillating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that Γ is ⃗C-nontrivial and ⃗C-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By the latter, ∆ :=  �  β∈Γ(acc(Cβ) ∪ {β}) is a stationary subset of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For each δ ∈ ∆, pick βδ ∈ Γ  such that sup(Cβδ ∩ δ) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Next, to verify that Γ is ⃗C-oscillating, let D ⊆ sup(Γ) be a given club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' There exists δ ∈ ∆ such that sup(D ∩ δ \\ Cβδ) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, for every δ ∈ ∆, ǫδ := sup(D ∩ δ \\ Cβδ) is smaller than  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Fix ǫ < sup(Γ) for which S := {δ ∈ ∆ | ǫδ < ǫ < δ} is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, consider  the club D′ := D \\ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, for every α ∈ Γ, letting δ := min(S \\ α), it is the case  that D′ ∩α ⊆ D ∩[ǫ, δ) ⊆ Cβδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This contradicts the fact that Γ is ⃗C-nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Let δ be given by the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As sup(D ∩δ \\Cβδ) = δ = sup(Cβδ ∩δ), it is easy to  recursively construct an increasing sequence ⟨δj | j < ω⟩ of ordinals in D ∩ δ \\ Cβδ  such that (δj, δj+1) ∩ Cβδ ̸= ∅ for all j < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For two disjoint sets of ordinals y and z, we say that P is a  y-convex subset of z iff one of the following occurs:  P = {ζ ∈ z | ζ < α} and α = min(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  P = {ζ ∈ z | β < ζ} and β = max(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  P = {ζ ∈ z | α < ζ < β} and α < β are two consecutive elements of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Note that if P and Q are nonempty y-convex subsets of z, then either P < Q or  Q < P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5 (Todorˇcevi´c, [Tod07, §8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For an ordinal ε < κ, define a function  Oscε : [P(κ)]2 → P(P(κ)) via  Oscε(x, y) :=  �  {P | P is a nonempty y-convex subset of x \\ ε},  if y ∩ x ⊆ ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ∅,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then the oscillation mapping oscε : [P(κ)]2 → CARD(κ + 1) is defined via  oscε(x, y) := | Oscε(x, y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (1) If we omit the subscript ε, then Osc(x, y) and osc(x, y) are  understood to be Oscε(x, y) and oscε(x, y) for ε := ssup(x ∩ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) For all ε < α < β < κ such that Cα∩Cβ ⊆ ε, Cα\\ε and Cβ have no common  accumulation points, and hence Oscε(Cα, Cβ) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In this case, we shall  identify Oscε(Cα, Cβ) with its increasing enumeration ⟨P0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' , Pn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The next lemma makes explicit some of the features that are present in the proof  of [Tod07, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7 (Todorˇcevi´c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that Γ is a cofinal subset of some θ ≤ κ of  uncountable cofinality, and that Γ is ⃗C-oscillating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every cofinal E ⊆ θ, there  26  IDO FELDMAN AND ASSAF RINOT  exists β ∈ Γ such that for every positive integer n, there are α ∈ Γ∩β and ε ∈ E ∩α  such that all of the following hold:  oscε(Cα, Cβ) = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  for every j < n, there is a pair ǫ < ǫ′ of ordinals in E \\ Cα for which  Oscε(Cα, Cβ)(j) = Cα ∩ (ǫ, ǫ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set µ := cf(θ), and fix a map ψ : µ → θ whose image is cofinal in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let M  be a continuous ∈-chain of length µ consisting of elementary submodels M ≺ HΥ  with M ∩µ ∈ µ and {ψ, ⃗C, Γ, E} ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' It follows that D := {sup(M ∩θ) | M ∈ M}  constitutes a club in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Recalling that Γ is ⃗C-oscillating, pick β ∈ Γ and an  increasing sequence ⟨δj | j < ω⟩ of ordinals in D \\ Cβ such that (δj, δj+1) ∩ Cβ ̸= ∅  for all j < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By possibly replacing δj by δj+1, we may assume that Cβ ∩ δ0  is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For every j < ω, since δj ∈ D \\ Cβ, pick Mj ∈ M such that  sup(Mj ∩ θ) = δj, and note that γj := sup(Cβ ∩ δj) and Ωj := min(Mj ∩ θ \\ γj) are  both smaller than δj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, for every j < ω:  0 < sup(Cβ ∩ δj) = γj ≤ Ωj < δj < γj+1 < β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For each k < ω, let Ik denote the collection of all increasing sequences ⃗I = ⟨Ij |  j ≤ k⟩ of closed intervals in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, let n be a positive integer and we shall find  α ∈ Γ ∩ β and ε ∈ E ∩ α as in the conclusion of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Define a sequence of collections ⟨Fn−i | i ≤ n⟩ by recursion on i ≤ n, as follows:  ◮ For i = 0, let Fn be the set of all ⟨Ij | j ≤ n⟩ ∈ In such that the following  two hold:  (1) I0 = [0, Ω0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) α := max(In) belongs to Γ, Cα ⊆ I0 ∪ · · · ∪ In, and Cα ∩ Ij ̸= ∅ for every  j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ For every i < n such that Fn−i has already been defined, let Fn−i−1 be the  collection of all ⃗I ∈ In−i−1 with the property that for every ǫ < θ there exists a  closed interval I ⊆ (ǫ, θ) such that ⃗I⌢⟨I⟩ ∈ Fn−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' ⟨[0, Ω0]⟩ ∈ F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every j ≤ n, define:  Ij :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  [0, Ω0],  if j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  [δj−1, Ωj],  if 0 < j < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  [δn−1, β],  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  We shall prove by induction on i ≤ n that ⟨Ij | j ≤ n− i⟩ ∈ Fn−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The base case  is immediate, since ⟨Ij | j ≤ n⟩ satisfies requirements (1) and (2), with β playing  the role of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Next, suppose that we are given i < n for which ⟨Ij | j ≤ n− i⟩ ∈ Fn−i has been  established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note:  ⟨Fk | k ≤ n⟩ ∈ M0 ⊆ Mn−i−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ⟨Ij | j ≤ n − i − 1⟩ ∈ Mn−i−1 ∩ Fn−i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In−i ∈ Mn−i \\ Mn−i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So, by elementarity of Mn−i−1, ⟨Ij | j ≤ n − i − 1⟩ ∈ Fn−i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  It follows that we may recursively construct a sequence ⟨Ij | j ≤ n⟩ such that:  (3) I0 = [0, Ω0], so that ⟨I0⟩ ∈ F0 ∩ M0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  27  (4) ⟨Ij | j ≤ k + 1⟩ ∈ Fk+1 ∩ Mk for every k < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (5) Ij+1 ⊆ (min(E \\ Ωj + 1), θ) for every j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For each j < n, denote ǫj := min(E \\Ωj +1), and note that since Ij+1 and E are  in Mj, ǫ′  j := min(E \\ max(Ij+1) + 1) is < δj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote γ′  j := min(Cβ \\ γj + 1) so that  γj < γ′  j are two consecutive elements of Cβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since sup(Cβ ∩ δj) = γj, altogether,  Ij+1 ⊆ (ǫj, ǫ′  j) ⊆ (Ωj, δj) ⊆ (γj, γ′  j) ⊆ (γj, γj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Now, put α := max(In).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then α ∈ Γ ∩ δn−1 ⊆ Γ ∩ β, Cα ⊆ I0 ∪ · · · ∪ In and  Cα ∩ Ij ̸= ∅ for every j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So (Cα \\ I0) ⊆ �  j<n(Ωj, δj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' On the other hand,  (Cβ \\ I0) ∩ (�  j<n(Ωj, δj)) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Therefore, for ε := ǫ0, we get that  Cα ∩ Cβ ⊆ (Ω0 + 1) ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' {ǫj, ǫ′  j | j < n} ∩ Cα = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose not, and fix j < n such that {ǫj, ǫ′  j} ∩ Cα ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As max(I0) = Ω0 ≤  Ωj < ǫj < ǫ′  j, we may fix some i < n such that {ǫj, ǫ′  j} ∩ Ii+1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Recalling that  Ij+1 ⊆ (ǫj, ǫ′  j), it must be the case that i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note:  ◮ If i < j, then γi+1 ≤ Ωi+1 ≤ Ωj < ǫj < ǫ′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ If i > j, then ǫj < ǫ′  j < δj < γj+1 ≤ γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So, both options contradict the fact that Ii+1 ⊆ (γi, γi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  By Clause (2), for every j ≤ n, Cα ∩ Ij+1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Altogether, for every ς ∈ (Ω0, ε]:  Oscς(Cα, Cβ) = ⟨Cα ∩ Ij+1 | j < n⟩  = ⟨Cα ∩ (γj, γ′  j) | j < n⟩  = ⟨Cα ∩ (ǫj, ǫ′  j) | j < n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In particular, oscε(Cα, Cβ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In the preceding proof, in the special case that κ = θ or κ = (cf(θ))+,  one can secure that Ω0 be equal to γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, in this case, we would get that max(Cα ∩  Cβ) = Ω0, meaning that the conclusion of the lemma remains valid also after  omitting the subscript ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Define χ : [κ]3 → ω by letting for all α < β < κ:  χ(α, β, γ) := max{k < ω | Tr(α, γ)(k) = Tr(β, γ)(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='10 ([Tod07, Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' A subset A ⊆ κ is said to be stable  if χ“[A]3 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Otherwise, we say that A is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Similar to [Tod07, Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3], we use χ to derive the following stepping-up  of the two-dimensional oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The three-dimensional oscillation mapping, osc : [κ]3 → ω is  defined on the basis of the two-dimensional oscillation defined in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5 via:  osc(α, β, γ) := oscα(CTr(α,β)(χ(α,β,γ)), CTr(α,γ)(χ(α,β,γ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  We now verify a rectangular version of [Tod07, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4]:  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12 (Todorˇcevi´c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that B is a cofinal subset of some θ ≤ κ of  uncountable cofinality, and that every cofinal subset of B is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then, for every cofinal A ⊆ θ and every positive integer n, there exists (α, β, γ) ∈  A ⊛ B ⊛ B such that osc(α, β, γ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  28  IDO FELDMAN AND ASSAF RINOT  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For each δ < θ, let βδ := min(B \\ (δ + 1)) and Λδ := λ2(δ, βδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Fodor’s  lemma, fix Λ < θ, k < ω and a stationary S ⊆ acc(θ) such that, for all δ ∈ S:  (1) Λδ ≤ Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) ρ2(ðδ,βδ, βδ) = k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) for every ¯δ < δ, β¯δ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every δ ∈ S and every ordinal α with Λ < α < ðδ,βδ:  Tr(α, βδ) ↾ (k + 1) = Tr(δ, βδ) ↾ (k + 1), and  Tr(α, βδ)(k) = ðδ,βδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Let Γ := {ðδ,βδ | δ ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For each ξ ∈ Γ, pick δ(ξ) ∈ S such that ξ = ðδ(ξ),βδ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Note that δ(ξ) ≤ ξ ≤ βδ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Γ is ⃗C-oscillating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As δ ∈ acc(ðδ,βδ)∪{ðδ,βδ} for every δ ∈ S, we have S ⊆ �  ξ∈Γ acc(Cξ)∪{ξ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So Γ is ⃗C-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3, it thus suffices to prove that Γ is ⃗C-nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Towards a contradiction, suppose this is not so, and fix a club D ⊆ θ such that,  for every α ∈ Γ there exists β ∈ Γ with D ∩ α ⊆ Cβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As sup(Γ) = θ, we may then  recursively construct a sparse enough cofinal subset X ⊆ Γ with the property that  for every pair ξ < ξ′ of ordinals from X, all of the following hold:  Λ < δ(ξ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  D ∩ (βδ(ξ), δ(ξ′)) ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  D ∩ βδ(ξ) ⊆ Cξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As B′ := {βδ(ξ) | ξ ∈ X} is a cofinal subset of B, it must be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We shall  reach a contradiction by showing that χ“[B′]3 = {k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To this end, let α < β < γ  be a triple of ordinals from B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Fix a triple ξ < ξ′ < ξ′′ of ordinals from X such  that α = βδ(ξ), β = βδ(ξ′), and γ = βδ(ξ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then:  Λ < δ(ξ) < α < δ(ξ′) < β < δ(ξ′′) ≤ ξ′′ ≤ γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  D ∩ (α, δ(ξ′)) ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  D ∩ β ⊆ Cξ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Pick ι ∈ D ∩ (α, δ(ξ′)) ̸= ∅, so that ι ∈ D ∩ (α, β) ⊆ Cξ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Appealing to  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1 with δ′′ := δ(ξ′′), we infer that:  Tr(α, γ) ↾ (k + 1) = Tr(δ′′, γ) ↾ (k + 1) = Tr(β, γ) ↾ (k + 1), and  Tr(α, γ)(k) = ξ′′ = Tr(β, γ)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Therefore  Tr(α, γ)(k + 1) = min(Cξ′′ \\ α) ≤ ι < β ≤ min(Cξ′′ \\ β) = Tr(β, ξ)(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Recalling Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9, this indeed means that χ(α, β, γ) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Now, given a cofinal A ⊆ θ and a positive integer n, appeal to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7 with  E := acc+(A \\ Λ) to find a pair (ξ, ζ) ∈ Γ ⊛ Γ and an ordinal ε ∈ E ∩ ξ such that:  oscε(Cξ, Cζ) = n + 1, and  for every j < n + 1, there is a pair ǫ < ǫ′ of ordinals in E \\ Cξ for which  Oscε(Cξ, Cζ)(j) = Cξ ∩ (ǫ, ǫ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  29  Let ǫ < ǫ′ be a pair of ordinals witnessing the case j = 0 of the preceding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Clearly,  Oscε(Cξ, Cζ)(0) = Cξ ∩ [ǫ, ǫ′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Since osc(Cξ, Cζ) > 1 and ξ < ζ, we may fix two consecutive elements ¯α < ¯β of Cζ  such that  Oscε(Cξ, Cζ)(1) = Cξ ∩ (¯α, ¯β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So, ǫ < ǫ′ ≤ ¯α < ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Since Cξ is a closed subset of ξ, and ǫ′ ∈ acc+(A \\ Λ) ∩ (ξ \\ Cξ), we may pick a  large enough α ∈ A ∩ (Λ, ǫ′) such that  Oscε(Cξ, Cζ)(0) ⊆ (ǫ, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In particular, Cξ ∩ Cζ ⊆ ε ⊆ α, and  oscα(Cξ, Cζ) = oscε(Cξ, Cζ) − 1 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Denote ¯δ := δ(ξ) and δ := δ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then (¯δ, δ) ∈ [S]2, ξ = ð¯δ,β¯δ and ζ = ðδ,βδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set  β := β¯δ and γ := βδ, so that (β, γ) ∈ [B]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note that  Λ < α < ǫ′ ≤ ¯α < ξ ≤ β < δ ≤ ζ ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1, then:  Tr(α, γ) ↾ (k + 1) = Tr(δ, γ) ↾ (k + 1) = Tr(β, γ) ↾ (k + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Tr(α, γ)(k) = ζ = Tr(β, γ)(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Tr(α, β)(k) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Therefore,  Tr(α, γ)(k + 1) = min(Cζ \\ α) ≤ ¯α < β ≤ min(Cζ \\ β) = Tr(β, γ)(k + 1),  and χ(α, β, γ) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Summing all up, (α, β, γ) ∈ A ⊛ B ⊛ B, and  osc(α, β, γ) = oscα(CTr(α,β)(χ(α,β,γ)), CTr(α,γ)(χ(α,β,γ)))  = oscα(CTr(α,β)(k), CTr(α,γ)(k))  = oscα(Cξ, Cζ) = n,  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  The ending of the proof of Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2 makes it clear that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose:  λ2(δ, γ) < α < β < δ < γ < κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Cðδ,γ ∩ [α, β) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then χ(α, β, γ) = ρ2(ðδ,γ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  The next lemma extracts features present in the proof of [Tod07, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that X is a stable cofinal subset of some θ ≤ κ of uncount-  able cofinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then there exist a cofinal X′ ⊆ X, a club D ⊆ θ, and a positive  integer k satisfying all of the following:  (1) for every (δ, γ) ∈ D ⊛ X′, D ∩ δ ⊆ Cðδ,γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) for every (δ, α, δ′, β, δ′′, γ) ∈ D ⊛ θ ⊛ D ⊛ θ ⊛ D ⊛ X′:  χ(α, β, γ) = k, and  D ∩ δ′′ ⊆ CTr(α,γ)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  30  IDO FELDMAN AND ASSAF RINOT  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Set µ := cf(θ), and fix a map ψ : µ → θ whose image is cofinal in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let M  be a continuous ∈-chain of length µ consisting of elementary submodels M ≺ HΥ  with M ∩ µ ∈ µ and {ψ, ⃗C, X} ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For each M ∈ M, denote θM := sup(M ∩ θ),  so that E := {θM | M ∈ M} is a club in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let N ≺ HΥ be such that {ψ, ⃗C, X, M} ∈ N and N ∩ µ ∈ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Denote θN := sup(N ∩ θ) and let γ ∈ X \\ (θN + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7  If sup(E∩θN\\CðθN ,γ) = θN, then there exists γ′ ∈ X\\(θN+1) with ρ2(ðθN ,γ′, γ′) >  ρ2(ðθN,γ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote ¯γ := ðθN,γ, n := ρ2(¯γ, γ), and  Mγ := {M ∈ M ∩ N | θM /∈ C¯γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Now, assuming that sup(E ∩ θN \\ C¯γ) = θN, we infer that  sup{θM | M ∈ Mγ} = θN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Pick M ∈ Mγ with θM > λ2(θN, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Since θM /∈ C¯γ, it is the case that  ρ2(θM, ¯γ) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13,  tr(θM, γ) = tr(¯γ, γ)⌢ tr(θM, ¯γ),  and  ρ2(θM, γ) > ρ2(¯γ, γ) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Thus, Im(tr(θM, γ)) contains not only Im(tr(¯γ, γ)) but also min(C¯γ\\θM) which is  strictly above θM, therefore, ρ2(ðθM,γ, γ) > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick Λ ∈ M ∩ θ above λ2(ðθM,γ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then, the following set belongs to M and γ witnesses that it has θM as an element:  S := {δ < θ | ∃γ′ ∈ X \\ (θ + 1) [ρ2(ðδ,γ′, γ′) > n & λ2(δ, γ′) ≤ Λ]},  so that S is stationary in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick δ ∈ S above θN, along with a witnessing γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As  λ2(δ, γ′) ≤ Λ < θM < θN < δ, we get from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13 that  tr(θN, γ′) = tr(ðδ,γ′, γ′)⌢ tr(θN, ðδ,γ′),  and hence ρ2(θN, γ′) ≥ ρ2(ðδ,γ′, γ′) + 1 > n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Therefore ρ2(ðθN,γ′, γ′) > n, as  sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' ∆ := {δ < θ | ∃γ ∈ X \\ (δ + 1) [E ∩ δ ⊆∗ Cðδ,γ]} covers a club in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let S be a stationary subset of θ, and we shall prove that S ∩ ∆ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let  N ≺ HΥ be such that {ψ, ⃗C, X, M} ∈ N, N ∩ µ ∈ µ, with θN := sup(N ∩ θ)  in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Using the fact that X is stable, fix m < ω such that χ“[X]3 ⊆ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now,  if θN /∈ ∆, then by iterating Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1 finitely many times, we may find a  γ ∈ X \\ (θN + 1) such that ρ2(ðθN,γ, γ) > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick α, β ∈ X with λ2(θN, γ) < α <  β < θN such that CðθN ,γ ∩ (α, β) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13, χ(α, β, γ) > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' This  is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  For each δ ∈ ∆, fix γδ ∈ X \\ (δ + 1), ǫδ < δ and kδ < ω such that:  E ∩ δ \\ ǫδ ⊆ Cðδ,γδ ,γδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  λ2(δ, γδ) ≤ ǫδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ρ2(ðδ,γδ, γδ) = kδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  7As cf(θ) = µ > |N|, X \\ θN is co-bounded in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  31  Find ǫ < θ and k < ω for which the following set is stationary:  S := {δ ∈ ∆ | ǫδ = ǫ & kδ = k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Consider the club D := {δ ∈ acc(E \\ ǫ) | ∀¯δ ∈ ∆ ∩ δ (γ¯δ < δ)}, and the set  X′ := {γδ | δ ∈ S ∩ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' We shall verify that X′, D and k satisfy the requirements  of the two clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (1) Let δ ∈ D and γ ∈ X′ \\ (δ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick δ∗ ∈ D such that γ = γδ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If δ∗ = δ,  then D ∩ δ∗ ⊆ E ∩ δ \\ ǫ ⊆ Cðδ∗,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Otherwise, λ2(δ∗, γ) ≤ ǫ < δ < δ∗ < γ = γδ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So,  by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13,  tr(δ, γ) = tr(ðδ∗,γ, γ)⌢ tr(δ, ðδ∗,γ),  with E∩δ∗\\ǫ ⊆ Cðδ∗,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Since δ ∈ acc(E)∩(ǫ, δ∗), it is the case that δ ∈ acc(Cðδ∗,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Altogether,  tr(ðδ,γ, γ) = tr(ðδ∗,γ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In particular, D ∩ δ ⊆ E ∩ (ǫ, δ∗) ⊆ Cðδ∗,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) Let (δ, α, δ′, β, δ′′, γ) ∈ D ⊛ θ ⊛ D ⊛ θ ⊛ D ⊛ X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Fix δ∗ ∈ S ∩ D such that  γ = γδ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As δ∗ ∈ ∆ and γ¯δ < δ′′ < γ for every ¯δ ∈ ∆ ∩ δ′′, it must be the case that  δ∗ ≥ δ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So λ2(δ∗, γ) ≤ ǫ < α < β < δ∗ < γ and δ′ ∈ D ∩ (α, β) ⊆ Cðδ∗,γ ∩ (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then, Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13 implies that χ(α, β, γ) = ρ2(ðδ∗,γ, γ) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' While we will not be needing this fact, we point out that the proofs of  Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14 together show that for every θ ≤ κ of uncountable cofinality,  and every cofinal subset X ⊆ θ, the following are equivalent:  Every cofinal subset of X is unstable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For every stationary S ⊆ θ and every ⟨βδ | δ ∈ S⟩ in �  δ∈S(X \\ (δ + 1)),  the set Γ := {ðδ,βδ | δ ∈ S} is ⃗C-nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The following is an easy strengthening of [Tod07, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2]:  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='16 (Todorˇcevi´c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that κ = λ+ for some regular uncountable  cardinal λ, and let ρ : [κ]2 → λ be the corresponding map given by Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Suppose also that X is a stable subset of κ of order-type λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then there exists a  cofinal subset X′ ⊆ X such that ρ ↾ [X′]2 witnesses U(λ, λ, λ, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote θ := sup(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let X′ ⊆ X and D ⊆ θ be given by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To  see that ρ ↾ [X′]2 witnesses U(λ, λ, λ, ω), let A ∈ [X′]σ be some λ-sized pairwise  disjoint, with σ < ω, and let τ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As A consists of λ-many pairwise disjoint subsets of X′ and otp(X′) = λ, for  every δ ∈ D, we may pick bδ ∈ A with min(bδ) > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then put Λδ := max{λ2(δ, β) |  β ∈ bδ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Recalling that X′ and D were given by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14, for all δ ∈ D and  β ∈ bδ,  D ∩ δ ⊆ Cðδ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Next, fix some Λ < θ and a stationary set S ⊆ D such that for every δ ∈ S:  (1) Λδ ≤ Λ < δ,  (2) otp(D ∩ δ) > τ, and  (3) For every γ ∈ D ∩ δ, sup(bγ) < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Evidently, B := {bδ | δ ∈ S} is a λ-sized subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, given a ̸= b in B, we  may find a pair γ ̸= δ of ordinals from S such that a = bγ and b = bδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Without loss  of generality, γ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let (α, β) ∈ a × b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then  Λδ < γ < α < δ < β,  32  IDO FELDMAN AND ASSAF RINOT  so by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='13, ðδ,β ∈ Im(tr(α, β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As the map ρ was given by Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14,  ρ(α, β) ≥ otp(Cðδ,β ∩ α) ≥ otp(Cðδ,β ∩ γ) ≥ otp(D ∩ γ) > τ,  as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  It is clear that every subset of a stable set is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The next corollary addresses  the question of closure under unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that A, B are cofinal stable subsets of some θ ≤ κ of  uncountable cofinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, there exist cofinal subsets A′ ⊆ A and B′ ⊆ B such  that:  A′ ∪ B′ is stable, and  for every (α, β, γ) ∈ [A′∪B′]3, if (β, γ) ∈ [A′]2∪[B′]2, then osc(α, β, γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Appeal to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14 with A to get a cofinal A′ ⊆ A, a club D1 ⊆ θ and  an integer k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Likewise, appeal to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14 with B to get a cofinal B′ ⊆ B,  a club D2 ⊆ θ and an integer k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Consider the club D := D1 ∩ D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick sparse  enough cofinal subsets A′′ ⊆ A′ and B′′ ⊆ B′ such that, letting X := A′′ ∪ B′′, for  every (α, β) ∈ [X]2, there are ι < θ and δ, δ′, δ′′ ∈ D with δ < α < δ′ < ι < δ′′ < β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' X is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Furthermore, max(χ“[X]3) = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let (α, β, γ) ∈ [X]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Pick δ, δ′, δ′′ such that  (δ, α, δ′, β, δ′′, γ) ∈ D ⊛ X ⊛ D ⊛ X ⊛ D ⊛ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then χ(α, β, γ) = k1 if γ ∈ A′, and χ(α, β, γ) = k2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let Y ∈ {A′′, B′′} and (α, β, γ) ∈ X⊛Y ⊛Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then osc(α, β, γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If Y = A′′, then denote k := k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Otherwise, denote k := k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, pick  ι, δ, δ′, δ′′, δ′′′ such that  (δ, α, δ′, ι, δ′′, β, δ′′′, γ) ∈ D ⊛ X ⊛ D ⊛ θ ⊛ D ⊛ Y ⊛ D ⊛ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In particular,  {(δ, α, δ′, ι, δ′′, β), (δ, α, δ′′, β, δ′′′, γ)} ⊆ D ⊛ θ ⊛ D ⊛ θ ⊛ D ⊛ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then D ∩ δ′′ ⊆ CTr(α,β)(k) and D ∩ δ′′′ ⊆ CTr(α,γ)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In addition, χ(α, β, γ) = k,  so that  δ′ ∈ D ∩ (α, δ′′) ⊆ CTr(α,β)(χ(α,β,γ)) ∩ CTr(α,γ)(χ(α,β,γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Recalling Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5, this means that  Oscα(CTr(α,β)(χ(α,β,γ)), CTr(α,γ)(χ(α,β,γ))) = ∅,  and hence osc(α, β, γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  So A′′ and B′′ are as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that:  (1) µ < λ < λ+ = κ are infinite regular cardinals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) Eλ  µ admits a nonreflecting stationary set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (3) there exists a weak µ-Kurepa tree with at least κ-many branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then κ  sup  �−→ [λ, λ]3  ω holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  33  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let c : [κ]3 → λ be the map given by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2 with respect to the  subadditive coloring ρ : [κ]2 → λ of Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Define cω : [κ]3 → ω via  cω(α, β, γ) :=  �  c(α, β, γ),  if c(α, β, γ) < ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Let T ⊆ <µ2 be a weak µ-Kurepa tree with at least κ-many branches, and let  ⟨bξ | ξ < κ⟩ be an injective sequence consisting of elements of B(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For notational  simplicity, we shall write ∆(α, β) for ∆(bα, bβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Define a coloring d : [κ]3 → ω by  letting for all α < β < γ < κ:  d(α, β, γ) :=  �  osc(α, β, γ) − 1,  if osc(α, β, γ) > 0 and ∆(α, β) < ∆(β, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  cω(α, β, γ),  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  To see that d witnesses κ  sup  �−→ [λ, λ]3  ω, let A, B be disjoint subsets of κ of order-  type λ with sup(A) = sup(B), and let n < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4 and by possibly  passing to cofinal subsets, we may assume the existence of s ∈ T and i ̸= i′ such  that s⌢⟨i⟩ ⊑ bα for all α ∈ A, and s⌢⟨i′⟩ ⊑ bβ for all β ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let (α, β, γ) ∈ [A ∪ B]3 \\ ([A]3 ∪ [B]3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then (β, γ) ∈ ([A]2 ∪ [B]2) iff ∆(α, β) < ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every (ǫ, δ) ∈ (A ⊛ B) ∪ (B ⊛ A), ∆(ǫ, δ) = ∆(s⌢⟨i⟩, s⌢⟨i′⟩) = dom(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For every (ǫ, δ) ∈ (A ⊛ A) ∪ (B ⊛ B), ∆(ǫ, δ) ≥ dom(s + 1) > dom(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  By the hypothesis on (α, β, γ), there are three cases to consider:  ◮ If (α, β, γ) ∈ A ⊛ B ⊛ B, then ∆(α, β) = dom(s) < ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ If (α, β, γ) ∈ B ⊛ A ⊛ A, then ∆(α, β) = dom(s) < ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ If (β, γ) ∈ (A ⊛ B) ∪ (B ⊛ A), then ∆(β, γ) = dom(s) ≤ ∆(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  There are three cases to consider:  ◮ Suppose that there exist A′ ∈ [A]λ and B′ ∈ [B]λ such that A′ and B′ are  stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17, we may moreover assume that A′ ∪ B′ is  stable, and that for every (α, β, γ) ∈ [A′ ∪ B′]3 with (β, γ) ∈ [A′]2 ∪ [B′]2,  osc(α, β, γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='16, then, ρ ↾ [A′∪B′]2 witnesses U(λ, λ, λ, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As c : [κ]3 → λ was given by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2, we may find (α, β, γ) ∈ [A′ ∪  B′]3 \\ ([A′]3 ∪ [B′]3) such that c(α, β, γ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, if (β, γ) ∈ [A′]2 ∪ [B′]2,  then osc(α, β, γ) = 0, and if (β, γ) /∈ [A′]2 ∪ [B′]2, then ∆(α, β) ≥ ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  It thus follows that d(α, β, γ) = cω(α, β, γ) = c(α, β, γ) = n, as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ Suppose that every cofinal subset of B is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By appealing to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12  with A and B, we may find (α, β, γ) ∈ A ⊛ B ⊛ B such that osc(α, β, γ) =  n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  As (β, γ) ∈ [B]2, it is the case that ∆(α, β) < ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So  d(α, β, γ) = osc(α, β, γ) − 1 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ Otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, every cofinal subset of A is unstable, and then the argument  is similar to that of the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  We are finally in conditions to prove the rectangular extension of [Tod07, Theo-  rem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that λ = µ+ for an infinite cardinal µ = µ<µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then λ+ sup  �−→ [λ, λ]3  ω holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  34  IDO FELDMAN AND ASSAF RINOT  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote κ := λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If 2µ > µ+, then since µ<µ = µ, there is a weak µ-Kurepa  tree with κ-many branches, and Eλ  µ constitutes a nonreflecting stationary set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So,  by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='18, we may assume here that 2µ = µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, λ ↛ [µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' λ]2  λ  holds by a theorem of Sierpi´nski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Let c : [κ]3 → λ be the map given by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3  with respect to the subadditive coloring ρ : [κ]2 → λ of Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Derive cω from  c as in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Also, denote T := <λ2, and let ⟨bξ | ξ < κ⟩  be the injective sequence of elements of B(T ) used in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3 to  define the coloring c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For notational simplicity, we shall write ∆(α, β) for ∆(bα, bβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Likewise, for B ⊆ κ, we write T ⇝B for T ⇝{bβ|β∈B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Finally, define a coloring d : [κ]3 → ω by letting for all α < β < γ < κ:8  d(α, β, γ) :=  �  osc(α, β, γ) − 1,  if osc(α, β, γ) > 0 and ∆(α, β) < ∆(β, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  cω(α, β, γ),  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  To see that d witnesses κ  sup  �−→ [λ, λ]3  ω, let A, B be disjoint subsets of κ of order-  type λ with sup(A) = sup(B), and let n < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Using Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='17 and by possibly  passing to cofinal subsets, we may assume that one of the following holds:  (I) A ∪ B is stable, and for every (α, β, γ) ∈ [A ∪ B]3 with (β, γ) ∈ [A]2 ∪ [B]2,  osc(α, β, γ) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (II) every cofinal subset of B is unstable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (III) every cofinal subset of A is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Let us dispose of Case (I) right away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In this case, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='16, ρ ↾ [A ∪ B]2  witnesses U(λ, λ, λ, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So by the choice of c, we may find (α, β, γ) ∈ [A ∪ B]3 \\  ([A]3∪[B]3) such that c(α, β, γ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Going over the division into cases in the proof  of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3, we see that ∆(α, β) ≥ ∆(β, γ) in all subcases but to Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So, in all of these cases, d(α, β, γ) = cω(α, β, γ) = c(α, β, γ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Finally, looking  at Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1, we see that the provided triple (α, β, γ) is an element of A ⊛ B ⊛ B  (or an element of B ⊛ A ⊛ A, once lifting the initial “without loss of generality”  assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So (β, γ) ∈ [A]2 ∪ [B]2, and hence osc(α, β, γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Therefore, again  d(α, β, γ) = cω(α, β, γ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Moving on to handling Cases (II) and (III), we shall need the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' There are cofinal subsets A′ ⊆ A and B′ ⊆ B such that, for every  (α, β, γ) ∈ (A ⊛ B ⊛ B) ∪ (B ⊛ A ⊛ A), ∆(α, β) < ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By possibly passing to a cofinal subset of B, we may assume that B = B′  in the sense of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So let θB ≤ λ be such that T ⇝B′ is a normal tree in  T (λ, θB) for every B′ ∈ [B]λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Likewise, we may assume that A = A′ in the sense  of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='7, and let θA ≤ λ be such that T ⇝A′ is a normal tree in T (λ, θA) for  every A′ ∈ [A]λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  If min{θA, θB} < λ, then the proof of Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2 provides χ < min{θA, θB} and  a pair (t, t′) ∈ (T ⇝A)χ+1 ×(T ⇝B)χ+1 such that ∆(t, t′) = χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Thus letting A′ := At  and B′ := Bt′, we see that ∆(α, β) = χ whenever (α, β) ∈ (A′ ⊛ B′) ∪ (B′ ⊛ A′),  and ∆(α, β) > χ whenever (α, β) ∈ (A′ ⊛ A′) ∪ (B′ ⊛ B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Thus, we may assume  that θA = θB = λ, so that T ⇝A, T ⇝B ∈ T (λ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, if (T ⇝A) ⊈ (T ⇝B) and  (T ⇝A) ⊈ (T ⇝B), then by normality of the two trees there must exist χ < λ,  8It may appear that this is the same map from the proof of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note, however,  that there ∆ was a map from [κ]2 to µ, whereas here ∆ is a map from [κ]2 to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  SUMS OF TRIPLES IN ABELIAN GROUPS  35  t ∈ (T ⇝A)χ \\ (T ⇝B)χ and t′ ∈ (T ⇝B)χ \\ (T ⇝A)χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Clearly, A′ := At and B′ := Bt′  are as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Thus, the only nontrivial case is in which θA = θB = λ and T ⇝A ⊆ T ⇝B or  T ⇝A ⊆ T ⇝B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Without loss of generality, assume that T ⇝A ⊆ T ⇝B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, there  are two options:  ◮ If B(T ⇝A) is nonempty, then there is b : λ → 2 that constitutes a branch  through both T ⇝A and T ⇝B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In this case, it is easy to recursively simultaneously  construct cofinal subsets A′ ⊆ A and B′ ⊆ B such that for all triple α < β < γ of  ordinals from A′ ∪ B′, it is the case that ∆(α, β) < ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ Otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So T ⇝A is a λ-Aronszajn tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' In particular, we may find χ < λ  and t ̸= t′ in (T ⇝A)χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Altogether, t ∈ (T ⇝A)χ, t′ ∈ (T ⇝B)χ and ∆(t, t′) < χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Then A′ := At and B′ := Bt′ are as sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  At this point, the proof is similar to that of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Succinctly, in  Case (II), we appeal to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='12 with A and B, to find (α, β, γ) ∈ A ⊛ B ⊛ B  such that osc(α, β, γ) = n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the preceding claim, it is the case that ∆(α, β) <  ∆(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So d(α, β, γ) = osc(α, β, γ) − 1 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The handling of Case (III) is simi-  lar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Connecting the dots  The next result implies Theorem A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every infinite cardinal µ satisfying µ<µ < µ+ < 2µ:  (1) S3(µ++, µ+, ω) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) G ↛ [µ+]FS3  ω  holds for every Abelian group (G, +) of size µ++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that µ is an infinite cardinal satisfying µ<µ < µ+ < 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Denote  λ := µ+ and κ := λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, T := <µ2 is a weak µ-Kurepa tree with at least  κ-many branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' So, by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='18, κ  sup  �−→ [λ, λ]3  ω holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As T witnesses that  κ ∈ T (λ, µ), by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='23, Extract3(κ, λ, µ, ω) holds, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Together with  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='22, this yields Clause (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Then, Clause (2) follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='19  and the fact (see [FBR17, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2]) that every Abelian group is a well-behaved  magma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  By [FBR17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='8], for every infinite cardinal µ = 2<µ, S2(2µ, 2µ, ω)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By the upcoming corollary, for every infinite cardinal µ = 2<µ, S2(2µ, µ+, 2)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' While it is easy to get S2(2µ, µ+, µ) from 2µ = µ+, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4 shows that  S2(2µ, µ+, µ) is also compatible with 2µ > µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Note, however, that by a theorem  of Shelah [She88, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1], one cannot prove S2(2µ, µ+, 3) in ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Thus, in  view of the number of colors, the following corollary is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every infinite cardinal µ, S2(µθ, µ+, 2) holds, for θ := logµ(µ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  In particular, S2(2µ, µ+, 2) holds for every strong limit cardinal µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Appeal to the upcoming theorem with λ := µ+ and κ := µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that θ < λ ≤ κ are infinite cardinals, with λ being regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  If κ ∈ T (λ, θ), then:  (1) κ  sup  �−→ [λ, λ]2  2 holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (2) S2(κ, λ, 2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  36  IDO FELDMAN AND ASSAF RINOT  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that κ ∈ T (λ, θ), and fix T ∈ T (λ, θ) admitting an injective se-  quence ⟨bξ | ξ < κ⟩ consisting of elements of B(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  (1) Consider the Sierpi´nski map c : [κ]2 → 2 defined by letting, for all α < β < κ:  c(α, β) := 1 iff bα <lex bβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' c witnesses κ  sup  �−→ [λ, λ]2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Suppose that we are given two disjoint subsets A, B of κ with otp(A) =  otp(B) = λ and sup(A) = sup(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4(2) (using µ := λ), we may find  s ∈ T and i ̸= i′ such that A′ := {α ∈ A | s⌢⟨i⟩ ⊑ bα} and B′ := {β ∈ B |  s⌢⟨i′⟩ ⊑ bβ} are both of size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' As sup(A′) = sup(B′), we may now fix (α, β, γ) ∈  A′ ⊛ B′ ⊛ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' To see that {c(α, β), c(β, γ)} = 2, consider the following cases:  ◮ If i < i′, then bα, bγ <lex bβ and hence c(α, β) = 1 > 0 = c(β, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  ◮ If i′ < i, then bβ < bα, bγ and hence c(α, β) = 0 < 1 = c(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  (2) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='9 (again, using µ := λ), in particular, Extract2(κ, λ, ω, ω) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  So, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='22, Clause (1) implies Clause (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' If there exists a weak µ-Kurepa tree with κ-many branches, then  S2(κ, µ+, 2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  Theorem B now follows (using µ := 2):  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' For every infinite set G, for every map ϕ : G → [G]<ω, and for  every pair of cardinals µ, θ such that µ<θ < |G| ≤ µθ, there exists a corresponding  coloring c : G → 2 satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  For every binary operation ∗ on G, if ϕ witnesses that (G, ∗) is well-behaved,  then for every X ⊆ G of size (µ<θ)+ and every i ∈ {0, 1}, there are x ̸= y in X  such that c(x ∗ y) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Given G, ϕ, µ and θ as above, denote κ := |G| and λ := (µ<θ)+, so that  λ ≤ κ ≤ µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Evidently, T := <θµ witnesses that κ ∈ T (λ, θ), so S2(κ, λ, 2) holds  by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Now, appeal to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Acknowledgments  The results of this paper stemmed from the first author’s M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' thesis written  under the supervision of the second author at Bar-Ilan University, and supported  by the Israel Science Foundation (grant agreement 2066/18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  The first author was partially supported by the Israel Science Foundation (grant  agreement 203/22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' The second author was partially supported by the Israel Science  Foundation (grant agreement 203/22) and by the European Research Council (grant  agreement ERC-2018-StG 802756).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  References  [BR21]  Ari Meir Brodsky and Assaf Rinot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' A microscopic approach to Souslin-tree constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Part II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
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+page_content=' Walks on ordinals and their characteristics, volume 263 of Progress  in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content=' Birkh¨auser Verlag, Basel, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='  URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='assafrinot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf'}